Last Annotation: 04/11/2019

- a ring if (I. N. Herstein 142)
- (a) (b) (c) (d) (e) (I. N. Herstein 142)
- (f) (g) (I. N. Herstein 142)
- (h) (I. N. Herstein 142)
- a . b = 0 we concluded that a = 0 or b = O. (I. N. Herstein 143)
- . When it does hold, (I. N. Herstein 143)
- name; it is called a domain. (I. N. Herstein 143)
- is an integral domain if a . b = 0 in (I. N. Herstein 143)
- R implies that a = 0 or b = O. (I. N. Herstein 143)
- e a division ring if for every (I. N. Herstein 143)
- Definition. A ring R with unit is sa a =I=0 in R there is an element b E R ( (I. N. Herstein 143)
- s a-I) (I. N. Herstein 143)
- a . a-I = a-I. a = 1. (I. N. Herstein 143)
- A ring R is said to be afield if R is a commutative division (I. N. Herstein 143)
- rlng. (I. N. Herstein 143)
- , R is a field if the nonzero ele- (I. N. Herstein 143)
- ments of R form an abelian group under· , the product in R. (I. N. Herstein 143)
- e examples (I. N. Herstein 144)
- of rings. W (I. N. Herstein 144)
- y 7L, (I. N. Herstein 144)
- , 7L is an example of an integral domain. (I. N. Herstein 144)
- o Q is a field. (I. N. Herstein 144)
- s, IR, also give us an example of a field. (I. N. Herstein 144)
- s, C, form a field. (I. N. Herstein 144)
- t R = 7L 6 , (I. N. Herstein 144)
- 7L 6 is not an integral domain, (I. N. Herstein 144)
- . R is a commutative ring with unit. (I. N. Herstein 144)
- t a =1= 0 in a ring R is a zero-divisor in R if ab =0 (I. N. Herstein 144)
- for some b =1= 0 in R. (I. N. Herstein 144)
- n 7L p is clearly a (I. N. Herstein 145)
- commutative ring with 1. (I. N. Herstein 145)
- t 7L p is a field. T (I. N. Herstein 145)
- 7L p has only a finite number of elements, it is called a finite (I. N. Herstein 145)
- field. (I. N. Herstein 145)
- et R be (I. N. Herstein 145)
- the set of all a E Q in whose reduced form the denominator is odd. (I. N. Herstein 145)
- t R forms a ring. It is an inte- (I. N. Herstein 145)
- gral domain with unit but is not a field, f (I. N. Herstein 145)
- l a E Q in whose reduced form the denominator is (I. N. Herstein 145)
- not divisible by a fixed prime p. (I. N. Herstein 145)
- , is an integral domain but is not a field. (I. N. Herstein 145)
- ), R is a ring u (I. N. Herstein 145)
- t R be the set of all real-valued continuous functions on the closed unit (I. N. Herstein 145)
- interval [0, 1]. F (I. N. Herstein 145)
- e (f + g)(x) = f (x) + g (x), (I. N. Herstein 145)
- (f· g)(x) = f(x)g (x). F (I. N. Herstein 145)
- s R is a commuta- (I. N. Herstein 145)
- tive rin (I. N. Herstein 145)
- g. It is not an integral domain. F (I. N. Herstein 145)
- For S to be a subring, it is necessary and sufficient that S be nonempty (I. N. Herstein 145)
- and that ab, a + b E S for all a, b E S. (I. N. Herstein 145)
- a subring of R is a subset S of R which (I. N. Herstein 145)
- is a ring if the operations ab and a + b are just the operations of R applied to (I. N. Herstein 145)
- the elements a, b E S. (I. N. Herstein 145)
- n i 2 = j2 = (I. N. Herstein 148)
- k 2 = -1,ij=k,jk=i,ki=jan (I. N. Herstein 148)
- dji= -k,kj= -i,ik= -j.If (I. N. Herstein 148)
- i i @ S 2. S (I. N. Herstein 149)
- s the s ¢ quaternions guaiernions (I. N. Herstein 149)
- o form POO N JLIETTL o a& noncommutative A o Selifelasiaa N it e VO division RORCOMMEIaiive A givision ring. QR W @A W VL singe -y FeiiX. S s s AN TR Wt LAY oy ¢ o L= (I. N. Herstein 149)
- @F 8. F (I. N. Herstein 150)
- 10. (I. N. Herstein 150)
- 14. S 15. F (I. N. Herstein 151)
- 14. S 15. F 16. V (I. N. Herstein 151)
- 19. S q (I. N. Herstein 151)
- q 20. I (I. N. Herstein 151)
- 21. S 22. i (I. N. Herstein 151)
- 33. L d (I. N. Herstein 152)
- ety 36. I 37. I (I. N. Herstein 153)
- 39. I 39. I (40, P 40. P (I. N. Herstein 153)
- n integral domain (I. N. Herstein 155)
- ) cp(a + b) = cp (a) + cp(b) a (I. N. Herstein 155)
- cp(ab) = cp(a)cp(b) f (I. N. Herstein 155)
- , the image of R under (I. N. Herstein 156)
- a homomorphism from R to R’, is a subring of R’, (I. N. Herstein 156)
- Ker cp is an additive subgroup of R. (I. N. Herstein 156)
- . So Ker cp swallows up multiplication (I. N. Herstein 156)
- from the left and the right by arbitrary ring elements. (I. N. Herstein 156)
- ideal of R if: (I. N. Herstein 156)
- (a) I (b) G (I. N. Herstein 156)
- Ker cp is an (I. N. Herstein 156)
- ideal of R. (I. N. Herstein 156)
- t every ideal can be made the kernel of a homo- (I. N. Herstein 156)
- morphism. (I. N. Herstein 156)
- t well-defined. (I. N. Herstein 157)
- e, the (I. N. Herstein 157)
- So R/K is now endowed with a sum and a product. Furthermore, the mapping cp: R ~ R/K defined by cp(a) = a + K for a E R is a homomor- (I. N. Herstein 157)
- phism of R onto R/K with kernel K. ( (I. N. Herstein 157)
- R/K is a ring, (I. N. Herstein 157)
- So R/K is a homo- (I. N. Herstein 157)
- morphic image of R. (I. N. Herstein 157)
- n R’ ~ RIK; (I. N. Herstein 158)
- ‘P: R ~ R’ b (I. N. Herstein 158)
- tfJ: RIK ~ R’ (I. N. Herstein 158)
- y tfJ(a + K) = ’P(a) (I. N. Herstein 158)
- isomorphism of RIK onto R’. (I. N. Herstein 158)
- f I’ is an ideal of R’, (I. N. Herstein 158)
- I = {a E R I ‘P(a) E I’}. Then I is an ideal of R, I => K and 11K ~ I’. T (I. N. Herstein 158)
- a 1-1 correspondence between all the ideals of R’ and those ideals of R (I. N. Herstein 158)
- that contain K. (I. N. Herstein 158)
- n A + I = { (I. N. Herstein 158)
- subring of R, I is an ideal of A + I, and (A + 1)1I ~ A I(A n I). (I. N. Herstein 158)
- n RII ~ R’II’. (I. N. Herstein 158)
- d I => K is (I. N. Herstein 158)
- n RII ~ (RIK)/(IIK). (I. N. Herstein 158)
- 7L, the ring of integers, f (I. N. Herstein 158)
- t In be the set of all multiples of n; t (I. N. Herstein 158)
- t R be the ring of all rational numbers having odd denominators in their (I. N. Herstein 158)
- reduced form (I. N. Herstein 158)
- t I b (I. N. Herstein 158)
- even numerator; (I. N. Herstein 159)
- e cp : R ~» : ~ 7.2 , 7L _ Ay ! < (I. N. Herstein 159)
- v ’ y@{a’b) == 00 if cp(a/b) iy Y if aa is g d is even even (a {a (I. N. Herstein 159)
- efalb)y cp (a/b) == 11 if a¢ is 15 odd.W / W (I. N. Herstein 159)
- &, 2 ===== &/{. s 7L R/I. G (I. N. Herstein 159)
- t & bethe RIy be E ey e oy sud e 1T e st ot T oh T 1 et T ey vy Nur I the ring of all rational numbers whose denominators ey (I. N. Herstein 159)
- ivisibleby p, are not divisible p, (I. N. Herstein 159)
- t I§§ bebe those t el those elements in (I. N. Herstein 159)
- & nuimerator is R whose numerator Pi e TR Y S e s divisible by p; ¥ L p; I (I. N. Herstein 159)
- t {I == {f E t &€ Rif{&) R If(~) IJ \... = OJ. = (]. (I. N. Herstein 159)
- T % ‘L..q o & i « < 9. . § o .° § m . ¢ _ N o < 5 . t R t FANE be VIR the B Ll ring 3§ 5524 of all Al real-valued FEA-Vailugcyd continuous CONTIMIONUS functions FICTIONS on O the 1830 closed Ci08Sa unit 7Oy TS TNy N e al xPAITISO OSOantien e\ 3-’ TIQ M :} ; O io e"§ I t | w1y (I. N. Herstein 159)
- interval w interval w (I. N. Herstein 159)
- What is ’lll/ 1s R/I? 8/{?7 (I. N. Herstein 159)
- cp(f) «{f) == f(~).f{i). (I. N. Herstein 159)
- t cp: ¢: &R— ~ & IR (I. N. Herstein 159)
- N 33 ~ ~) @yL a, bb E& IR} ; ¥ N RR {’g {3,} E 5 i~ 31 e ?“‘\4‘ é . t R = {( 33 R VU sy 3§ 5 : ls. (I. N. Herstein 160)
- &g 3 i Y (~ ~).1 b&R}li 8 eN v Xi Vi 5 y & RV eti=4{{, et I ={ R & 83 v b E IR} I i i N CVU ideal of R? Consider (I. N. Herstein 160)
- at R/I at &/{ “’” == IR ¥ aa FRPT R (I. N. Herstein 160)
- at I == Ker cpo So R/I = image of cp == IR (I. N. Herstein 161)
- et R = {( _: ~) a, b E IR} an (I. N. Herstein 161)
- by l/J ( _: ~ ) = a + bi. W sm of R onto C. So R is (I. N. Herstein 161)
- t R be any commutative ring with 1. I (I. N. Herstein 161)
- t (a) is an ideal of R. T (I. N. Herstein 161)
- Let R be a commutative ring with unit whose only (I. N. Herstein 164)
- Lemma 4.4.1. Let R be a commu ideals are (0) and itself. Then R is a field. (I. N. Herstein 164)
- A proper ideal M of R is a maximal ideal of R if the only (I. N. Herstein 164)
- ideals of R that contain Mare M itself and R. (I. N. Herstein 164)
- t M be a (I. N. Herstein 165)
- maximal ideal of R. Then RIM is a field. (I. N. Herstein 165)
- t R be a commutative ring with 1, (I. N. Herstein 165)
- the greatest common divisor always be a (I. N. Herstein 173)
- monic polynomial. (I. N. Herstein 173)
- The polynomial p (x) E F[x] is irreducible if p (x) is of pos- (I. N. Herstein 175)
- itive degree and given any polynomial f(x) in F[x], then either p(x) I f(x) or (I. N. Herstein 175)
- p(x) is relatively prime to f(x). (I. N. Herstein 175)
- If p(x) E F[x] , then the ideal (p(x» generated by (I. N. Herstein 176)
- p(x) in F[x] is a maximal ideal of F[x] if and only if p(x) is irreducible in F[x]. (I. N. Herstein 176)
- e it tells us exactly what the maximal (I. N. Herstein 177)
- ideals of F[x] are, namely the ideals generated by the irreducible polynomi- (I. N. Herstein 177)
- als. (I. N. Herstein 177)
- . If M is a maximal ideal of F[x], F[x]/M is a field, and this field contains (I. N. Herstein 177)
- F ( (I. N. Herstein 177)
- An integral domain R is a Euclidean ring if there is a (I. N. Herstein 179)
- function d from the nonzero elements of R to the nonnegative integers that (I. N. Herstein 179)
- satisfies: (I. N. Herstein 179)
- (a) (b) (I. N. Herstein 179)

Last Annotation: 04/11/2019

- For many years, Algebraic Topology rests on three legs: “ordinary” Cohomology, K-theory, and Cobordism (Budyak 6)
- The pioneering work of Pontryagin and Thom forged a deep connection between certain geometric problems (such as the classiﬁcation of manifolds) and homotopy theory, through the medium of the Thom space (Budyak 6)
- Computations become possible upon stabilization, and this provided some of the ﬁrst and most compelling examples of “spectra.” (Budyak 6)
- This international tradition was continued with the more or less simultaneous work by Novikov and Milnor on complex cobordism, and later by Quillen. More recently Dennis Sullivan opened the way to the study of “manifolds with singularities,” a study taken up most forcefully by the Russian school, notably by Vershinin, Botvinnik, and Rudyak. (Budyak 6)
- There is a ﬁne introduction to the stable homotopy category. The subtle and increasingly important issue of phantom maps is addressed here with care. Equally careful is the treatment of orientability (Budyak 6)
- The contents of this book are concentrated around Thom spaces (spectra), orientability theory and (co)bordism theory (including (co)bordism with singularities), framed by (co)homology theories and spectra. (Budyak 10)
- we consider the (inter)connections between geometry and homotopy theory, since Thom spectra and related matters are now the main tools for this interplay (Budyak 11)
- of the ﬁrst results in this area was the Gauss–Bonnet formula, relating a geometrical invariant (the curvature) to a homotopical one (the Euler characteristic). Proceeding, we can recall the Riemann–Roch Theorem, the Poincaré integrality theory, relationships between critical points of a smooth function on a smooth manifold and its homotopy type (Lusternik– Schnirelmann, Morse), the de Rham Theorem, etc. (Budyak 11)
- The term “cohomology theory” is used for what was previously called “generalized” or “extraordinary” cohomology theory, i.e., for functors which satisfy all the Eilenberg– Steenrod axioms except the dimension axioms. (Budyak 11)
- Every homology theory h∗ (−) yields a so-called dual cohomology theory h ∗ (−), and vice versa. They are connected via the equality h h i (X) = h hn−i (Y ) where Y is n-dual to X (and tilde denotes the reduced (co)homology). (Budyak 11)
- Thom spaces. The Thom space T ξ of a locally trivial Rn -bundle ξ = {p : E → B} is deﬁned as follows. Let ξ • be the S n -bundle obtained from ξ by the ﬁberwise one-point compactiﬁcation, and let E • be the total space of ξ . Then the “inﬁnities” of the ﬁbers form a section s : B → E • , and we deﬁne T ξ := E • /s(B). Furthermore, the Thom space of a spherical ﬁbration {p : E → B} is the cone C(p) of the projection (Budyak 12)
- p. For example, the Thom space of the Rn -bundle over a point is S n , the Thom space of the open Möbius band (considered as the R1 -bundle over S 1 ) is the real projective plane RP 2 , the Thom space of the Hopf bundle S 3 → S 2 (with ﬁber S 1 ) is the complex projective plane CP 2 . We use Thom’s notation M On for the Thom space T γ n of the universal n-dimensional vector bundle γ n over the classifying space BOn , i.e., M On := T γ n ; e.g., M O1 = RP ∞ (Budyak 12)
- .C. Whitehead observed the importance of the structure on the normal bundle in classifying structures on manifolds. It turns out that Thom spaces establish an adequate context for this. Namely, for every closed smooth manifold M n , the set of (diﬀeomorphism classes of) smooth manifolds homotopy equivalent to M is controlled by the group πn+N (T ν), where ν is the normal bundle of an embedding of M in Rn+N with N large enough (Budyak 12)
- This is closely related to the Milnor–Spanier–Atiyah Duality Theorem, which asserts that T ν and M/∂M are stable N -dual for every compact manifold M . This theorem clariﬁes connections between manifolds and their normal bundles and enables us to transmit properties of bundles to properties of manifolds. For example, we have the Thom isomorphism ϕ : H i (X; Z/2) → H H i+n (T ξ; Z/2) for every locally trivial Rn -bundle ξ over a space X, and the above theorem transforms it to the Poincaré duality H i (M ; Z/2) ∼ = ∼ = Hn−i (M, ∂M ; Z/2) for every compact n-dimensional manifold M . (Budyak 12)
- Turning to another example, I recall the Thom formula wi (ξ) = ϕ −1 Sq i uξ where ξ is an n-dimensional vector bundle over a space X, wi (ξ) is its i-th Stiefel–Whitney class, ϕ : H i (X; Z/2) → H H i+n (T ξ; Z/2) is the Thom isomorphism and uξ ∈ H (T ξ; Z/2) is the Thom class of ξ. This formula expands n a geometric invariant (the Stiefel–Whitney class) via the Steenrod operation which is a purely homotopic thing. Moreover, we can use the formula in order to deﬁne the Stiefel–Whitney classes of spherical ﬁbrations. In particular, it becomes clear that the Stiefel–Whitney classes are invariants of the ﬁber homotopy type of a vector bundle (Budyak 12)
- Generalizing, we can consider an arbitrary natural transformation τ : h ∗ → k ∗ of cohomology theories instead of Sq i . Then, under suitable conditions on ξ, there is a generalized Thom class u hξ ∈ h h n (T ξ) and a generalized Thom isomorphism ϕk : k i (X) → k k i+n (T ξ), and so we can form the class K(ξ) = ϕ −1 k τ u h ξ which is an analogue and generalization of the Stiefel–Whitney class (Budyak 13)
- So, we have a large source of invariants of Rn -bundles. For example, the Todd genus and the A genus are particular cases of this construction. Moreover, the wellA known integrality theorems which are related to Todd and A genera can be A generalized for the class K (Budyak 13)
- Pontrjagin [1] proved that if a manifold bounds then all its characteristic numbers are trivial. In particular, RP 2 does not bound because w2 (RP 2 ) = 0. So, N2 = 0, i.e., some groups Nk are non-trivial (Budyak 13)
- Well, but how to compute Nk ? Clearly, N0 = Z/2, N1 = 0. Using the classiﬁcation of closed surfaces, one can prove that N2 = Z/2: every orientable surface bounds, and every non-orientable surface either bounds or is bordant to RP 2 ; and RP 2 does not bound (Budyak 13)
- Rokhlin [1] proved that N3 = 0, using complicated and tricky geometry. The further computation of Nk looked absolutely hopeless; however this was done by Thom [2] via an exciting and successful application of homotopy theory. Namely, Thom proved that N k = πk+N (M ON ) for N large enough (Budyak 13)
- he answer is N∗ = Z/2 [xi ], dim xi = i, i ∈ N, i = 2s − 1 (Budyak 13)
- where N∗ = ⊕Nk is the graded ring with the multiplication induced by the direct product of manifolds (Budyak 13)
- The above constructions can be generalized: we can consider oriented manifolds or, more generally, manifolds equipped with some extra structures. As above, there arise certain bordism groups, and they can be interpreted as homotopy groups of certain Thom spaces (Budyak 14)
- We use a category of spectra proposed by Adams [5]. So, a spectrum E is a sequence {En , sn }∞ n=−∞ of pointed CW -spaces En and pointed CW embeddings sn : SEn → En+1 where S denotes the pointed suspension. (Budyak 14)
- (1) For every pointed space X we have the spectrum Σ ∞ X = {S n X, sn } where sn : SS n X → S n+1 X is the identity map. (2) For every pointed space X and every spectrum E = {En , sn } we have the spectrum X ∧ E = {X ∧ En , 1 ∧ sn }. (Budyak 14)
- (3) Let θ 1 be the trivial 1-dimensional vector bundle over BOn , and let the map BOn → BOn+1 (assuming it to be an embedding) classify the vector bundle γ n ⊕ θ1 . Then we have a map sn : T (γ n ⊕ θ1 ) → T γ n+1 . Moreover, (Budyak 14)
- Clearly, Nk = Nk (pt). Moreover, N k (X) = πk+N (X + ∧ M ON ) for N large enough, where X + is the disjoint union of the space X and a point. (Budyak 14)
- one can prove that T (γ n ⊕ θ1 ) = ST γ n = SM On , and so we have the Thom spectrum M O = {M On , sn (Budyak 15)
- Given a spectrum E, we have the homomorphisms hk,n : πk (En ) → πk+1 (SEn ) (sn )∗ (sn )∗ −−−→ πk+1 (En+1 ). We deﬁne the homotopy group πk (E) to be the direct limit of the sequence ··· − → πk+n (En ) hk+n,n hk+n,n −−−−→ πk+n+1 (En+1 ) − → ··· , i.e., πk (E) = lim n→∞ πi+n (En ). (Budyak 15)
- N k = πk (M O), Nk (X) = πk (X + ∧ M O) and so get rid of “N large enough” (Budyak 15)
- Every spectrum E yields a homology theory E∗ (−) and a cohomology theory E ∗ (−) by the formulae Ei (X) := lim n→∞ πi+n (X + ∧ En ), E i (X) := lim n→∞ [S n X + , Ei+n ]. Moreover, E∗ (−) and E ∗ (−) are dual to each other. Conversely, every (co)homology theory can be represented by a spectrum via the above formulae. (Budyak 15)
- Note that, in particular, the spectrum M O yields the bordism (resp. cobordism) theory N∗ (−) (resp. N∗ (−)). (Budyak 15)
- Orientability. We consider in this book orientability with respect to arbitrary cohomology theories (Budyak 15)
- Similarly, we deﬁne a locally trivial Rn -bundle ξ over a connected base to be orientable if H n (T ξ) = Z, and an orientation of ξ is a generator of the group H n (T ξ (Budyak 16)
- For example, an h-orientation of an Rn -bundle ξ is a suitable element uξ ∈ h h n (T ξ), an horientation of a closed manifold M n is an element [M ] ∈ hn (M n ). (Budyak 16)
- Furthermore, one can develop an elegant theory of characteristic classes taking values in h∗ (−) provided that all complex vector bundles are horientable; these classes generalize the classical Chern classes. (Budyak 16)
- As the last example, we mention that general orientabilty theory provides a formal group input to algebraic topology; this matter is completely degenerate for classical cohomology, and so this remarkable theory was able to appear only under the general approac (Budyak 16)
- oach. (Budyak 16)
- (Co)bordism with singularities. (Co)bordism with singularities is now a common and convenient notion, being a favorite tool as well as subject of research in algebraic topology. Roughly speaking, we take a class of manifolds and extend it to a class of suitable polyhedra (manifolds with singularities) where a notion of a boundary is reasonably deﬁned (Budyak 16)
- Moreover, the famous Morava k-theories are also constructed as certain cobordism with singularities (Budyak 17)
- ories are also constructed as certain cobordism with singularities. I also want to mention an application of (co)bordism with singularities to the topological quantum ﬁeld theory: for example, the elliptic (co)homology can be constructed as (co)bordism with singularities (Budyak 17)
- Finally, (co)bordism with singularities gives a natural geometric ﬂavor to algebraic or homotopical matters. For example, the Adams resolution of certain spectra can be interpreted in terms of (co)bordism with singularities, and this enables us to get useful information about some classical (co)bordism theories, like M SU and M Sp (Budyak 17)
- paradigm of algebraic topology, and it freshly demonstrated the power and usefulness of the relations between homotopy theory and geometry. In order to exhibit relatively rece (Budyak 17)
- The paper Thom [2] made a revolution and formed the contemporary paradigm of algebraic topology, and it freshly demonstrated the power and usefulness of the relations between homotopy theory and geometry. In order to exhibit relatively recent advantages of this matter, I just write down a list (unavoidably incomplete) of certain geometric problems which were (partially or completely) solved via an application of homotopy theory. 1 (1) When can a manifold M be immersed in a manifold N , and how can one classify these immersions? (Smale [1], Hirsch [1].) (2) When can a homology class in a space be realized by a map of a closed manifold? (Thom [2].) (3) When is a closed manifold a boundary of a compact manifold with boundary? (Thom [2].) (4) Which spaces are homotopy equivalent to closed smooth manifolds? (Browder [1,2], Novikov [2,3].) (5) How can one classify manifolds up to diﬀeomorphism (PL isomorphism, homeomorphism)? (Smale [1], Kervaire–Milnor [1], Browder [1,2], Novikov [2,3], Hirsch–Mazur [1], Sullivan [1], Kirby–Siebenmann [1], Freedman [1], Donaldson [1].) (6) How many pointwise linearly independent tangent vector ﬁelds exist on the n-dimensional sphere? (Adams [3].) (7) Which smooth manifolds admit a Riemannian metric of positive scalar curvature? (Gromov–Lawson [1], Stolz [1].) (Budyak 17)
- The category consisting of sets (as objects) and functions (as morphisms) is denoted by E ns. The category of pointed sets and pointed functions is denoted by E ns• . (Budyak 19)
- partially ordered set, or a poset, is a quasi-ordered set with the following condition: if λ ≤ μ ≤ λ then λ = μ (Budyak 20)
- A chain in a poset is a family {ai } such that, for every pair i, j of indices, either ai ≤ aj or aj ≤ ai . An upper bound of the chain is any a such that a i ≤ a for every i. A poset is called inductive if every chain in it has an upper bound. (Budyak 20)
- 1.3. Zorn’s Lemma. Every inductive set has a maximal element. (Budyak 20)
- K → E ns is represented by a certain object B of K if there exists a natural equivalence F ∼ = TB . In this case B is called a classifying or representing object for F . Furthermore, F is called representable if it can be represented by some B. (Budyak 20)
- 1.4. Deﬁnition. We say that a contravariant functor F : K → E ns is represented by a certain object B of K if there exists a natu (Budyak 20)
- Let F, G : K → E ns be represented by B, C respectively. It is obvious that every morphism f : B → C yields a natural transformation Tf : TB → TC and hence F → G. The converse is also true. ∼ = ∼ = ∼ = ∼ = 1.5. Lemma (Yoneda). Fix natural equivalences b : F −→ TB , c : G −→ TC . For every natural transformation ϕ : F → G there exists a morphism f : B → C such that for every object X of K the diagram ϕ F (X) −−−−→ G(X) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ b T f T Tf B (X) −−−−→ TC (X) commutes, and such a morphism f is unique. In particular, the representing object B for F is determined by F uniquely up to isomorphism. (Budyak 20)
- The category of abelian groups and homomorphisms is denoted by A G . Note that the usual direct product of abelian groups is the categorical product in A G , while the usual direct sum is the categorical coproduct in A G . (Budyak 21)
- In algebraic context, we reserve the word “unit” for the neutral element of a monoid (group). In particular, the multiplicative identity element of a ring is also called the unit. (Budyak 21)
- Given a commutative ring R, we denote by R[x, y, . . . , z] the polynomial ring of indeterminates x, y, . . . , z. The corresponding power series ring is denoted by R[[x, y, . . . , z]]. If R is a graded ring, we assume that x, y, . . . , z are homogeneous indeterminates. Furthermore, ΛR (x, y, . . . , z) denotes the free exterior algebra (with a unit) over R of indeterminates x, y, . . . , z, and for a graded R we assume that x, y, . . . , z have odd degrees. We use the notation Λ(x, y, . . . , z) for the ring ΛZ (x, y, . . . , z). (Budyak 21)
- Let ρ : A → B be a ring homomorphism, and let M be a right Amodule. The homomorphism ρ turns B into a left A-module ρ B, where a·b = ρ(a)b for a ∈ A, b ∈ B, cf. Cartan–Eilenberg [1]. We can therefore form the tensor product over A of A-modules M, B. This tensor product is denoted by M ⊗ρ B (Budyak 21)
- (a) Let K be a category. A direct system over Λ, or brieﬂy, a direct Λsystem, in K is a covariant functor M : Λ → K . In other words, M is a family M = {Mλ , j λμ }λ , μ ∈ Λ, where Mλ ∈ K and where jλμ : Mμ → Mλ for μ ≤ λ are morphisms such that jλμ jμν = jλν for ν ≤ μ ≤ λ and jλλ = 1Mλ . (b) A morphism f : {Mλ , jλμ } → {Nλ , hμλ } of direct Λ-systems is a natural transformation of functors, i.e., a family {fλ : Mλ → Nλ } with hμλ fμ = fλ jλμ . (Budyak 22)
- 2.4. Deﬁnition. Let Λ be a quasi-ordered set, and let {Aλ }λ∈Λ be a direct Λsystem of abelian groups. Let iλ : Aλ → ⊕λ Aλ be the inclusion, and let B ⊂ ⊕λ Aλ be the subgroup generated by all elements of the form (iμ aμ − iλ jλμ aμ ). The quotient group (⊕λ Aλ )/B is called the direct limit of the direct system {Aλ } and is denoted by −→{Aλ }. lim (Budyak 22)
- 2.5. Theorem. Let G be an abelian group, and let ϕλ : Aλ → G be a family of homomorphisms such that ϕμ = ϕλ jλμ for every μ ≤ λ. Then there exists a homomorphism ϕ : −→{Aλ } → G such that ϕkλ = ϕλ for every λ. lim (Budyak 22)
- Given two pairs (X, A), (Y, B) of spaces and a map f : A → B, the space X ∪f Y is deﬁned to be the quotient space (X Y )/ ∼, where ∼ is the smallest equivalence relation generated by the following relation: a ∼ b if f (a) = b for a ∈ A, b ∈ B. We say that the space X ∪f Y is obtained from X by adjoining, or gluing, Y via f . For instance, if Y = pt = B then X ∪f Y ∼ = X/A. (Budyak 24)
- A triad (X; A, B) is a topological space and two of its closed subspaces A, B such that X = A ∪ B. A ﬁltration of a topological space X is a sequence {· · · ⊂ X0 ⊂ · · · ⊂ Xn ⊂ · · · ⊂ X} such that: (1) X = ∪n Xn . (2) Every Xn is closed in X. (3) X inherits the direct limit topology, i.e., U is open in X iﬀ U ∩ Xn is open in Xn for every n. (Budyak 24)
- Given a space X, we denote by X + the disjoint union of X and a point, and the added point is assumed to be the base point. (Budyak 24)
- Algebraic topologists prefer to deal with “nice” spaces, such as CW spaces. However, a class of spaces in which algebraic topologists work should be closed under standard operations which topologists use. In other words, the suitable category of spaces should be large enough to accommodate operations and small enough to rule out pathologies at the same time. One such category was suggested by Steenrod [2] and improved by McCord [1]3 , and is known as the category of weak Hausdorﬀ compactly generated spaces (Budyak 24)
- 3.2. Deﬁnition. (a) A topological space X is called weak Hausdorﬀ if, for every map ϕ : C → X of a compact space C, the set ϕ(C) is closed in X. (Budyak 24)
- (b) A subset U of a topological space X is called compactly open if ϕ−1 (U ) is open for every map ϕ : C → X of a compact space C. A topological space X is called compactly generated if each of its compactly open sets is open. (Budyak 25)
- Note that every point of a weak Hausdorﬀ space is closed, and that every Hausdorﬀ space is weak Hausdorﬀ. Thus, the weak Hausdorﬀ property lies between T1 and T2 . (Budyak 25)
- Generally speaking, the usual Cartesian product of two spaces from W is not in W . See Dowker [1], §5. Nevertheless, the category W admits products (Budyak 26)
- 3.6. Deﬁnition. Given a family {Xi } of topological spaces, we deﬁne their compactly generated direct product c Xi := k Xi i i where is the usual Cartesian product of topological spaces (Budyak 26)
- Deﬁne the compact-open topology as follows: let ϕ : C → X be a map of a compact space C, and let U be an open set in Y . We denote by W (ϕ, U ) the set of all maps f : X → Y such that f ϕ(C) ⊂ U . Then the family {W (ϕ, U )} for all such pairs (ϕ, U ) forms a subbasis of the compact-open topology on the set of maps from X to Y . (Budyak 27)
- (c) The loop space Ω(X, ∗) of a pointed space (X, ∗) is just the pointed space (X, ∗)(S 1 ,∗) where S 1 is the circle. (Budyak 27)
- 3.11. Convention. Throughout the book we will assume that all spaces belong to W unless somthing else is said explicitly, i.e., the word “space” means “weak Hausdorﬀ compactly generated space”. (Budyak 27)
- 3.12. Basic homotopy theory. (a) Two maps f, g : X → Y are called homotopic if there is a map (homotopy, or deformation) H : X × I → Y such that H|X × {0} = f and H|X × {1} = (Budyak 28)
- (b) A map f : X → Y is called a homotopy equivalence if there is a map g : Y → X such that gf 1X and f g 1Y . In this case we say that f and g are homotopy inverse to each other. Two spaces X, Y are called homotopy equivalent if there is a homotopy equivalence X → Y , and we write X Y . The homotopy type of a space X is the class of all spaces homotopy equivalent to X. (Budyak 28)
- (c) By saying that two maps f, g : (X, A) → (Y, B) are homotopic we mean that there exists a homotopy (X × I, A × I) → (Y, B). Furthermore, we say that two maps f, g : (X, A) → (Y, B) are homotopic relative to A, and write f g rel A, if there is a homotopy H : f g such that H(a, t) = f (a) for every a ∈ A, t ∈ I (Budyak 28)
- 3.13. Deﬁnition. We say that a map is essential if it is not homotopic to a constant map. (Budyak 28)
- 3.14. Deﬁnition. Let H W denote the category whose objects are the same as those of W but whose morphisms are the homotopy classes of maps. Clearly, every diagram in W yields a diagram in H W . We say that a diagram in W is homotopy commutative if the corresponding diagram in H W is commutative (Budyak 28)
- (a) The mapping cylinder, or just the cylinder, of f is the space M f := X × [0, 1] ∪f Y, f where f is considered as the map X × {0} = X − → Y . Recall that there is a standard deformation F : M f × I → Y where F ((x, t), s) = (x, st) if (x, t) ∈ X × (0, 1] and s > 0 F ((x, t), 0) = f (x) if (x, t) ∈ X × (0, 1] F (y, s) = y if y ∈ Y. Note that F M f × {0} : M f → Y is a retraction and F M f × {1} = 1Mf , i.e., Y is a deformation retract of M f . (Budyak 29)
- (b) The mapping cone, or the coﬁber, or just the cone, of f is the space Cf := M f /(X × {1}). (Budyak 29)
- 3.18. Deﬁnition. (a) Given two maps f : X → Y and g : X → Z, the double f g mapping cylinder of the diagram Y ← −X − → Z is the space D := X × [0, 2] ∪ϕ (Y Z) (Budyak 29)
- For instance, Cf is (homeomorphic to) the double mapping cylinder of f the diagram Y ← −X − → pt. (Budyak 30)
- (b) The mapping cone of the constant map X → pt is called the suspension over a space X and denoted by SX (Budyak 30)
- . Thus, the suspension is the double mapping cylinder of the diagram pt ← − X − → p (Budyak 30)
- (c) The mapping cylinder of the trivial map X → pt is denoted by CX. So, Cf = CX ∪f Y , and SX = CX/X × {1}. (d) The join X ∗ (Budyak 30)
- ∗ Y of the spaces X, Y is deﬁned to be the double mapping cylinder of the diagram p1 p2 X ←− X × Y −→ Y. For instance, X ∗ S 0 = SX (Budyak 30)
- fn−1 fn fn+1 3.19. Deﬁnition. Given a sequence X = {· · · −−−→ Xn −→ Xn+1 −−−→ · · · } of maps, deﬁne its telescope T X to be the space T X := (Xn × [n, n + 1]) ∼, where under ∼, (x, n + 1) ∈ Xn × [n, n + 1] is identiﬁed with (fn (x), n + 1) ∈ X n+1 × [n + 1, n + 2]. (Budyak 30)
- 3.20. Deﬁnition. Let {(Xi , xi )} be a family of pointed spaces. (a) The pointed direct product is the pointed space (X i , xi ) := Xi , ∗ where ∗ is the point {xi }. (b) The wedge is the pointed space i Xi (X Xi i , xi ) := i ,∗ . ∪i {xi } i where ∗ is the image of ∪i {xi }. (Budyak 31)
- (c) The obvious injective maps (Xi , xi ) → Xi , ∗ yield an injective map (∨i Xi , ∗) → Xi , ∗ . Generally speaking, this map is not closed, but it is closed for a ﬁnite set of spaces. So, given two pointed spaces (X, ∗), (Y, ∗), we deﬁne the smash product (X, ∗) × (Y, ∗) (X, ∗) × (Y, ∗) (X, ∗) ∧ (Y, ∗) := . (X, ∗) ∨ (Y, ∗) (Budyak 31)
- 3.21. Deﬁnition. Let {(Xi , xi )} be a family of copies of a pointed space (X, x). We deﬁne the folding map π : ∨(Xi , xi ) → X, to be the unique map π such that π|Xi = 1X . (Budyak 31)
- 3.23. Deﬁnition. (a) The reduced mapping cylinder of a pointed map f : (X, ∗) → (Y, ∗) is the space M f = (X × [0, 1] ∪f Y )/(∗ × [0, 1]). Note that the base points of X and Y yield the same point ∗ ∈ M f ; we agree that ∗ is the base point of M f . (Budyak 31)
- (b) The reduced mapping cone of f is deﬁned to be Cf = M f /(X × {1}). It is a pointed space in the obvious way: its base point is the image of the base point of M f . (Budyak 32)
- (c) The reduced mapping cone of the constant map (X, ∗) → (pt, ∗) is called the reduced suspension over a space X and denoted by SX. Furthermore, we can deﬁne the iterated reduced suspension S n X, and S n turns out to be a functor on W • , see 3.18(e). (Budyak 32)
- (d) The reduced telescope of a sequence f n−1 fn fn+1 fn−1 fn fn+1 X = {· · · −−−→ Xn −→ Xn+1 −−−→ · · · } of pointed maps is deﬁned to be the pointed space T X := (Xn × [n, n + 1]) ∼, where (x, n + 1) ∈ Xn × [n, n + 1] is identiﬁed with (fn (x), n + 1) ∈ Xn+1 × [n + 1, n + 2] and all the points of the form (∗, t) are identiﬁed. (Budyak 32)
- (In fact, the reduced and unreduced cone (cylinder, etc.) of any map(s) of CW -spaces are homotopy equivalent (Budyak 32)
- Prove as an exercise that SX ∼ = S 1 ∧ X for every X ∈ W • . (Budyak 32)
- 3.24. Deﬁnition. Given a pair (X, A), the inclusion i : A → X is called a coﬁbration if it satisﬁes the homotopy extension property, i.e., given maps g : X → Y and F : A × I → Y such that F |A × {0} = g|A, there is a map G : X × I → Y such that G|X × {0} = g and G|A × I = F . In this case we also say that (X, A) is a coﬁbered pair. (Budyak 32)
- 3.25. Proposition. (i) (X, A) is a coﬁbered pair iﬀ every map h : X × {0} ∪ A × I → Y can be extended to a map X × I → Y . (ii) (X, A) is a coﬁbered pair iﬀ X × {0} ∪ A × I is a retract of X × I (Budyak 33)
- 3.26. Proposition. (i) For every map f : X → Y , the inclusion i : X = X × {1} → M f is a coﬁbration. In particular, every map is homotopy equivalent to a coﬁbration. (ii) Let (X, A) be a coﬁbered pair. Then Ci X/A. (iii) Let (X, A) be a coﬁbered pair. If A is contractible then the collapsing map c : X → X/A is a homotopy equivalence. (Budyak 33)
- 3.28. Deﬁnition. A pointed space (X, x0 ) is called well-pointed if (X, {x0 }) is a coﬁbered pair. (Budyak 34)
- 3.29. Lemma (Puppe [1]). Let f : (X, x0 ) → (Y, y0 ) be a pointed map of well-pointed spaces. If f : X → Y is a homotopy equivalence then f : (X, x0 ) → (Y, y0 ) is a pointed homotopy equivalence. (Budyak 34)
- (c) A long coﬁber sequence is a sequence (ﬁnite or not) ··· − → Xi − → Xi+1 − → Xi+2 − → ··· where every pair of adjacent morphisms forms a coﬁber sequence. (Budyak 36)
- u v 3.38. Deﬁnition. (a) A strict coﬁber sequence is a diagram A − →B − →C where u : A → B is a map and v is the canonical inclusion as in (3.17). f g (b) A sequence X − →Y − → Z is called a coﬁber sequence if there exists a homotopy commutative diagram f g X −−−−→ Y −−−−→ Z ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ c ⏐ a b c u v A −−−−→ B −−−−→ C such that all the vertical arrows are homotopy equivalences and the bottom row is a strict coﬁber sequenc (Budyak 36)
- 3.39. Proposition. Let f : X → Y be an arbitrary map of pointed spaces, and let g : Y → Cf = Z be the canonical inclusion. Then C(g) SX = S 1 ∧ X. Moreover, there is a long coﬁber sequence n f Sn f g Sf Sg Sn f Sn g X− →Y − →Z− → SX −−→ SY −→ · · · − → S n X −−→ S n Y −−→ · · · . (Budyak 37)
- Proposition 3.39 was originally proved by Puppe [1]. Because of this, the long coﬁber sequence is often refered to as the Puppe sequence (Budyak 37)
- Recall that a map f : X → Y of CW -complexes is called cellular if f (X (n) ) ⊂ Y (n) for every n. (Budyak 37)
- 3.41. Theorem. (i) Let i : X (n) → X be the inclusion. Then i∗ : π i (X (n) , ∗) → πi (X, ∗) is an isomorphism for i < n and an epimorphism for i = n. (ii) Every map f : X → Y of CW -complexes is homotopic to a cellular map. (Budyak 37)
- We recall that i : A → X is a coﬁbration for every CW -pair (X, A). In particular, every pointed CW -space is well-pointed, and so we can safely omit base points from notation. (Budyak 37)
- Every CW -space is Hausdorﬀ (and so weak Hausdorﬀ) and compactly generated. Thus, when we talk about products (or smash products) of CW complexes we follow 3.6. Then the direct product X × Y and the smash product X ∧ Y of two CW -spaces X, Y are also CW -spaces. Note that, for every cellular map f : X → Y , the spaces M f and Cf are CW -complexes in an obvious canonical way, see e.g. Fritsch–Piccinini [1]. In particular, the suspension SX of a CW -complex X is a CW -complex. (Budyak 38)
- (b) A map f : X → Y is called a Whitehead equivalence if f∗ : π n (X, x0 ) → πn (Y, f (x0 )) is a bijection for every n ≥ 0 and every x0 ∈ X. (Budyak 38)
- We say that two spaces X, Y are CW -equivalent if there is a sequence a 0 a1 ai−1 ai an−1 a0 a1 ai X = X0 −−−− X1 −−−− · · · −−−−− − Xi −−−− · · · −−−−− − Xn = Y where every ai is a Whitehead equivalence (Budyak 38)
- 3.43. Remark. Traditionally, CW -equivalences, as well as Whitehead equivalences, are called weak equivalences. We refrain from using this terminology in this book because these names are not quite compatible with the concept of weak homotopy (Budyak 38)
- 3.44. Proposition–Deﬁnition. For every topological space X, there is a Whitehead equivalence f : Y → X where Y is a CW -space. Every such CW space Y is called a CW -substitute for X. (Budyak 38)
- 3.45. Lemma. Let h : Y → Z be a Whitehead equivalence. (i) Let (X, A) be a CW -pair, and let f : A → Y, u : X → Z be two maps such that hf = u|A. Then there is a map g : X → Y such that g|A = f and hg u. (Budyak 39)
- Proof. (i) This is an exercise in elementary obstruction theory, see e.g. Switzer [1], 6.30. (Budyak 39)
- 3.47. Corollary (the Whitehead Theorem). If h : Y → Z is a Whitehead equivalence of CW -spaces then h is a homotopy equivalence. (Budyak 40)
- 1.1. Deﬁnition. (a) A spectrum E is a sequence {En , sn }, n ∈ Z, of CW complexes En and CW -embeddings sn : SEn → En+1 (i.e., sn (SEn ) is a subcomplex of En+1 ). (Budyak 42)
- (e) For every CW -complex X the spectrum Σ∞ X is deﬁned as follows: pt, if n < 0, ∞ (Σ∞ X)n = S n X, if n ≥ 0, and sn : S(S n X) → S n+1 X, for n ≥ 0, are the identity maps. For example, the spectrum Σ∞ S 0 is the sphere spectrum {S n , in }, where i n : SS = n −→ S n+1 . (Budyak 42)
- 1.2. Deﬁnition. (a) A cell of a spectrum E is a sequence {e, Se, . . . , S k e, . . . } where e is a cell of any En such that e is not the suspension of any cell of E n−1 . If e is a cell of En of dimension d then the dimension of the cell {e, Se, . . . , S k e, . . . } of E is d − n. Furthermore, the base points of En ’s yield the cell of dimension −∞. (Budyak 43)
- (b) A subspectrum F of a spectrum E is coﬁnal (in E) if every cell of E is eventually in F , i.e., for every cell e of En there exists m such that S m e belongs to Fn+m . (Budyak 43)
- (g) A suspension spectrum is a spectrum of the form Σk Σ∞ X where X is a pointed space and k ∈ Z. (Budyak 43)
- 1.3. Deﬁnition. (a) Let E = {En , sn } and F = {Fn , tn } be two spectra. A map f from E to F (i.e., a map f : E → F ) is a family of pointed cellular maps fn : En → Fn such that fn+1 sn = tn ◦Sfn for all n. (Budyak 43)
- Let S = Σ∞ S 0 be the spectrum of spheres. The group [Σk S, E] is called the k-th homotopy group of E and denoted by πk (E). It is easy to see that π k (E) = lim πk+N (EN ) where the direct limit is that of the direct system N →∞ (sN )∗ · · · → πk+N (EN ) → πk+N +1 (SEN ) −−−→ πk+N +1 (EN +1 ) → · · · , (Budyak 45)
- 1.9. Deﬁnition. (a) Two maps g0 , g1 : E → F of spectra are called homotopic if there exists a map G : E ∧ I + → F (called a homotopy) such that G coincides with gi on the subspectrum E ∧ {i, ∗}, i = 0, 1, of E. In this case we write g0 g1 or G : g0 g1 . (Budyak 45)
- (b) Two morphisms ϕ0 , ϕ1 : E → F of spectra are called homotopic, if there exists a coﬁnal subspectrum E of E and two maps gi : E → F, gi ∈ ϕ i , i = 0, 1, such that g0 |E g1 |E . It is straightforward to show that homotopic morphisms form equivalence classes, and in particular we can deﬁne the homotopy class [ϕ] of a morphism ϕ to be the set of all morphisms homotopic to ϕ. The set of all homotopy classes of morphisms E → F is denoted by [E, F ]. (Budyak 45)
- Thus, we can deﬁne a category H S with spectra as objects and sets [E, F ] as sets of morphisms. Isomorphisms of H S are called equivalences (of spectra), and we use the notation E F when E is equivalent to (Budyak 45)
- o F . (Budyak 45)
- In particular, if E = Σ∞ X then π k (E) is just the stable homotopy group Πk (X) (denoted also by πkst (X)) (Budyak 46)
- An analog of the Whitehead Theorem is valid for spectra. 1.10. Theorem. A morphism ϕ : E → F is an equivalence iﬀ the induced homomorphism ϕ∗ : πk (E) → πk (F ) is an isomorphism for every integer k. (Budyak 46)
- One of the important advantages of the category H S is that the suspension operator is invertible there. 1.11. Proposition. The spectra S 1 ∧ E and ΣE are equivalent. (Budyak 46)
- 1.12. Deﬁnition. (a) A strict coﬁber sequence of spectra is a diagram ϕ ψ E− →F − → Cϕ where ϕ : E → F is a morphism of spectra (resp. map of spaces) and ψ is a canonical inclusion as in 1.7(b). (Budyak 46)
- 1.14. Proposition. For every spectrum F the function {i∗ λ } : [∨λ E(λ), F ] → [E(λ), F ], where {i∗λ }(f ) = {f iλ }, λ is a bijection. (Budyak 47)
- f g 1.13. Lemma. (i) If X − → Y − → Z is a coﬁber sequence of spectra, then f g h there exists a map h : Z → ΣX such that X − →Y − →Z − → ΣX is a long coﬁber sequence. (Budyak 47)
- Since [E, F ] = [Σ2 E, Σ2 F ] = [S 2 ∧ E, Σ2 F ] (the last equality follows from 1.11), [E, F ] admits a natural structure of an abelian group (Budyak 48)
- Thus, H S is an additive category. (Budyak 48)
- f g In view of 1.11 and 1.13(i), every coﬁber sequence X − →Y − → Z yields a long coﬁber sequence −1 . . . → Σ −1 Y Σ−1 g f g Σf −−−→ Σ−1 Z − →X − →Y − → Z → ΣX −−→ ΣY − → ··· . (Budyak 48)
- −1 . . . → Σ −1 Y Σ−1 g f g Σf −−−→ Σ−1 Z − →X− →Y − → Z → ΣX −−→ ΣY − → ··· yields the exact sequences ∗ g∗ f∗ g∗ − [Σ−1 Z, E] ← ··· ← − [X, E] ←− [Y, E] ←− [Z, E] ← − [ΣX, E] ← − ··· → [E, Σ ··· − −1 Z] f∗ g∗ − → [E, X] −→ [E, Y ] −→ [E, Z] − → [E, ΣX] − → ··· of abelian groups and homomorphisms (Budyak 48)
- The ﬁrst of the above sequences is similar to a sequence which holds for a coﬁbration X → Y with coﬁber Z , while the second one is similar to a sequence which holds for a ﬁbration X → Y with ﬁber Σ−1 Z . Thus, the diﬀerence between ﬁbrations and coﬁbrations disappears in the category H S . For this reason we call Σ −1 Cϕ the fiber of a morphism ϕ. (Budyak 48)
- ular, π∗ (∨λ E(λ)) ∼ = ∼ = ⊕λ π∗ (E(λ)) (Budyak 49)
- Hence, the element 1X ⊕ 1X of the right hand side yields a (unique up to homotopy) morphism ∇ : X → X ∨ X. We leave it to the reader to show that addition in [X, E] is given by the composition ∼ = ∇∗ ∼ = ∇∗ [X, E] ⊕ [X, E] −→ [X ∨ X, E] −−→ [X, E]. Because of this, we call ∇ coaddition (Budyak 49)
- f g 1.17. Proposition. Let X − →Y − → Z be a coﬁber sequence of spectra. The following two conditions are equivalent: (i) The morphism g is inessential; (ii) There is a morphism s : Y → X such that f s 1Y . (Budyak 49)
- Furthermore, if these conditions hold then X Σ−1 Z ∨ Y . (Budyak 50)
- 1.18. Deﬁnition. A prespectrum is a family {Xn , tn }, n ∈ Z, of pointed spaces Xn and pointed maps tn : SXn → Xn+1 . A CW -prespectrum is a prespectrum {Xn , tn } such that every Xn is a CW -complex and every tn is a cellular map. 1.19. Lemma–Deﬁnition. For every prespectrum {Xn , tn }, there exist a spectrum E = {En , sn } and pointed homotopy equivalences fn : En → Xn such that the diagram SE Sfn n −−−−→ SXn ⏐ ⏐ ⏐ s ⏐ ⏐ ⏐ ⏐t n n E fn+1 n+1 −−−−→ Xn+1 commutes. Every such spectrum E is called a spectral substitute of the prespectrum X (Budyak 50)
- 1.20. Deﬁnition. A prespectrum X = {Xn , tn } is called an Ω-prespectrum if for every n the map τn : Xn → ΩXn+1 adjoint to tn is a homotopy equivalence. A spectrum is called an Ω-spectrum if it is an Ω-prespectrum. 1.21. Proposition (cf. Adams [5]). Every spectrum E = {En , sn } is equivalent to some Ω-spectrum. (Budyak 51)
- Some authors have developed a ﬁner theory by indexing terms of a spectrum not by integers but by ﬁnite dimensional subspaces of R∞ . This approach was suggested by Puppe [2] and May [3]. Such spectra are very useful for working with some ﬁne geometry. However, the foundations of this theory are quite complicated. For our purposes, the mass of preliminaries outweighs the gain; thus we do not use these spectra here and so do not dwell on them. However, they seem to be very useful for advanced homotopy theory (Budyak 54)
- One can introduce a smash product E ∧ F of spectra E, F as a generalization of the smash product E ∧ X of a spectrum and a space. (Budyak 54)
- 2.3. Deﬁnition. (a) A morphism u : S → A ∧ A⊥ is called a duality morphism, or simply a duality, between spectra A and A⊥ if for every spectrum E the homomorphisms u E : [A, E] → [S, E ∧ A ⊥ ], uE (ϕ) = (ϕ ∧ 1A⊥ )u and u E : [A⊥ , E] → [S, A ∧ E], uE (ϕ) = (1A ∧ ϕ)u are isomorphisms. (b) A spectrum A⊥ is called dual to a spectrum A if there exists a duality S → A ∧ A⊥ . By 2.1(ii), in this case A is dual to A⊥ . So, “to be dual” is a symmetric relation. (Budyak 56)
- 2.6. Remarks. (a) Some authors deﬁne duality to be a morphism v : A ∧ A ⊥ → S such that F vE : [E, A⊥ ∧ F ] → [A ∧ E, F ] and F v E : [E, F ∧ A] → [E ∧ A⊥ , F ] are isomorphisms (Budyak 58)
- 2.9. Corollary. (i) Every ﬁnite CW -space X admits an n-dual ﬁnite CW space X for n large enough. (ii) Every ﬁnite spectrum A admits a dual ﬁnite spectrum A⊥ . (Budyak 60)
- f g 2.10. Proposition. If A − →B− → C is a coﬁber sequence of ﬁnite spectra, ⊥ ⊥ f ⊥ ⊥ g f then C ⊥ −−→ B ⊥ −−→ A⊥ is a coﬁber sequence. (Budyak 60)
- 2.12. Deﬁnition. (a) A ring spectrum is a triple (E, μ, ι) where E is a spectrum and μ : E ∧ E → E (the multiplication) and ι : S → E (the unit morphism, or the unit) are certain morphisms with the following properties: (Budyak 60)
- 2.13. Deﬁnition. (a) A module spectrum over a ring spectrum (E, μ, ι), or an E-module spectrum, is a pair (F, m) where F is a spectrum and m : E ∧ F → F is a morphism such that the following diagrams commute up to homotopy: (Budyak 61)
- it was noticed that many useful constructions of algebraic topology (K-functor, (co)bordism, etc.) are formally similar to (co)homology theories. Afterwards the reason for this phenomenon was clariﬁed: namely, most of these constructions satisfy all the Eilenberg–Steenrod axioms except the so-called dimension axiom. So, it seemed reasonable to consider the objects satisfying these axioms. These objects were called extraordinary (co)homology theories. However, later mathematicians came to call these objects just (co)homology theories, while H(−; G) got the name ordinary (co)homology theory. 5 Now this terminology is commonly accepted, and we use it in this book. (Budyak 62)
- It is easy to see that we have a category of homology theories and their morphisms. In particular, the equivalence (isomorphism) of homology theories is deﬁned in the usual way: it is a morphism of homology theories which is also a natural equivalence of functors. (Budyak 63)
- The exact sequences as in 3.1(2), 3.2(iv), and 3.2(v) are known as the exact sequence of a pair, the exact sequence of a triple, and the Mayer– Vietoris exact sequence (Budyak 65)
- The family {hn , sn } is called a reduced homology theory (on K ) corresponding • to {hn , ∂n }. (Budyak 65)
- This proposition shows that there is a bijective correspondence between unreduced and reduced homology theories. In other words, every unreduced homology theory is completely determined by its reduced form (Budyak 68)
- The groups hi (pt, ∅) = hi (S hi (S 0 , ∗) are called the coeﬃcient groups of the homology theory {hn , ∂n }. To justify this term, note that H∗ (pt, ∅; A) = A for every abelian group A. (Budyak 68)
- It is possible and useful to introduce (co)homology theories on spectra. Consider the following full subcategories of S : Sfd : its objects are all ﬁnite dimensional spectra; Ss : its objects are all suspension spectra; Ssfd : its objects are all spectra of the form Σn Σ∞ X, n ∈ Z, X ∈ Cfd • ; Sf : its objects are all ﬁnite spectra. 3.10. Deﬁnition. Let L be one of the categories S , Sfd , Ss , Ssfd , Sf , and let Σ : L → L be the functor deﬁned in 1.1(d). (a) A homology theory on L is a family {hn , sn }, n ∈ Z of covariant functors hn : L → A G and natural transformations sn : hn → hn+1 Σ satisfying the following axioms: (Budyak 70)
- 3.12. Proposition. Let L be as in 3.10, and let {hn , ŝn } be a cohomology theory on L . Then: (i) hn (X ∨ Y ) ∼ = ∼ = hn (X) ⊕ hn (Y ). f g (ii) For every coﬁber sequence X − →Y − → Z of spectra there is a natural exact sequence ∗ f∗ g∗ f∗ → hn−1 (X) − ··· − → hn (Z) −→ hn (Y ) −→ hn (X) − → hn+1 (Z) − → ··· . (iii) Let A, B be two subspectra of a spectrum X. Set C = A ∩ B. Then there is a natural (Mayer-Vietoris) exact sequence ··· − → hn (X) − → hn (A) ⊕ hn (B) − → hn (C) − → hn−1 (X) − → ··· . (Budyak 72)
- 3.22. Construction. Let E be an arbitrary spectrum. (a) Deﬁne covariant functors En : S → A G where En (X) := πn (E ∧ X) for every X ∈ S and En (f ) := πn (1E ∧ f ) for every morphism f : X → Y of spectra. Furthermore, deﬁne sn : En → En+1 Σ to be the composition En (X) = πn (E ∧ X) = πn+1 (Σ(E ∧ X)) πn+1 (E ∧ ΣX) = En+1 (ΣX) for every X ∈ S . By 1.15 and 2.1(vi), (En , sn ) is a homology theory on S , and, by 1.16(iii) and 2.1(v), it is additive. (Budyak 78)
- hus, every spectrum yields a (co)homology theory on S (Budyak 78)
- . So, we have a functor from spectra to (co)homology theories. In particular, equivalent spectra yield isomorphic (co)homology theories. According to 3.22, one can assign a (co)homology theory to a spectrum. This situation turns out to be invertible, see Ch. III, §3 below. ∗ 3.23. Proposition. For every spectrum E, the cohomology theory E ∗ is dual to the homology theory E∗ . (Budyak 78)
- 3.25. Proposition. If E = {En } is an Ω-spectrum, then for every space X i (X) ∼ there is a natural equivalence E = [X, Ei ]• . (Budyak 79)
- 3.26. Corollary. For every spectrum E and every i, the functor i : E H C • → A G , X → E i (X), E is representable. (Budyak 80)
- .27. Proposition-Deﬁnition. Given a spectrum E, let Ω∞ E denote a 0 : representing space for E H C • → A G (i.e., Ω∞ E = F0 for some Ωspectrum F equivalent to E). This space Ω∞ E is called the inﬁnite delooping of E and has the following properties: (Budyak 80)
- this section we develop the homotopy theory of spectra. Namely, we discuss Postnikov towers, Cartan killing constructions, Serre theory of classes of abelian group, etc., for spectra. (We assume that the reader knows these notions in the case of spaces; otherwise he can ﬁnd them e.g. in Mosher– Tangora [1].) Closely related material is exposed in Dold [3] and Margolis [1]. (Budyak 88)
- 4.9. Theorem (the Universal Coeﬃcient Theorem). For every spectrum E and every abelian group G, there are exact sequences 0 → Ext(Hn−1 (E), G) → H n (E; G) → Hom(Hn (E), G) → 0 and 0 → Hn (E) ⊗ G → Hn (E; G) → Tor(Hn−1 (E), G) → 0. In particular, H0 (H(A); B) ∼ = A ⊗ B, H 0 (H(A); B) ∼ = ∼ = Hom(A, B). (Budyak 91)
- 4.11. Theorem (the Künneth Theorem). Let k be a ﬁeld, and let E, F be a pair of spectra. (i) The homomorphism μ E,F : H∗ (E; k) ⊗ H∗ (F ; k) → H∗ (E ∧ F ; k) is an isomorphism. In particular, H n (E ∧ F ; k) ∼ = ∼ = Hi (E; k) ⊗k Hj (F ; k). i+j=n (ii) Assume that E is bounded below and F has ﬁnite type. Then the homomorphism μ E,F : H ∗ (E; k) ⊗ H ∗ (F ; k) → H ∗ (E ∧ F ; k) is an isomorphism. In particular, H n (E ∧ F ; k) ∼ = ∼ = H i (E; k) ⊗k H j (F ; k). i+j=n (Budyak 93)
- 4.12. Deﬁnition. A Postnikov tower of a spectrum E is a homotopy commutative diagram of spectra (Budyak 94)
- 4.13. Theorem. Every spectrum E has a Postnikov tower (Budyak 94)
- 4.19. Deﬁnition. The element κn ∈ H n+1 (E(n−1) ; πn (E)) is called the n-th Postnikov invariant of E. (Budyak 97)
- Now we apply the Serre class theory (see Serre [1], Mosher–Tangora [1]) to spectra. C (Budyak 98)
- 4.21. Deﬁnition. (a) A Serre class is a family of abelian groups C satisfying the following axiom: If 0 → A → A → A → 0 is a short exact sequence, then A is in C iﬀ both A and A are in C. (Budyak 98)
- C iﬀ both A and A are in C. (b) Let H(A) denote the Eilenberg–Mac Lane spectrum of an abelian group A. A Serre class is called stable if it satisﬁes the following axiom: If A ∈ C, then Hi (H(A)) ∈ C for every i. (Budyak 98)
- 4.23. Proposition. (i) The class of all ﬁnite abelian groups is a stable Serre class. (ii) The class of all ﬁnitely generated abelian groups is a stable Serre class. (Budyak 98)
- (iii) Given a prime p, let C be the class of all abelian groups having pprimary exponents (i.e., for every A ∈ C there exists k such that pk A = 0). Then C is a stable Serre class. (iv) Given a prime p, the class of all ﬁnite p-primary abelian groups is a stable Serre class. (Budyak 99)
- 4.32. Theorem-Deﬁnition. For every abelian group A, there exists a spectrum M (A) with the following properties: (i) πi (M (A)) = 0 for i < 0; (ii) π0 (M (A)) = A = H0 (M (A)); (iii) Hi (M (A)) = 0 for i = 0. (Budyak 104)
- Moreover, these properties determine M (A) uniquely up to equivalence. This spectrum M (A) is called the Moore spectrum of the abelian grou (Budyak 104)
- Let Q be the ﬁeld of rational numbers. Let p be a prime, and let Z[p] be the subring of Q consisting of all irreducible fractions with denominators relatively prime to p. The Z[p]-localization of an abelian group A is the homomorphism A → A ⊗ Z[p], a → a ⊗ 1. The group A ⊗ Z[p] is simpler than A in a certain sense: for example, it has no q-torsion if (p, q) = 1. On the other hand, if we know the groups A ⊗ Z[p] for all p then we can obtain a lot of information about A; for example, if A is ﬁnitely generated then it is completely determined by the groups A ⊗ Z[p], where p runs through all primes. So, we can describe an abelian group A via descriptions of the simpler groups A ⊗ Z[p], and this trick is very eﬀective. For example, it is ∗ very convenient to describe the ring H ∗ (HZ) of cohomology operations via the rings H ∗ (HZ[p]; Z[p]). Also, localization enables us to ignore the torsions which are irrelevant to a particular problem. (Budyak 106)
- More generally, it makes sense to consider subrings Λ of Q. In this case the localization A → A ⊗ Λ deletes the q-torsion with q ∈ S, where S is the set of denominators of all irreducible fractions of Λ. (Budyak 106)
- It is remarkable that the localization can be transferred from algebra to topology, and, in particular, one can consider the Z[p]-homotopy type of a space and a spectrum (Budyak 106)
- Let Λ be a subring of Q ; its additive group is also denoted by Λ. Let π be an abelian group. π 5.1. Deﬁnition. The homomorphism l = lΛ : π → π ⊗ Λ, a → a ⊗ 1 is called the Λ-localization of π. (Budyak 106)
- f g 5.3. Proposition. If E − → F − → G is a coﬁber sequence of spectra then E fΛ gΛ Λ −→ FΛ −→ GΛ is. In particular, C(fΛ ) = (Cf )Λ for every morphism f : E → F of spectra. (Budyak 107)
- 5.5. Corollary. There are natural isomorphisms π i (EΛ ) ∼ = πi (E) ⊗ Λ, Hi (EΛ ) ∼ = Hi (E) ⊗ Λ j∗ such that the homomorphisms πi (E) −→ πi (EΛ ) ∼ πi (E) = ⊗ Λ, Hi (E) −j→ ∗ H i (EΛ ) ∼ = ∼ = Hi (E)⊗Λ have the form x → x⊗1. So, j Λ-localizes homotopy and homology groups. In particular, every Λ-local spectrum has Λ-local homotopy and homology groups. (Budyak 108)
- 5.7. Lemma. Let E be an arbitrary spectrum, and let Cj be the cone of the localization j : E → EΛ . Then H ∗ (Cj; π) = 0 for every Λ-local group π. (Budyak 110)
- This proposition enables us to construct localizations via Postnikov towers. (Budyak 112)
- This approach enables us to construct the localization of spaces also. The main results of this theory are Theorems 5.12 and 5.13 below (Budyak 112)
- 5.12. Theorem–Deﬁnition. For every simple space X there exist a simple space XΛ and a map j = jΛX : X → XΛ such that the homomorphisms π i (X) −→ πi (XΛ ) ∼ = j∗ ∼ = π i (X) ⊗ Λ and Hi (X) −→ Hi (XΛ ) j∗ ∼ = Hi (X) ⊗ Λ have the form x → x ⊗ 1. So, j localizes homotopy and homology groups. Every such a map j is called localization of X. (Budyak 112)
- As in 5.2, a simple space X is called Λ-local if j : X → XΛ is a homotopy equivalence. (Budyak 112)
- 5.13. Theorem–Deﬁnition. For every two simple spaces X, Y the following conditions are equivalent: (i) The map f : X → Y Λ-localizes homotopy groups; (ii) The map f : X → Y Λ-localizes homology groups; (iii) For every Λ-local space Z the map f ∗ : [Y, Z] → [X, Z] is a bijection. If some (and hence all) of these conditions hold then there exists a homotopy equivalence h : XΛ → Y with f = hjΛX . Furthermore, in this case we say that f localizes X. (Budyak 113)
- 5.14. Lemma. For every two spectra E, F there exists an equivalence ϕ : (E ∧ F )Λ → EΛ ∧ FΛ such that the diagram E∧F j E∧F E∧F −−−−→ (E ∧ F )Λ ⏐ ⏐ ⏐ ⏐ j E ∧j F ⏐ ⏐ ⏐ ϕ ⏐ ∧j E Λ ∧ FΛ EΛ ∧ FΛ is homotopy commutative. (Budyak 113)
- 6.1. Deﬁnition. (a) An algebra over R (or simply an R-algebra) is a triple (A, μ, η), where A is an R-module and μ : A ⊗ A → A, η : R → A are R-homomorphisms such that the diagrams μ⊗1 η⊗1 1⊗η A ⊗ A ⊗ A −−−−→ A ⊗ A R ⊗ A −−−−→ A ⊗ A ←−−−− A ⊗ R ⏐ ⏐ ⏐ ⏐ ⏐ 1⊗μ ⏐ ⏐ ⏐ ⏐ ⏐μ ⏐ ⏐μ ⏐∼ =⏐ ∼ ⏐= ⏐ ⏐μ ⏐∼ =⏐ ∼ ⏐= μ A⊗A −−−−→ A A A A ∼ commute. Here = denotes the canonical isomorphisms R ∼ ⊗ A = ∼ ∼ = A = A ⊗ R, e.g., A ∼ = A ⊗ R has the form a → a ⊗ 1. Furthermore, μ is called the multiplication and η is called the unit homomorphism. An algebra (A, μ, η) is commutative if μTA,A = μ. (Budyak 116)
- (b) A (left) module over an R-algebra (A, μ, η) is a pair (M, ϕ), where M is an R-module and ϕ : A ⊗ M → M is an R-homomorphism such that the following diagrams commute: μ⊗1 A ⊗ A ⊗ M −−−−→ A ⊗ M ⏐ ⏐ ⏐ 1⊗ϕ ⏐ ⏐ ⏐ ⏐ϕ ϕ A⊗M −−−−→ M ϕ A ⊗ M −−−−→ M ⏐ ⏐ ⏐∼ η⊗1⏐ ⏐= R⊗M R ⊗ M. As usual, we shall simply say “algebra A” or “A-module M”, omitting μ, η, ϕ, and we shall write ab instead of μ(a ⊗ b) and am instead of ϕ(a ⊗ m) (Budyak 116)
- ote that every ring is a Z-algebra. (Budyak 116)
- 6.2. Deﬁnition. A homomorphism f : A → B of R-algebras is an Rhomomorphism such that the ﬁrst two of the three diagrams below commute. (Budyak 116)
- A homomorphism h : M → N of A-modules is an R-homomorphism such that the third diagram commutes. (Budyak 117)
- 6.4. Deﬁnition. An augmented R-algebra is a quadruple (A, μ, η, ε), where (A, μ, η) is an R-algebra and ε : A → R is a homomorphism of R-algebras (called the augmentation). An augmented algebra (A, μ, η, ε) is called connected if Ai = 0 for i < 0 and ε|A0 : A0 → R is an isomorphism. (Budyak 117)
- 6.5. Deﬁnition. Let A be a connected R-algebra, and let Ā be the Rsubmodule consisting of all elements of positive degrees. The ideal ĀĀ is denoted by Dec A, and its elements are called decomposable elements of A. Given an A-module M , let GM denote the factor module M/ĀM . Furthermore, GĀ := Ā/ Dec A usually is denoted by QA and is called (not very aptly) the set of indecomposable elements of A. (Budyak 117)
- 6.7. Deﬁnition. (a) A coalgebra over R is a triple (C, Δ, ε), where C is an R-module and Δ : C → C ⊗ C, ε : C → R are R-homomorphisms such that the diagrams Δ C −−−−→ C ⊗C C C C ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Δ ⏐ ⏐ ⏐ ⏐ Δ⊗1 ∼ = ⏐ ⏐ ⏐ ⏐ ⏐ ⏐∼ Δ = 1⊗Δ ε⊗1 1⊗ε C ⊗ C −−−−→ C ⊗ C ⊗ C R ⊗ C ←−−−− C ⊗ C −−−−→ C ⊗ R commute. Here Δ is called the comultiplication, or the diagonal map, or just the diagonal, and ε is called the augmentation, or the counit homomorphism. A coalgebra (C, Δ, ε) is cocommutative if TC,C Δ = Δ. (Budyak 117)
- (b) A comodule over a coalgebra (C, Δ, ε) (or, brieﬂy, a C-comodule) is a pair (V, ψ), where V is a graded R-module and ψ = ψV is an Rhomomorphism V → C ⊗ (Budyak 118)
- c) A homomorphism h : C → D of R-coalgebras is an R-homomorphism such that ΔD ◦h = (h ⊗ h)◦ΔC and εD ◦h = εC . A homomorphism f : V → W of comodules over a coalgebra (C, Δ, ε) is an R-homomorphism such that (1 C ⊗ f )◦ψV = ψW ◦f . (Budyak 118)
- The duality between algebras and coalgebras is exhibited not only in the deﬁning diagrams. For instance, let R be a ﬁeld k. Given a k-vector space C, ∗ consider the dual vector space C ∗ = Homk (C, k). If C is a k-algebra (C, μ, η) ∗ obtains a natural k-coalgebra structure (C ∗ then C ∗ obtains a natural k-coalgebra structure (C ∗ , Δ, ε) provided that every component Cn of C is a ﬁnite dimensional k-vector space (Budyak 118)
- Namely, Δ(f )(a ⊗ b) = f (ab), ε = η ∗ : C ∗ → k ∗ = k. (Budyak 118)
- 6.8. Lemma. For every C-comodule (V, ψ), the function t : HomC (V, C ⊗ M ) → HomR (V, M ), t(f ) = (ε ⊗ 1)f, is bijective. (Budyak 118)
- 6.9. Deﬁnition. A coalgebra (C, Δ, ε) is called connected if Ci = 0 for i < 0 and ε|C0 : C0 → R is an isomorphism. In this case the element v = ε−1 (1) is called the counit of C. (Budyak 119)
- 6.11. Examples. (a) Let p be a prime, and let (E, μ, ι) be a ring spectrum of ﬁnite Z[p]-type. Then H ∗ (E; Z/p) has the natural structure of a Z/pcoalgebra. (Budyak 119)
- (b) If the ring spectrum (E, μ, ι) in (a) is commutative then H ∗ (E; Z/p) is a cocommutative coalgebra. (Budyak 119)
- (c) If E is a spectrum as in (a) and F is any E-module spectrum of ﬁnite Z[p]-type, then H ∗ (F ; Z/p) admits a structure of a comodule over H ∗ (E; Z/p). (Budyak 119)
- (e) For every CW -space X and every ﬁeld k we have a coalgebra H∗ (X; k) (Budyak 120)
- ∗ (f) Dually to (e), H ∗ (X; k) is a k-algebra for every ﬁeld k and CW -space X of ﬁnite type. (Budyak 120)
- 6.12. Deﬁnition. Let (C, Δ, ε) be a connected coalgebra with counit v. (a) An element m ∈ C is called primitive if Δ(m) = m ⊗ v + v ⊗ m. The set (in fact, R-submodule) of all primitive elements of C is denoted by Pr C. (b) Let (V, ψ) be a C-comodule. An element m ∈ V is called simple if ψ(m) = v ⊗ m. The R-submodule of simple elements of V is denoted by Si V . (Budyak 120)
- 6.13. Remarks. (a) Under the duality between algebras and coalgebras over a ﬁeld, Pr C is dual to QC ∗ . (b) Sometimes simple elements are also called primitive, but we do not like this because of the danger of confusion: Pr C = Si C where C is regarded as a coalgebra on the left and as comodule on the right. (Budyak 120)
- 6.14. Lemma. Let h : C → D be a homomorphism of connected coalgebras over a ﬁeld k. If the map h| Pr C is injective in dimensions ≤ d then h is. (Budyak 120)
- 6.15. Construction. (Boardman [1]). Given a connected coalgebra C, we deﬁne a ﬁltration Fm C by setting C n if n ≤ m, (F m C)n = 0 otherwise. (Budyak 121)
- m Fm V = V for every V . (ii) F0 V = Si V . (iii) f (Fm V ) ⊂ Fm W for every C-comodule homomorphism f : V → W (Budyak 121)
- 6.18. Deﬁnition. Let M be a free R-module. Given a coalgebra (C, Δ, ε), deﬁne its cofree M -extension to be the C-comodule (V, ψ) where V = C ⊗ M and ψ(c ⊗ x) = Δ(c) ⊗ x, c ∈ C, x ∈ M . A C-comodule V is called cofree if there is M such that V is isomorphic to the cofree M -extension of C. (Budyak 121)
- 6.19. Lemma. For every m, the homomorphism ψ : Fm V /Fm−1 V → C m ⊗ Si V is a monomorphism, and it is an isomorphism if V is a cofree C- (Budyak 122)
- comodule (Budyak 122)
- 6.22. Deﬁnition. A Hopf algebra over R is a quintuple (A, μ, η, Δ, ε) such that (A, μ, η, ε) is an augmented R-algebra, (A, Δ, ε) is an R-coalgebra, Δ : A → A ⊗ A and ε : A → R are homomorphisms of algebras, and η : R → A is a homomorphism of coalgebras. (Budyak 122)
- Note that if A is a Hopf algebra over a ﬁeld k and if dimk An < ∞ for every n then A∗ = Hom(A, k) is a Hopf algebra also (Budyak 122)
- 6.24. Deﬁnition. Let A be a Hopf algebra over R. (a) An A-module algebra is a quadruple (M, μ, η, ϕ), where (M, μ, η) is an R-algebra and (M, ϕ) is an A-module such that μ : M ⊗ M → M is a homomorphism of A-modules and ϕ : A ⊗ M → M is a homomorphism of R-algebras. A homomorphism of A-module algebras is a homomorphism of A-modules which is at the same time a homomorphism of R-algebras. (b) An A-comodule algebra is a quadruple (M, μ, η, ψ), where (M, μ, η) is an R-algebra and (M, ψ) is an A-comodule such that μ : M ⊗ M → M is a homomorphism of A-comodules and ψ : M → A ⊗ M is a homomorphism of R-algebras. (c) An A-module coalgebra is a quadruple (V, Δ, ε, ϕ), where (V, Δ, ε) is an R-coalgebra and (V, ϕ) is an A-module such that Δ : V → V ⊗ V is a homomorphism of A-modules and ϕ : A ⊗ V → V is a homomorphism of R-coalgebras. (Budyak 123)
- 6.25. Recollection. A very important example of a Hopf algebra over the ﬁeld Z/p, p prime, is the Steenrod algebra Ap = H ∗ (HZ/p; Z/p) = [HZ/p, Σd HZ/p]. (Budyak 123)
- All that we need to know about Ap can be found in Steenrod–Epstein [1], Margolis [1]. (Budyak 124)
- . The Steenrod algebra A2 is generated by elements Sq i of dimension i, where i = 1, 2, . . . , and all relations among Sq i follow from the Adem relations [a/2] [a/2] b−c−1 b = Sq a+b−c Sq c (6.26) Sq a Sq b = Sq a+b−c Sq c for a < 2b, a − 2c c=0 k where Sq 0 := 1. The comultiplication has the form Δ(Sq k ) = Sq i ⊗Sq k−i . (Budyak 124)
- The Steenrod algebra Ap , p > 2, is generated by elements β (the Bockstein homomorphism) and P i , dim β = 1, dim P i = 2i(p − 1), i = 1, 2, . . . (Budyak 124)
- Note that the Hopf algebra Ap∗ can be described as H∗ (H), where H = HZ/p. T (Budyak 125)
- 27. Examples. (a) For every H-space X and every ﬁeld k, we have a Hopf algebra H∗ (X; k): t (Budyak 125)
- Similarly, H ∗ (X; k) is a Hopf algebra for every H-space X of ﬁnite ∗ type, and H∗ (X; k) is the Hopf algebra dual to H ∗ (X; k). (Budyak 125)
- (b) Let H(−) denote H(−; Z/p). For every spectrum E, the group H ∗ (E) admits a natural Ap -module struct (Budyak 125)
- Furthermore, if E is a ring spectrum of ﬁnite Zor Z[p]-type then H ∗ (E) is a coalgebra over the Hopf algebra Ap , (Budyak 126)
- d this homomorphism turns H∗ (E) into a comodule over the Hopf algebra A p∗ . (Budyak 126)
- 6.29. Theorem. Let A be a connected Hopf algebra over a ﬁeld k, let C be a connected module coalgebra over A, and let v be the counit of C. Let p : C → GC be the canonical epimorphism (see 6.5), and let λ : GC → C be a k-homomorphism with pλ = 1GC . Let Bm be the subspace of A ⊗ GC generated by all elements a⊗x with |a| ≤ m. If the map ν : A → C, ν(a) = av, is monic for |a| ≤ m then the A-homomorphism η : A ⊗ GC → C, η(a ⊗ x) = a(λx) is monic on Bm . Furthermore, η is epic and thus is an isomorphism in dimensions ≤ m. Proof. This theorem is a version of the Milnor–Moore Theorem, (Budyak 126)
- 6.30. Corollary (The Milnor–Moore Theorem). Let A and C be as in 6.29. If ν : A → C, ν(a) = av, is monic then there is an isomorphism of A-modules C∼ = A ⊗ GC. In particular, C is a free A-module (Budyak 127)
- a canonical antiautomorphism c : A → A (called also an antipode) as follows (Budyak 128)
- 6.35. Deﬁnition. Given a connected Hopf algebra (A, μ, η, Δ, ε), we deﬁne a canonical antiautomorphism c : A → (Budyak 128)
- Milnor–Moore [1]. The ﬁrst example of such an antiautomorphism was found by Thom [1]; this was the canonical antiautomorphism χ : Ap → Ap of the Steenrod algebra Ap . (Budyak 128)
- Hopf [1] found that ordinary (co)homology rings of Lie groups (in fact, H-spaces) had certain speciﬁc algebraic properties. (For example, the rational cohomology ring of a Lie group is a free commutative algebra.) Afterwards Borel [1] clariﬁed the situation: every algebra A (over a ﬁeld) admitting a diagonal Δ : A → A ⊗ A, Δ(ab) = Δ(a)Δ(b), has such properties. Borel suggested the name “Hopf algebra” for such object; (Budyak 129)
- Milnor [2] discovered that the Steenrod algebra is a Hopf algebra. In this way he got a new description of Ap , and this enabled him (and some others) to compute initial terms of certain Adams spectral sequences. (Budyak 129)
- Every graded abelian group G can be realized as the total homotopy group π∗ (H(G)) of the graded Eilenberg–Mac Lane spectrum H(G), but not every spectrum E is (equivalent to) the graded Eilenberg–Mac Lane spectrum H(π∗ (E)). (Budyak 130)
- . (For example, the sphere spectrum S is not, because otherwise H∗ (HZ) would be a direct summand of H∗ (S).) It is clear that it is useful to know whether a spectrum is a graded Eilenberg–Mac Lane spectrum. (Budyak 130)
- For example, Thom [2] proved that the spectrum M O of the non-oriented (co)bordism is a graded Eilenberg–Mac Lane spectrum, and this enabled him to compute the group π∗ (M O) (i.e., non-oriented cobordism group) and to prove the realizability of all Z/2-homology classes by singular manifold (Budyak 130)
- In this section we give some suﬃcient conditions for a spectrum E to be a graded Eilenberg–Mac Lane spectrum, i.e., E H(π∗ (E) (Budyak 130)
- 7.6. Corollary. Every HZ-module spectrum is a graded Eilenberg–Mac Lane spectrum. (Budyak 132)
- 7.9. Corollary. For every spectrum E, the spectrum HZ ∧ E is a graded Eilenberg–Mac Lane spectrum. (Budyak 132)
- 7.8. Corollary. If a ring spectrum E admits a ring morphism HZ → E then E is a graded Eilenberg–Mac Lane spectrum. (Budyak 132)
- 7.11. Theorem. (i) The Q -localization of the sphere spectrum S is HQ. In particular, the Hurewicz homomorphism h : π∗ (E) ⊗ Q → H∗ (E) ⊗ Q is an isomorphism for every spectrum E. (Budyak 132)
- 7.13. Theorem-Deﬁnition (cf. Dold [1]). For every ring spectrum E there exists a ring equivalence E[0] → H(π∗ (E) ⊗ Q). This equivalence is called the Chern–Dold character with respect to E and is denoted by chE . (Budyak 134)
- 7.14. Theorem. Let p be an odd prime, and let E be a Z[p]-local spectrum of ﬁnite Z[p]-type. If E ∧ M (Z/p) is a graded Eilenberg–Mac Lane spectrum then so is E. (Budyak 134)
- 7.16. Theorem. Let E be a spectrum of ﬁnite Z[p]-type. If H ∗ (E) is a free Ap -module then E is a graded Eilenberg–Mac Lane spectrum. (Budyak 136)
- 7.28. Corollary. If a commutative ring spectrum E is a graded Eilenberg– Mac Lane spectrum with pπ∗ (E) = 0, then there is an isomorphism H∗ (E) ∼ = A p∗ ⊗ π∗ (E) of Ap -comodule algebras. (Budyak 141)
- 7.30. Theorem (Boardman [1]). Let E, F be two commutative ring spectra. Suppose that E, F are graded Eilenberg–Mac Lane spectra with pπ∗ (E) = 0 = pπ∗ (F ). Then every ring homomorphism r : π∗ (E) → π∗ (F ) is induced by a ring morphism f : E → F . So, if there exists a ring isomorphism π ∗ (E) ∼ = ∼ = π∗ (F ) then there exists a ring equivalence E F . In particular, there is a ring equivalence E H(π∗ (E)). (Budyak 142)
- A phantom, or a phantom map, is an essential map f : X → Y of a CW complex X such that f |X (n) is inessential for every n (Budyak 143)
- 1.1. Deﬁnition. A map f : X → Y of spaces (or a morphism of spectra) is called an X -phantom if σ[f ] = ∗ while [f ] = ∗. Similarly, an element a ∈ E ∗ (X) is an X -phantom if σ(a) = 0 while a = 0. This deﬁnition is given for an arbitrary family X , but really interesting are families with ∪Xα = X (Budyak 143)
- The classes of X homotopic maps (or X -equivalent elements) form a set [X, Y ]X (or a group E ∗ X (X)) with the distinguished element ∗ given by a constant map . There are the obvious quotient functions σ : [X, Y ] → [X, Y ]X and σ : E ∗ (X) → (Budyak 143)
- E ∗ X (X). (Budyak 143)
- 1.5. Example of a weak phantom. Let X = S n [1/3] be a Z[1/3]-localized sphere S n , n > 1, i.e., the telescope of a sequence f f f f Sn − → Sn − → ··· − → Sn − → ··· , where f : S n → S n is a map of degree 3. If we regard S n as a CW -complex with two cells, we obtain a cellular decomposition of X with 0-, nand (n+1)dimensional cells. This gives us a chain complex {C∗ (X), ∂∗ }, where Cn (X) has Z-basis {a1 , . . . , ai , . . . }, and Cn+1 (X) has Z-basis {b1 , . . . , bn , . . . }, and ∂ n+1 bi = ai − 3ai+1 . Let X = {Xλ } be the family of all ﬁnite CW n+1 subcomplexes of X. It is clear that HX (X) = H n+1 (S n ) = 0. On the other hand, H n+1 (X) = 0 because the cocycle ϕ : Cn+1 (X) → Z, ϕ(bi ) = 1 for every i, is not a coboundary. Indeed, if ϕ = δψ for some ψ : Cn (X) → Z, then ψ(ai ) − 3ψ(ai+1 ) = 1. In particular, (Budyak 144)
- ψ(a 1 ) = + 3k ψ(ak+1 ) 2 for every k. Hence, 3k divides 2ψ(a1 ) + 1 for every k, and so 2ψ(a1 ) = −1. This is a contradiction. Thus, the subgroup of weak phantoms of H n+1 (X) is nontrivial (and even uncountable, see 5.1 below). (Budyak 145)
- 1.6. Example of a phantom (Adams–Walker [1]). Let X = S 1 ∧ CP ∞ . Consider the space T = S 3 [0], the telescope of the sequence S 13 ϕ1 ϕ2 ϕn −→ S23 −→ · · · − → Sn3 −−→ Sn+1 3 − → ··· , where Sn3 is a copy of S 3 and deg ϕn = n. As in 1.5, we have C3 (T ) = {a1 , . . . , an , . . . }, C4 (T ) = {b1 , . . . , bn , . . . } (Budyak 145)

Last Annotation: 04/11/2019

- Proofs for video (J. Peter May 18)
- Rewrite with diagrams and injections (J. Peter May 36)
- Write out simultaneous replacement for cofibration (J. Peter May 58)
- Homotopy with squares! (J. Peter May 66)
- Find some problems that demonstrate the usefulness of relative groups (J. Peter May 79)
- How to build a CW complex with prescribed homotopy (J. Peter May 85)
- Write this as a commutator (J. Peter May 100)
- I thought this wasn’t true (J. Peter May 108)
- The sphere is dual to a point in homology theories (J. Peter May 117)
- Can probably work this result backwards (J. Peter May 172)
- Write this as a ring of manifolds (J. Peter May 175)
- Motivation for spectra (J. Peter May 183)
- Rewrite as a graded space (J. Peter May 184)
- Diagram of Postnikov tower (J. Peter May 191)
- How do you use them? (J. Peter May 193)
- Example computation with characteristic classes (J. Peter May 200)
- Application of spectral sequences (J. Peter May 203)
- The LES in homotopy is exactly why you want contractible spaces in fibrations. (J. Peter May 208)

- A treatment more closely attuned to the needs of algebraic geometers and analysts would include Č Čech cohomology on the one hand and de Rham cohomology and perhaps Morse homology on the other. A treatment more closely attuned to the needs of algebraic topologists would include spectral sequences and an array of calculations with them (J. Peter May 9)
- These changes reflect in part an enormous internal development of algebraic topology over this period, one which is largely unknown to most other mathematicians, even those working in such closely related fields as geometric topology and algebraic geometry (J. Peter May 9)
- The study of generalized homology and cohomology theories pervades modern algebraic topology. These theories satisfy the excision axiom. One constructs most such theories homotopically, by constructing representing objects called spectra, and one must then prove that excision holds. There is a way to do this in general that is no more difficult than the standard verification for singular homology and cohomology. (J. Peter May 9)
- A defect of nearly all existing texts is that they do not go far enough into the subject to give a feel for really substantial applications: the reader sees spheres and projective spaces, maybe lens spaces (J. Peter May 10)
- A function p : X −→ Y is continuous if it takes nearby points to nearby points. Precisely, p−1 (U ) is open if U is open. If X and Y are metric spaces, this means that, for any x ∈ X and ε > 0, there exists δ > 0 such that p(Uδ (x)) ⊂ Uε (p(x)). (J. Peter May 13)
- The further one goes in the subject, the more elaborate become the constructions A and the more horrendous become the relevant calculational techniques (J. Peter May 13)
- Let X be a space. Two paths f, g : I −→ X from x to y are equivalent if they are homotopic through paths from x to y (J. Peter May 14)
- Now suppose given two maps p, q : X −→ Y and a homotopy h : p ≃ q. We would like to conclude that p∗ = q∗ , but this doesn’t quite make sense because homotopies needn’t respect basepoints (J. Peter May 15)
- Theorem. π1 (S 1 , 1) ∼ = Z. (J. Peter May 16)
- the first map induces the evident identification of I/∂I with S 1 (J. Peter May 17)
- This lifting of paths works generally. For any path f : I −→ S 1 with f (0) = 1, there is a unique path ˜ f˜ : I −→ R such that ˜ f˜(0) = 0 and p ◦ f ˜ f˜ = f . (J. Peter May 17)
- The Brouwer fixed point theorem (J. Peter May 18)
- Proposition. There is no continuous map r : D 2 −→ S 1 such that r ◦ i = id. (J. Peter May 18)
- Theorem (Brouwer fixed point theorem). Any continuous map f : D 2 −→ D 2 has a fixed point. Proof. Suppose that f (x) 6= x for all x. Define r(x) ∈ S 1 to be the intersection with S 1 of the ray that starts at f (x) and passes through x. Certainly r(x) = x if x ∈ S 1 . By writing an equation for r in terms of f , we see that r is continuous. This contradicts the proposition. (J. Peter May 18)
- Theorem (Fundamental theorem of algebra). Let f (x) = x n + c1 x n−1 + · · · + cn−1 x + cn be a polynomial with complex coefficients ci , where n > 0. Then there is a complex number x such that f (x) = 0. Therefore there are n such complex numbers (counted with multiplicities). Proof. Using f (x)/(x−c) for a root c, we see that the last statement will follow by induction from the first (J. Peter May 18)
- Then h is a homotopy from the constant map at f (0)/|f (0)| to f ˆ fˆ, and we conclude that deg( ˆ fˆ) = 0. (J. Peter May 18)
- 3. Natural transformations A natural transformation α : F −→ G between functors C −→ D is a map of functors. It consists of a morphism αA : F (A) −→ G(A) for each object A of C such that the following diagram commutes for each morphism f : A −→ B of C : (J. Peter May 22)
- For example, if F : S −→ A b is the functor that sends a set to the free Abelian group that it generates and U : A b −→ S is the forgetful functor that sends an Abelian group to its underlying set, then we have a natural inclusion of sets S −→ U F (S). (J. Peter May 22)
- The functors F and U are left adjoint and right adjoint to each other, in the sense that we have a natural isomorphism A b(F (S), A) ∼ = ∼ = S (S, U (A)) (J. Peter May 22)
- for a set S and an Abelian group A. This just expresses the “universal property” of free objects: a map of sets S −→ U (A) extends uniquely to a homomorphism of groups F (S) −→ A. (J. Peter May 22)
- The fundamental group is a homotopy invariant functor on T , in the sense that it factors through a functor hT −→ G . (J. Peter May 22)
- When we have a (suitable) relation of homotopy between maps in a category C , we define the homotopy category hC to be the category with the same objects as C but with morphisms the homotopy classes of maps (J. Peter May 22)
- A homotopy equivalence in U is an isomorphism in hU . (J. Peter May 22)
- Less mysteriously, a map f : X −→ Y is a homotopy equivalence if there is a map g : Y −→ X such that both g ◦ f ≃ id and f ◦ g ≃ id. (J. Peter May 22)
- Functors carry isomorphisms to isomorphisms, so we see that a based homotopy equivalence induces an isomorphism of fundamental groups. (J. Peter May 22)
- Proposition. If f : X −→ Y is a homotopy equivalence, then f∗ : π1 (X, x) −→ π1 (Y, f (x)) is an isomorphism for all x ∈ X. (J. Peter May 22)
- A space X is said to be contractible if it is homotopy equivalent to a point. (J. Peter May 23)
- For this purpose, we define the “fundamental groupoid” Π(X) of a space X to be the category whose objects are the points of X and whose morphisms x −→ y are the equivalence classes of paths from x to y. Thus the set of endomorphisms of the object x is exactly the fundamental group π1 (X, x). (J. Peter May 23)
- The term “groupoid” is used for a category all morphisms of which are isomorphisms. (J. Peter May 23)
- There is a useful notion of a skeleton skC of a category C . This is a “full” subcategory with one object from each isomorphism class of objects of C , “full” meaning that the morphisms between two objects of skC are all of the morphisms between these objects in C . The inclusion functor J : skC −→ C is an equivalence of categories (J. Peter May 23)
- Let D be a small category and let C be any category. A D-shaped diagram in C is a functor F : D −→ C . A morphism F −→ F ′ of D-shaped diagrams is a natural transformation, and we have the category D[C ] of D-shaped diagrams in C (J. Peter May 24)
- The colimit, colim F , of a D-shaped diagram F is an object of C together with a morphism of diagrams ι : F −→ colim F that is initial among all such morphisms. (J. Peter May 24)
- This means that if η : F −→ A is a morphism of diagrams, then there is a unique map η̃ : colim F −→ A in C such that η̃ ◦ ι = η. Diagrammatically, this property is expressed by the assertion that, for each map d : D −→ D′ in D, we have a commutative diagram F (J. Peter May 24)
- The limit of F is defined by reversing arrows: it is an object lim F of C together with a morphism of diagrams π : lim F −→ F that is terminal among all such morphisms (J. Peter May 24)
- If D is a set regarded as a discrete category (only identity morphisms), then colimits and limits indexed on D are coproducts and products indexed on the set D. Coproducts are disjoint unions in S or U , wedges (or one-point unions) in T , free products in G , and direct sums in A b. Products are Cartesian products in all of these categories; more precisely, they are Cartesian products of underlying sets, with additional structure. (J. Peter May 24)
- If D is the category displayed schematically as eo /f or // d′ , d d where we have displayed all objects and all non-identity morphisms, then the colimits indexed on D are called pushouts or coequalizers, respectively. Similarly, if D is displayed schematically as e /do f or // d′ , d (J. Peter May 24)
- then the limits indexed on D are called pullbacks or equalizers, respectively (J. Peter May 25)
- A category is said to be cocomplete if it has all colimits, complete if it has all limits. The categories S , U , T , G , and A b are complete and cocomplete. If a category has coproducts and coequalizers, then it is cocomplete, and similarly for completeness. The proof is a worthwhile exercise. (J. Peter May 25)
- The proof well illustrates how to manipulate colimits formally. (J. Peter May 25)
- Any compact surface is homeomorphic to a sphere, or to a connected sum of tori T 2 = S 1 × S 1 , or to a connected sum of projective planes RP 2 = S 2 /Z2 (where we write Z2 = Z/2Z) (J. Peter May 28)
- Lemma. For based spaces X and Y , π1 (X × Y ) ∼ = ∼ = π1 (X) × π1 (Y ). (J. Peter May 28)
- Proposition. Let X = U ∪V , where U , V , and U ∩V are path connected open neighborhoods of the basepoint of X and V is simply connected. Then π1 (U ) −→ π1 (X) is an epimorphism whose kernel is the smallest normal subgroup of π1 (U ) that contains the image of π1 (U ∩ V ). (J. Peter May 28)
- We shall later use the following application of the van Kampen theorem to prove that any group is the fundamental group of some space. We need a definition. Definition. A space X is said to be simply connected if it is path connected and satisfies π1 (X) = 0. (J. Peter May 28)
- However, I know of no published source for the use that we shall make of the orbit category O(π1 (B, b)) in the classification of coverings of a space B (J. Peter May 29)
- This point of view gives us the opportunity to introduce some ideas that are central to equivariant algebraic topology, the study of spaces with group actions (J. Peter May 29)
- While the reader is free to think about locally contractible spaces, weaker conditions are appropriate for the full generality of the theory of covering spaces. A space X is said to be locally path connected if for any x ∈ X and any neighborhood U of x, there is a smaller neighborhood V of x each of whose points can be connected to x by a path in U . This is equivalent to the seemingly more stringent requirement that the topology of X have a basis consisting of path connected open sets. In fact, if X is locally path connected and U is an open neighborhood of a point x, then the set V = {y | y can be connected to x by a path in U } is a path connected open neighborhood of x that is contained in U (J. Peter May 29)
- Observe that if X is connected and locally path connected, then it is path connected (J. Peter May 29)
- Definition. A map p : E −→ B is a covering (or cover, or covering space) if it is surjective and if each point b ∈ B has an open neighborhood V such that each component of p−1 (V ) is open in E and is mapped homeomorphically onto V by p (J. Peter May 29)
- Any homeomorphism is a cover. A product of covers is a cover. The projection R −→ S 1 is a cover (J. Peter May 29)
- Theorem (Unique path lifting) (J. Peter May 30)
- Definition. A covering p : E −→ B is regular if p∗ (π1 (E, e)) is a normal subgroup of π1 (B, b). It is universal if E is simply connected (J. Peter May 30)
- As we shall explain in §4, for a universal cover p : E −→ B, the elements of Fb are in bijective correspondence with the elements of π1 (B, b). (J. Peter May 30)
- Example. For n ≥ 2, S n is a universal cover of RP n . Therefore π1 (RP n ) has only two elements. There is a unique group with two elements, and this proves our earlier claim that π1 (RP n ) = Z2 . (J. Peter May 30)
- As e′ runs through Fb , the groups p∗ (π1 (E, e′ )) run through all conjugates of p∗ (π1 (E, e)) in π1 (B, b). (J. Peter May 30)
- A path f : I −→ B with f (0) = b lifts uniquely to a path g : I −→ E (J. Peter May 30)
- Let p : E −→ B be a covering (J. Peter May 30)
- The category xof objects under x has objects the maps f : x −→ y in C ; for objects f : x −→ y and g : x −→ z, the morphisms γ : f −→ g in xare the morphisms γ : y −→ z in C such that γ ◦ f = g : x −→ z. (J. Peter May 30)
- (ii) Let C be a small groupoid. Define the star of x, denoted St(x) or StC (x), to be the set of objects of x, that is, the set of morphisms of C with source x. (J. Peter May 31)
- (iii) Let E and B be small connected groupoids. A covering p : E −→ B is a functor that is surjective on objects and restricts to a bijection (J. Peter May 31)
- For an object b of B, let Fb denote the set of objects of E such that p(e) = b. Then p−1 (St(b)) is the disjoint union over e ∈ Fb of St(e). (J. Peter May 31)
- Proposition. If p : E −→ B is a covering of spaces, then the induced functor Π(p) : Π(E) −→ Π(B) is a covering of groupoids (J. Peter May 31)
- The fibers Fb of a covering of groupoids are related by translation functions (J. Peter May 31)
- Proposition. Any two fibers Fb and Fb′ of a covering of groupoids have the same cardinality. Therefore any two fibers of a covering of spaces have the same cardinality. (J. Peter May 31)
- The isotropy group Gs of a point s is the subgroup {g|gs = s} of G. An action is free if gs = s implies g = e, that is, if Gs = e for every s ∈ S. (J. Peter May 32)
- An action is transitive if for every pair s, s′ of elements of S, there is an element g of G such that gs = s′ . (J. Peter May 32)
- When G acts transitively on a set S, we obtain an isomorphism of G-sets between S and the G-set G/Gs for any fixed s ∈ S by sending gs to the coset gGs (J. Peter May 32)
- For a subgroup H of G, let N H denote the normalizer of H in G and define W H = N H/H. Such quotient groups W H are sometimes called Weyl groups (J. Peter May 32)
- Lemma. Let G act transitively on a set S, choose s ∈ S, and let H = Gs . Then W H is isomorphic to the group AutG (S) of automorphisms of the G-set S. (J. Peter May 32)
- Lemma. A G-map α : G/H −→ G/K has the form α(gH) = gγK, where the element γ ∈ G satisfies γ −1 hγ ∈ K for all h ∈ H. (J. Peter May 32)
- Definition. The category O(G) of canonical orbits has objects the G-sets G/H and morphisms the G-maps of G-sets. (J. Peter May 32)
- Lemma. The category O(G) is isomorphic to the category G whose objects are the subgroups of G and whose morphisms are the distinct subconjugacy relations γ −1 Hγ ⊂ K for γ ∈ G. (J. Peter May 32)
- A covering of groupoids is universal if and only if π(B, b) acts freely on Fb , and then Fb is isomorphic to π(B, b) as a π(B, b)-set. Specializing to covering spaces, this sharpens our earlier claim that the elements of Fb and π1 (B, b) are in bijective correspondence. (J. Peter May 33)
- Definition. A map g : E −→ E ′ of coverings of B is a functor g such that the following diagram of functors is commutative (J. Peter May 34)
- Lemma. A map g : E −→ E ′ of coverings is itself a covering (J. Peter May 34)
- Corollary. If it exists, the universal cover of B is unique up to isomorphism and covers any other cover. (J. Peter May 34)
- We have given an algebraic classification of all possible covers of B: there is at most one isomorphism class of covers corresponding to each conjugacy class of subgroups of π(B, b) (J. Peter May 35)
- We begin with the following result, which deserves to be called the fundamental theorem of covering space theory and has many other applications (J. Peter May 36)
- Definition. A map g : E −→ E ′ of coverings over B is a map g such that the following diagram is commutative: (J. Peter May 36)
- In particular, two maps of covers g, g ′ : E −→ E ′ coincide if g(e) = g ′ (e) for any one e ∈ E. (J. Peter May 37)
- Corollary. If it exists, the universal cover of B is unique up to isomorphism and covers any other cover. (J. Peter May 37)
- We hasten to add that the theorem above is atypical of algebraic topology. It is not usually the case that algebraic invariants like the fundamental group totally determine the existence and uniqueness of maps of topological spaces with prescribed properties (J. Peter May 37)
- Corollary. Let p : E −→ B be a covering and choose b ∈ B and e ∈ Fb . Write G = π1 (B, b) and H = p∗ (π1 (E, e)). Then Aut(E) is isomorphic to the group of automorphisms of the G-set Fb and therefore to the group W H. If p is regular, then Aut(E) ∼ = ∼ = G/H. If p is universal, then Aut(E) ∼ = ∼ = G (J. Peter May 37)
- We have now given an algebraic classification of all possible covers of B: there is at most one isomorphism class of covers corresponding to each conjugacy class of subgroups of π1 (B, b). (J. Peter May 38)
- We show here that all of these possibilities are actually realized. (J. Peter May 38)
- Again, while it suffices to think in terms of locally contractible spaces, appropriate generality demands a weaker hypothesis. We say that a space B is semi-locally simply connected if every point b ∈ B has a neighborhood U such that π1 (U, b) −→ π1 (B, b) is the trivial homomorphism (J. Peter May 38)
- Theorem. If B is connected, locally path connected, and semi-locally simply connected, then B has a universal cover. (J. Peter May 38)
- Definition. A G-space X is a space X that is a G-set with continuous action map G × X −→ X. Define the orbit space X/G to be the set of orbits {Gx|x ∈ X} with its topology as a quotient space of X. (J. Peter May 39)
- The definition makes sense for general topological groups G. However, our interest here is in discrete groups G, for which the continuity condition just means that action by each element of G is a homeomorphism (J. Peter May 39)
- Proposition. Let p : E −→ B be a cover such that Aut(E) acts transitively on Fb . Then the cover p is regular and E/ Aut(E) is homeomorphic to B. (J. Peter May 39)
- A map f : X −→ Y is said to be a local homeomorphism if every point of X has an open neighborhood that maps homeomorphically onto an open set in Y . (J. Peter May 40)
- Let X be a G-space, where G is a (discrete) group. For a subgroup H of G, define X H = {x|hx = x for all h ∈ H} ⊂ X; X H is the H-fixed point subspace of X (J. Peter May 40)
- We define graphs, describe their homotopy types, and use them to show that a subgroup of a free group is free and that any group is the fundamental group of some space (J. Peter May 43)
- Observe that a graph is a locally contractible space: any neighborhood of any point contains a contractible neighborhood of that point. Therefore a connected graph has all possible covers. (J. Peter May 43)
- A graph is finite if it has only finitely many vertices and edges or, equivalently, if it is a compact space (J. Peter May 43)
- A graph is locally finite if each vertex is a boundary point of only finitely many edges or, equivalently, if it is a locally compact space. (J. Peter May 43)
- A subspace A of a space X is a deformation retract if there is a homotopy h : X × I −→ X such that h(x, 0) = x, h(a, t) = a, and h(x, 1) ∈ A for all x ∈ X, a ∈ A, and t ∈ I. Such a homotopy is called a deformation of X onto A. (J. Peter May 43)
- for a suitably nice inclusion, called a “cofibration,” of a contractible space T in a space X, the quotient map X −→ X/T is a homotopy equivalence (J. Peter May 44)
- Theorem. Let X be a connected graph with maximal tree T . Then the quotient space X/T is the wedge of one circle for each edge of X not in T , and the quotient map q : X −→ X/T is a homotopy equivalence (J. Peter May 44)
- types of graphs has the following immediate implication. Corollary. If X is a connected graph, then π1 (X) is a free group with one generator for each edge not in a given maximal tree. If X is finite, then π1 (X) is free on 1 − χ(X) generators; in particular, χ(X) ≤ 1, with equality if and only if X is a tree. (J. Peter May 45)
- Theorem. If B is a connected graph with vertex set B 0 and p : E −→ B is a covering, then E is a connected graph with vertex set E 0 = p−1 (B 0 ) and with one edge for each edge j of B and point e ∈ Fj(0) . Therefore, if B is finite and p is a finite cover whose fibers have cardinality n, then E is finite and χ(E) = nχ(B). (J. Peter May 45)
- Theorem. A subgroup H of a free group G is free. If G is free on k generators and H has finite index n in G, then H is free on 1 − n + nk generators. (J. Peter May 45)
- Theorem. For any group G, there is a connected space X such that π1 (X) is isomorphic to G. (J. Peter May 45)
- Define the (unreduced) cone on E to be CE = (E × I)/(E × {1}) and define X = B ∪p CE/(∼), where (e, 0) ∼ p(e). (J. Peter May 46)
- The space X constructed in the proof is called the “homotopy cofiber” of the map p. (J. Peter May 46)
- The homotopy and homology groups of spaces are supported on compact subspaces, and it turns out that if one assumes a separation property that is a little weaker than the Hausdorff property, then one can refine the point-set topology of spaces to eliminate such pathology without changing these invariants. (J. Peter May 47)
- We shall understand compact spaces to be both compact and Hausdorff (J. Peter May 47)
- A space X is said to be “weak Hausdorff” if g(K) is closed in X for every map g : K −→ X from a compact space K into X (J. Peter May 47)
- A space X is “compactly generated” if it is a weak Hausdorff k-space (J. Peter May 47)
- Lemma. If X is a compactly generated space and Y is any space, then a function f : X −→ Y is continuous if and only if its restriction to each compact subspace K of X is continuous (J. Peter May 47)
- A subspace A of X is said to be “compactly closed” if g −1 (A) is closed in K for any map g : K −→ X from a compact space K into X. When X is weak Hausdorff, this holds if and only if the intersection of A with each compact subset of X is closed. A space X is a “k-space” if every compactly closed subspace is closed (J. Peter May 47)
- By definition, a space X is Hausdorff if the diagonal subspace ∆X = {(x, x)} is closed in X ×c X. The weak Hausdorff property admits a similar characterization. Lemma. If X is a k-space, then X is weak Hausdorff if and only if ∆X is closed in X × X. (J. Peter May 48)
- The interpretation is that a quotient space of a compactly generated space by a “closed equivalence relation” is compactly generated (J. Peter May 48)
- Proposition. If X and Y are compactly generated spaces, A is a closed subspace of X, and f : A −→ Y is any continuous map, then the pushout Y ∪f X is compactly generated. (J. Peter May 48)
- Another source of pathology is passage to colimits over sequences of maps Xi −→ Xi+1 . (J. Peter May 48)
- Proposition. If {Xi } is a sequence of compactly generated spaces and inclusions Xi −→ Xi+1 with closed images, then colim Xi is compactly generated. (J. Peter May 48)
- Limits of weak Hausdorff spaces are weak Hausdorff, but limits of k-spaces need not be k-spaces (J. Peter May 48)
- We construct limits of compactly generated spaces by applying the functor k to their limits as spaces. (J. Peter May 48)
- Point-set level colimits of weak Hausdorff spaces need not be weak Hausdorff (J. Peter May 48)
- However, if a point-set level colimit of compactly generated spaces is weak Hausdorff, then it is a k-space and therefore compactly generated. We shall only be interested in colimits in those cases where this holds. (J. Peter May 48)
- Proposition. For spaces X, Y , and Z in U , the canonical bijection Z (X×Y ) ∼ = ∼ = (Z Y ) X is a homeomorphism. (J. Peter May 49)
- Observe in particular that a homotopy X × I −→ Y can equally well be viewed as a map X −→ Y I (J. Peter May 49)
- Exact sequences that feature in the study of homotopy, homology, and cohomology groups all can be derived homotopically from the theory of cofiber and fiber sequences that we present in this and the following two chapters (J. Peter May 51)
- The theories of cofiber and fiber sequences illustrate an important, but informal, duality theory, known as Eckmann-Hilton duality (J. Peter May 51)
- It is based on the adjunction between Cartesian products and function spaces (J. Peter May 51)
- Definition. A map i : A −→ X is a cofibration if it satisfies the homotopy extension property (HEP). This means that if h ◦ i0 = f ◦ i in the diagram (J. Peter May 51)
- Pushouts of cofibrations are cofibrations (J. Peter May 51)
- Namely, we can let Y in our original test diagram be the “mapping cylinder (J. Peter May 52)
- As a matter of point-set topology, left as an exercise, it follows that a cofibration is an inclusion with closed image. (J. Peter May 52)
- M i ≡ X ∪i (A × I), which is the pushout of i and (J. Peter May 52)
- We can use the mapping cylinder construction to decompose an arbitrary map f : X −→ Y as the composite of a cofibration and a homotopy equivalence (J. Peter May 53)
- To see this, recall that M f = Y ∪f (X × I) and observe that f coincides with the composite X j − → Mf r r − → Y, (J. Peter May 53)
- where j(x) = (x, 1) and where r(y) = y on Y and r(x, s) = f (x) on X × I. (J. Peter May 53)
- We shall often consider pairs (X, A) consisting of a space X and a subspace A. Cofibration pairs will be those pairs that “behave homologically” just like the associated quotient spaces X/A. (J. Peter May 53)
- Definition. A pair (X, A) is an NDR-pair (= neighborhood deformation retract pair) if there is a map u : X −→ I such that u−1 (0) = A and a homotopy h : X × I −→ X such that h0 = id, h(a, t) = a for a ∈ A and t ∈ I, and h(x, 1) ∈ A if u(x) < 1; (X, A) is a DR-pair if u(x) < 1 for all x ∈ X, in which case A is a deformation retract of X. (J. Peter May 53)
- Theorem. Let A be a closed subspace of X. Then the following are equivalent: (i) (X, A) is an NDR-pair. (ii) (X × I, X × {0} ∪ A × I) is a DR-pair. (iii) X × {0} ∪ A × I is a retract of X × I. (iv) The inclusion i : A −→ X is a cofibration (J. Peter May 53)
- It is often important to work in the category of spaces under a given space A, and we shall later need a basic result about homotopy equivalences in this category. We shall also need a generalization concerning homotopy equivalences of pairs. (J. Peter May 54)
- A homotopy between maps under A is a homotopy that at each time t is a map under A. We then write h : f ≃ f ′ rel A (J. Peter May 54)
- A. Such an equivalence is called a “cofiber homotopy equivalence.” The name is suggested by the following result, whose proof illustrates a more substantial use of the HEP than we have seen before (J. Peter May 54)
- Proposition. Let i : A −→ X and j : A −→ Y be cofibrations and let f : X −→ Y be a map such that f ◦i = j. Suppose that f is a homotopy equivalence. Then f is a cofiber homotopy equivalence. (J. Peter May 54)
- Definition. A surjective map p : E −→ B is a fibration if it satisfies the covering homotopy property (CHP) (J. Peter May 57)
- This notion of a fibration is due to Hurewicz. There is a more general notion of a Serre fibration, in which the test spaces Y are restricted to be cubes I n . Serre fibrations are more appropriate for many purposes (J. Peter May 57)
- With this formulation, we can “dualize” the proof that pushouts of cofibrations are cofibrations to show that pullbacks of fibrations are fibrations. We often write A ×g E for the pullback of a given fibration p : E −→ B and a map g : A −→ B. (J. Peter May 57)
- Lemma. If p : E −→ B is a fibration and g : A −→ B is any map, then the induced map A ×g E −→ A is a fibration (J. Peter May 57)
- Although the CHP is expressed in terms of general test diagrams, there is a certain universal test diagram. Namely, we can let Y in our original test diagram be the “mapping path space” N p ≡ E ×p B I = {(e, β)|β(0) = p(e)} ⊂ E × B I . (J. Peter May 57)
- In general, path lifting functions are not unique. In fact, we have already studied the special kinds of fibrations for which they are unique. Lemma. If p : E −→ B is a covering, then p is a fibration with a unique path lifting function s. (J. Peter May 58)
- A map s : N p −→ E I such that k ◦ s = id, where k : E I −→ N p has coordinates p0 and pI , is called a path lifting function (J. Peter May 58)
- Lemma. If i : A −→ X is a cofibration and B is a space, then the induced map p = B i : B X −→ B A is a fibration. (J. Peter May 58)
- We can use the mapping path space construction to decompose an arbitrary map f : X −→ Y as the composite of a homotopy equivalence and a fibration. (J. Peter May 58)
- When restricted to the spaces U in a well chosen open cover O of the base space B, a covering is homeomorphic to the projection U × F −→ U , where F is a fixed discrete set (J. Peter May 59)
- The obvious generalization of this is the notion of a bundle. A map p : E −→ B is a bundle if, when restricted to the spaces U in a well chosen open cover O of B, there are homeomorphisms φ : U × F −→ p−1 (U ) such that p ◦ φ = π1 , where F is a fixed topological space. (J. Peter May 59)
- that every bundle is a fibration (J. Peter May 59)
- Theorem. Let p : E −→ B be a map and let O be a numerable open cover of B. Then p is a fibration if and only if p : p−1 (U ) −→ U is a fibration for every U ∈ O. (J. Peter May 59)
- Since pullbacks of fibrations are fibrations (J. Peter May 59)
- B. A space over B is a map p : E −→ B. A map of spaces over B is a commutative diagram (J. Peter May 60)
- A homotopy between maps over B is a homotopy that at each time t is a map over B. There results a notion of a homotopy equivalence over B. Such an equivalence is called a “fiber homotopy equivalence.” (J. Peter May 60)
- Proposition. Let p : D −→ B and q : E −→ B be fibrations and let f : D −→ E be a map such that q ◦ f = p. Suppose that f is a homotopy equivalence. Then f is a fiber homotopy equivalence (J. Peter May 60)
- Translation of fibers along paths in the base space played a fundamental role in our study of covering spaces (J. Peter May 61)
- Theorem. Lifting of equivalence classes of paths in B to homotopy classes of maps of fibers specifies a functor λ : Π(B) −→ hU . Therefore, if B is path connected, then any two fibers of B are homotopy equivalent. (J. Peter May 62)
- Just as the fundamental group π1 (B, b) of the base space of a covering acts on the fiber Fb , so the fundamental group π1 (B, b) of the base space of a fibration acts “up to homotopy” on the fiber (J. Peter May 62)
- The composite of homotopy equivalences is a homotopy equivalence, and composition defines a continuous product on Aut(X). With this product, Aut(X) is a “topological monoid,” namely a space with a continuous and associative multiplication with a two-sided identity element, but it is not a group. However, the path components of Aut(X) are the homotopy classes of homotopy equivalences of X, and these do form a group under composition (J. Peter May 62)
- Corollary. Lifting of equivalence classes of loops specifies a homomorphism π1 (B, b) −→ π0 (Aut(Fb )). (J. Peter May 62)
- We use cofibrations and fibrations in the category T of based spaces to generate two “exact sequences of spaces” from a given map of based spaces (J. Peter May 65)
- For based spaces X and Y , we let [X, Y ] denote the set of based homotopy classes of based maps X −→ Y . This set has a natural basepoint, namely the homotopy class of the constant map from X to the basepoint of Y (J. Peter May 65)
- The appropriate analogue of the Cartesian product in the category of based spaces is the “smash product” X ∧ Y defined by X ∧ Y = X × Y /X ∨ Y (J. Peter May 65)
- With these definitions, we have a natural homeomorphism of based spaces F (X ∧ Y, Z) ∼ = ∼ = F (X, F (Y, Z)) for based spaces X and Y . (J. Peter May 65)
- We define the cone on X to be CX = X ∧ I, where I is given the basepoint 1. That is, CX = X × I/({∗} × I ∪ X × {1}). (J. Peter May 65)
- We view S 1 as I/∂I, denote its basepoint by 1, and define the suspension of X to be ΣX = X ∧ S 1 . That is, ΣX = X × S 1 /({∗} × S 1 ∪ X × {1}). (J. Peter May 65)
- These are sometimes called the reduced cone and suspension, to distinguish them from the unreduced constructions, in which the line {∗} × I through the basepoint of X is not identified to a point (J. Peter May 65)
- Dually, we define the path space of X to be P X = F (I, X), where I is given the basepoint 0. Thus the points of P X are the paths in X that start at the basepoint. We define the loop space of X to be ΩX = F (S 1 , X). Its points are the loops at the basepoint. (J. Peter May 66)
- We have the adjunction F (ΣX, Y ) ∼ = ∼ = F (X, ΩY ). Passing to π0 , this gives that [ΣX, Y ] ∼ = ∼ = [X, ΩY ]. (J. Peter May 66)
- Lemma. [ΣX, Y ] is a group and [Σ2 X, Y ] is an Abelian group (J. Peter May 66)
- We say that X is “nondegenerately based,” or “well pointed,” if the inclusion of its basepoint is a cofibration in the unbased sense. (J. Peter May 66)
- Write Y+ for the union of a space Y and a disjoint basepoint and observe that we can identify X ∧ Y+ with X × Y /{∗} × Y . (J. Peter May 66)
- The space X ∧ I+ is called the reduced cylinder on X, and a based homotopy X × I −→ Y is the same thing as a based map X ∧ I+ −→ Y . We change notations and write M f for the based mapping cylinder Y ∪f (X ∧ I+ ) of a based map (J. Peter May 66)
- As in the unbased case, we conclude that a based map i : A −→ X is a cofibration if and only if M i is a retract of X ∧ I+ . (J. Peter May 66)
- For a based map f : X −→ Y , define the “homotopy cofiber” Cf to be Cf = Y ∪f CX = M f /j(X), (J. Peter May 67)
- where j : X −→ M f sends x to (x, 1) (J. Peter May 67)
- Let i : Y −→ Cf be the inclusion. It is a cofibration since it is the pushout of f and the cofibration X −→ CX that sends x to (x, 0) (J. Peter May 67)
- is called the cofiber sequence generated by the map f ; here (J. Peter May 67)
- These “long exact sequences of based spaces” give rise to long exact sequences of pointed sets, where a sequence S ′ f − →S g → S − ′′ of pointed sets is said to be exact if g(s) = ∗ if and only if s = f (s′ ) for some s (J. Peter May 67)
- Theorem. For any based space Z, the induced sequence · · · −→ [ΣCf, Z] −→ [ΣY, Z] −→ [ΣX, Z] −→ [Cf, Z] −→ [Y, Z] −→ [X, Z] is an exact sequence of pointed sets, or of groups to the left of [ΣX, Z], or of Abelian groups to the left of [Σ2 X, Z]. (J. Peter May 67)
- Lemma. If i : A −→ X is a cofibration, then the quotient map ψ : Ci −→ Ci/CA ∼ = ∼ = X/A is a based homotopy equivalence (J. Peter May 68)
- For a based map f : X −→ Y , define the “homotopy fiber” F f to be F f = X ×f P Y = {(x, χ)|f (x) = χ(1)} ⊂ X × P Y. Equivalently, F f is the pullback displayed in the diagram (J. Peter May 69)
- Theorem. For any based space Z, the induced sequence · · · −→ [Z, ΩF f ] −→ [Z, ΩX] −→ [Z, ΩY ] −→ [Z, F f ] −→ [Z, X] −→ [Z, Y ] is an exact sequence of pointed sets, or of groups to the left of [Z, ΩY ], or of Abelian groups to the left of [Z, Ω2 Y ]. (J. Peter May 70)
- It is often useful to know that cofiber sequences and fiber sequences can be connected to one another. The adjunction between loops and suspension has “unit” and “counit” maps η : X −→ ΩΣX and ε : ΣΩX −→ X (J. Peter May 71)
- Observe that πn (X) = πn−1 (ΩX) = · · · = π0 (Ω n X). (J. Peter May 73)
- For ∗ ∈ A ⊂ X, the (homotopy) fiber of the inclusion A −→ X may be identified with the space P (X; ∗, A) of paths in X that begin at the basepoint and end in A (J. Peter May 73)
- This is a group if n ≥ 2 and an Abelian group if n ≥ 3. Again, πn (X, A) = π0 (Ω n−1 P (X; ∗, A)). These are called relative homotopy groups. (J. Peter May 73)
- Using φ∗ to identify π∗ F with π∗ (F p), we may rewrite the long exact sequence of the bottom row of the diagram as · · · −→ πn (F ) −→ πn (E) −→ πn (B) ∂ ∂ − → πn−1 (F ) −→ · · · −→ π0 (E) −→ {∗}. (J. Peter May 74)
- This is one of the main tools for the computation of homotopy groups (J. Peter May 74)
- Lemma. If p : E −→ B is a covering, then p∗ : πn (E) −→ πn (B) is an isomorphism for all n ≥ 2. (J. Peter May 74)
- Lemma. For all spaces X and Y and all n, πn (X × Y ) ∼ = ∼ = πn (X) × πn (Y ). (J. Peter May 75)
- There are three standard bundles, called the Hopf bundles, that can be used to obtain a bit more information about the homotopy groups of spheres. (J. Peter May 75)
- That is, CP 1 = (C × C − {0})/(∼), where (z1 , z2 ) ∼ (λz1 , λz2 ) for complex numbers λ, z1 , and z2 . Write [z1 , z2 ] for the equivalence class of (z1 , z2 ). We obtain a homeomorphism CP 1 −→ S 2 by identifying S 2 with the one-point compactification of C and mapping [z1 , z2 ] to z2 /z1 if z1 6= 0 and to the point at ∞ if z1 = 0. The Hopf map η : S 3 −→ S 2 is specified by η(z1 , z2 ) = [z1 , z2 ], where S 3 is identified with the unit sphere in the complex plane C2 . (J. Peter May 75)
- Since we have complete information on the homotopy groups of S 1 , the long exact sequence of homotopy groups associated to η has the following direct consequence. Lemma. π2 (S 2 ) = ∼ = Z and πn (S ∼ 3 ) ∼ = ∼ = πn (S 2 ) for n ≥ 3. (J. Peter May 75)
- It is left as an exercise to show that the long exact sequence associated to ν implies that π7 (S 4 ) contains an element of infinite order, and σ can be used similarly to show the same for π15 (S 8 ). (J. Peter May 75)
- In fact, the homotopy groups πq (S n ) for q > n > 1 are all finite except for π4n−1 (S 2n ), which is the direct sum of Z and a finite group. (J. Peter May 75)
- The difficulty of computing homotopy groups is well illustrated by the fact that there is no non-contractible simply connected compact manifold (or finite CW complex) all of whose homotopy groups are known. We shall find many non-compact spaces whose homotopy groups we can determine completely (J. Peter May 75)
- Lemma. If X is the colimit of a sequence of inclusions Xi −→ Xi+1 of based spaces, then the natural map colimi πn (Xi ) −→ πn (X) is an isomorphism for each n. (J. Peter May 75)
- Corollary. A homotopy equivalence of spaces or of pairs of spaces induces an isomorphism on all homotopy groups. (J. Peter May 77)
- Definition. A map e : Y −→ Z is an n-equivalence if, for all y ∈ Y , the map e∗ : πq (Y, y) −→ πq (Z, e(y)) is an injection for q < n and a surjection for q ≤ n; e is said to be a weak equivalence if it is an n-equivalence for all n. (J. Peter May 77)
- Observe that πn+1 (X, x) can be viewed as the set of relative homotopy classes of maps (CS n , S n ) −→ (X, x). (J. Peter May 77)
- It gives a useful criterion for determining when a given map is an n-equivalence (J. Peter May 77)
- We introduce a large class of spaces, called CW complexes, between which a weak equivalence is necessarily a homotopy equivalence. Thus, for such spaces, the homotopy groups are, in a sense, a complete set of invariants. Moreover, we shall see that every space is weakly equivalent to a CW complex (J. Peter May 81)
- iv) A map of pairs f : (X, A) −→ (Y, B) between relative CW complexes is said to be “cellular” if f (X n ) ⊂ Y n for all n. (J. Peter May 81)
- Of course, pushouts and unions are understood in the topological sense, with the compactly generated topologies. A subspace of X is closed if and only if its intersection with each X n is closed. (J. Peter May 81)
- (ii) Via a homeomorphism I × I ∼ = ∼ = D2 , the standard presentations of the torus T = S × S , the projective plane RP 2 , and the Klein bottle K as quotients of a 1 1 square display these spaces as CW complexes with one or two vertices, two edges, and one 2-cell: (J. Peter May 81)
- Lemma. If (X, A) is a relative CW complex, then the quotient space X/A is a CW complex with a vertex corresponding to A and one n-cell for each relative n-cell of (X, A). (J. Peter May 82)
- Lemma. If A is a subcomplex of a CW complex X, Y is a CW complex, and f : A −→ Y is a cellular map, then the pushout Y ∪f X is a CW complex that contains Y as a subcomplex and has one cell for each cell of X that is not in A. The quotient complex (Y ∪f X)/Y is isomorphic to X/A. (J. Peter May 82)
- Lemma. The colimit of a sequence of inclusions of subcomplexes Xn −→ Xn+1 in CW complexes is a CW complex that contains each of the Xi as a subcomplex. (J. Peter May 82)
- In particular, if we take e to be the identity map of Y , we see that the inclusion A −→ X is a cofibration. Observe that, by passage to colimits, we are free to take n = ∞ in the theorem. (J. Peter May 83)
- Theorem (Whitehead). If X is a CW complex and e : Y −→ Z is an nequivalence, then e∗ : [X, Y ] −→ [X, Z] is a bijection if dim X < n and a surjection if dim X = n. (J. Peter May 83)
- Theorem (Whitehead). An n-equivalence between CW complexes of dimension less than n is a homotopy equivalence. A weak equivalence between CW complexes is a homotopy equivalence (J. Peter May 84)
- If X is a finite CW complex, in the sense that it has finitely many cells, and if dim X > 1 and X is not contractible, then it is known that X has infinitely many non-zero homotopy groups. The Whitehead theorem is thus surprisingly strong: in its first statement, if low dimensional homotopy groups are mapped isomorphically, then so are all higher homotopy groups (J. Peter May 84)
- Fortunately, any map between CW complexes is homotopic to a cellular map (J. Peter May 84)
- Definition. A space X is said to be n-connected if πq (X, x) = 0 for 0 ≤ q ≤ n and all x. A pair (X, A) is said to be n-connected if π0 (A) −→ π0 (X) is surjective and πq (X, A, a) = 0 for 1 ≤ q ≤ n and all a. It is equivalent that the inclusion A −→ X be an n-equivalence (J. Peter May 84)
- Lemma. A relative CW complex (X, A) with no m-cells for m ≤ n is nconnected. In particular, (X, X n ) is n-connected for any CW complex X. (J. Peter May 84)
- Theorem (Cellular approximation). Any map f : (X, A) −→ (Y, B) between relative CW complexes is homotopic relative to A to a cellular map. (J. Peter May 84)
- Corollary. For CW complexes X and Y , any map X −→ Y is homotopic to a cellular map, and any two homotopic cellular maps are cellularly homotopic. (J. Peter May 85)
- Theorem (Approximation by CW complexes). For any space X, there is a CW complex ΓX and a weak equivalence γ : ΓX −→ X. For a map f : X −→ Y and another such CW approximation γ : ΓY −→ Y , there is a map Γf : ΓX −→ ΓY , unique up to homotopy, such that the following diagram is homotopy commutative (J. Peter May 85)
- If X is n-connected, n ≥ 1, then ΓX can be chosen to have a unique vertex and no q-cells for 1 ≤ q ≤ n. (J. Peter May 85)
- A triad (X; A, B) is said to be excisive if X is the union of the interiors of A and B. Such triads play a fundamental role in homology and cohomology theory (J. Peter May 87)
- A CW triad (X; A, B) is a CW complex X with subcomplexes A and B such that X = A ∪ B. (J. Peter May 87)
- Define the double mapping cylinder M (i, j) = A ∪ (C × I) ∪ B (J. Peter May 88)
- to be the space obtained from C × I by gluing A to C × {0} along i and gluing B to C × {1} along j. (J. Peter May 88)
- Lemma. For a cofibration i : C −→ A and any map j : C −→ B, the quotient map q : M (i, j) −→ A ∪C B is a homotopy equivalence (J. Peter May 88)
- Because i is a cofibration, the retraction r : M i −→ A is a cofiber homotopy equivalence (J. Peter May 88)
- That is, there is a homotopy inverse map and a pair of homotopies under C. (J. Peter May 88)
- The fundamental obstruction to the calculation of homotopy groups is the failure of excision: for an excisive triad (X; A, B), the inclusion (A, A ∩ B) −→ (X, B) fails to induce an isomorphism of homotopy groups in general (J. Peter May 91)
- However, we do have such an isomorphism in a range of dimensions. This implies the Freudenthal suspension theorem, which gives that πn+q (Σn X) is independent of n if q is small relative to n (J. Peter May 91)
- Theorem (Homotopy excision). Let (X; A, B) be an excisive triad such that C = A ∩ B is non-empty. Assume that (A, C) is (m − 1)-connected and (B, C) is (n − 1)-connected, where m ≥ 2 and n ≥ 1. Then the inclusion (A, C) −→ (X, B) is an (m + n − 2)-equivalence (J. Peter May 91)
- This specializes to give a relationship between the homotopy groups of pairs (X, A) and of quotients X/A and to prove the Freudenthal suspension theorem (J. Peter May 91)
- Theorem. Let f : X −→ Y be an (n−1)-equivalence between (n−2)-connected spaces, where n ≥ 2; thus πn−1 (f ) is an epimorphism. Then the quotient map π : (M f, X) −→ (Cf, ∗) is a (2n − 2)-equivalence. In particular, Cf is (n − 1)connected. If X and Y are (n − 1)-connected, then π : (M f, X) −→ (Cf, ∗) is a (2n − 1)-equivalence (J. Peter May 91)
- Corollary. Let f : X −→ Y be a based map between (n − 1)-connected nondegenerately based spaces, where n ≥ 2. Then Cf is (n − 1)-connected and πn (M f, X) −→ πn (Cf, ∗) is an isomorphism. Moreover, the canonical map η : F f −→ ΩCf induces an isomorphism πn−1 (F f ) −→ πn (Cf ). (J. Peter May 92)
- Theorem. Let i : A −→ X be a cofibration and an (n − 1)-equivalence between (n − 2)-connected spaces, where n ≥ 2. Then the quotient map (X, A) −→ (X/A, ∗) is a (2n − 2)-equivalence, and it is a (2n − 1)-equivalence if A and X are (n − 1)connected. (J. Peter May 92)
- A specialization of the last result gives the Freudenthal suspension theorem. For a based space X, define the suspension homomorphism Σ : πq (X) −→ πq+1 (ΣX) by letting Σf = f ∧ id : S q+1 ∼ = ∼ = S q ∧ S 1 −→ X ∧ S 1 = ΣX. (J. Peter May 93)
- Theorem (Freudenthal suspension). Assume that X is nondegenerately based and (n − 1)-connected, where n ≥ 1. Then Σ is a bijection if q < 2n − 1 and a surjection if q = 2n − 1. (J. Peter May 93)
- Theorem. For all n ≥ 1, πn (S n ) = Z and Σ : πn (S n ) −→ πn+1 (S n+1 ) is an isomorphism. (J. Peter May 93)
- We saw by use of the Hopf bundle S 3 −→ S 2 that π2 (S 2 ) = Z, and the suspension theorem applies to give the conclusion for n ≥ 2. A little extra argument is needed to check that Σ is an isomorphism for n = 1; one can inspect the connecting homomorphism of the Hopf bundle or refer ahead to the observation that the Hurewicz homomorphism commutes with the corresponding suspension isomorphism in homology (J. Peter May 93)
- The dimensional range of the suspension theorem is sharp. We saw before that π3 (S 2 ) = π3 (S 3 ), which is Z. The suspension theorem applies to show that Σ : π3 (S 2 ) −→ π4 (S 3 ) is an epimorphism, and it is known that π4 (S 3 ) = Z2 . (J. Peter May 93)
- Applying suspension repeatedly, we can form a colimit π s qs (X) = colim πq+n (Σ n X). (J. Peter May 93)
- This group is called the qth stable homotopy group of X. For q < n − 1, the maps of the colimit system are isomorphisms and therefore π s qs (X) = πq+n (Σ n X) if q < n − 1. (J. Peter May 94)
- The calculation of the stable homotopy groups of spheres, πqs qs (S 0 ), is one of the deepest and most studied problems in algebraic topology. Important problems of geometric topology, such as the enumeration of the distinct differential structures on S q for q ≥ 5, have been reduced to the determination of these groups (J. Peter May 94)
- 3. Proof of the homotopy excision theorem This is a deep result, and it is remarkable that a direct homotopical proof, in principle an elementary one, is possible. Most standard texts, if they treat this topic at all, give a far more sophisticated proof of a significantly weaker result. (J. Peter May 94)
- We are trying to show that a certain map of pairs induces an isomorphism in a range of dimensions. We capture the relevant map as part of a long exact sequence, and we prove that the third term in the long exact sequence vanishes in the required range. (J. Peter May 94)
- The long exact sequence of the pair in the first form of the definition is · · · −→ πq+1 (X; A, B) −→ πq (A, C) −→ πq (X, B) −→ πq (X; A, B) −→ · · · . (J. Peter May 94)
- Definition. For a triad (X; A, B) with basepoint ∗ ∈ C = A ∩ B, define πq (X; A, B) = πq−1 (P (X; ∗, B), P (A; ∗, C)), (J. Peter May 94)
- A chain complex over R is a sequence of maps of R-modules · · · −→ Xi+1 di+1 −−−→ Xi d i d i −→ Xi−1 −→ · · · such that di ◦ di+1 = 0 for all i (J. Peter May 99)
- An element of the kernel of di is called a cycle and an element of the image of di+1 is called a boundary. We say that two cycles are “homologous” if their difference is a boundary. We write Bi (X) ⊂ Zi (X) ⊂ Xi (J. Peter May 99)
- for the submodules of boundaries and cycles, respectively, and we define the ith homology group Hi (X) to be the quotient module Zi (X)/Bi (X). We write H∗ (X) for the sequence of Rmodules Hi (X). We understand “graded R-modules” to be sequences of R-modules such as this (and we never take the sum of elements in different gradings). (J. Peter May 99)
- A chain homotopy s : f ≃ g between chain maps f, g : X −→ X ′ is a sequence ′ of homomorphisms si : Xi −→ Xi+1 such that d′ ′i ′i+1 ◦ si + si−1 ◦ di = fi − gi (J. Peter May 100)
- Lemma. Chain homotopic maps induce the same homomorphism of homology groups. (J. Peter May 100)
- d = d ⊗ id + id ⊗ d. (J. Peter May 100)
- Lemma. A chain homotopy s : f ≃ g between chain maps f, g : X −→ X ′ determines and is determined by a chain map h : X ⊗I −→ X ′ such that h(x, [0]) = f (x) and h(x, [1]) = g(x). (J. Peter May 100)
- We regard R-modules M as chain complexes concentrated in degree zero, and thus with zero differential. For a chain complex X, there results a chain complex X ⊗ M ; H∗ (X ⊗ M ) is called the homology of X with coefficients in M . (J. Peter May 100)
- Define a chain complex I by letting I0 be the free Abelian group with two generators [0] and [1], letting I1 be the free Abelian group with one generator [I] such that d([I]) = [0] − [1], and letting Ii = 0 for all other i. (J. Peter May 100)
- A sequence M ′ f − →M g g → M ′′ of modules is exact if im f = ker g. If M ′ = 0, this − means that g is a monomorphism; if M ′′ = 0, it means that f is an epimorphism (J. Peter May 101)
- Proposition. A short exact sequence of chain complexes naturally gives rise to a long exact sequence of R-modules · · · −→ Hq (X ′ ) f f∗ −→ Hq (X) g g∗ −→ Hq (X ′′ ) ∂ ∂ − → Hq−1 (X ′ ) −→ · · · . (J. Peter May 101)
- We define the “connecting homomorphism” ∂ : Hq (X ′′ ) −→ Hq−1 (X ′ ) by ∂[x′′ ] = [x′ ], where f (x′ ) = d(x) for some x such that g(x) = x′′ . There is such an x since g is an epimorphism, and there is such an x′ since gd(x) = dg(x) = 0. It is a standard exercise in “diagram chasing” to verify that ∂ is well defined and the sequence is exact (J. Peter May 101)
- Let 0 −→ π f − →ρ g − → σ −→ 0 be an exact sequence of Abelian groups and let C be a chain complex of flat (= torsion free) Abelian groups. Write H∗ (C; π) = H∗ (C ⊗ π). Construct a natural long exact sequence · · · −→ Hq (C; π) f f∗ −→ Hq (C; ρ) g g∗ −→ Hq (C; σ) β − → Hq−1 (C; π) −→ · · · . The connecting homomorphism β is called a Bockstein operation. (J. Peter May 101)
- Unlike homotopy groups, these are stable invariants, the same for a space and its suspension, and it is this that makes them computable. (J. Peter May 103)
- Fix an Abelian group π and consider pairs of spaces (X, A). We shall see that π determines a “homology theory on pairs (X, A).” (J. Peter May 103)
- 1. Axioms for homology (J. Peter May 103)
- Clearly, up to canonical isomorphism, this construction of a homology theory on pairs of spaces is independent of the choice of our CW approximation functor Γ. The reader may have seen singular homology before. As we shall explain later, the classical construction of singular homology amounts to a choice of a particularly nice CW approximation functor, one that is actually functorial on the point-set level, before passage to homotopy categories (J. Peter May 104)
- Let X be a CW complex. We shall define the cellular chain complex C∗ (X). We let Cn (X) be the free Abelian group with one generator [j] for each n-cell j. We must define a differential dn : Cn (X) −→ Cn−1 (X) (J. Peter May 105)
- It will be convenient to work with unreduced cones, cofibers, and suspensions in this section; that is, we do not choose basepoints and so we do not collapse out lines through basepoints (J. Peter May 105)
- We still have the basic result that if i : A −→ X is a cofibration, then collapsing the cone on A to a point gives a homotopy equivalence ψ : Ci −→ X/A. (J. Peter May 105)
- Our first definition of dn involves the calculation of the degrees of maps between spheres (J. Peter May 105)
- However, there are three models of S n that are needed in our discussion: the standard sphere S n ⊂ Dn+1 , the quotient Dn /S n−1 , and the (unreduced) suspension ΣS n−1 . We must fix suitably compatible homeomorphisms relating these “n-spheres.” We define a homeomorphism (J. Peter May 105)
- For a subcomplex A of X, define C∗ (X, A) = C∗ (X)/C∗ (A) ∼ = ∼ = C ̃ C̃∗ (X/A) (J. Peter May 108)
- and define H∗ (X, A) = H∗ (C∗ (X, A)) ∼ = ∼ = H ̃ H̃∗ (X/A). (J. Peter May 109)
- We have dealt so far with the case of integral homology. For more general coefficient groups π, we define C∗ (X, A; π) = C∗ (X, A) ⊗ π (J. Peter May 109)
- A nice fact about cellular homology is that the definition leads directly to an algebraic procedure for the calculation of the homology of Cartesian products (J. Peter May 109)
- Theorem. If X and Y are CW complexes, then X × Y is a CW complex such that C∗ (X × Y ) ∼ = ∼ = C∗ (X) ⊗ C∗ (Y ). (J. Peter May 109)
- We define an isomorphism of graded Abelian groups κ : C∗ (X) ⊗ C∗ (Y ) −→ C∗ (X × Y ) by setting κ([i] ⊗ [j]) = (−1) pq [i × j]. (J. Peter May 110)
- Cellular chains make some computations quite trivial. For example, since S n is a CW complex with one vertex and one n-cell, we see immediately that ̃ H̃n (S n ; π) ∼ = ∼ =π and ̃ H̃q (S n ; π) = 0 for q 6= n. (J. Peter May 111)
- if we look back at the CW decompositions of the torus T , the projective plane RP 2 , and the Klein bottle K and if we let j denote the unique 2-cell in each case, then we find the following descriptions of the cellular chains and integral homologies by quick direct inspections (J. Peter May 111)
- H∗ (T ; Z) = C∗ (T ). (J. Peter May 111)
- H0 (RP 2 ; Z) = Z (J. Peter May 111)
- H1 (RP 2 ; Z) = Z2 (J. Peter May 111)
- Therefore H0 (K; Z) = Z (J. Peter May 111)
- H1 (K; Z) = Z ⊕ Z2 with Z generated by the class (J. Peter May 111)
- While homology groups are far easier to compute than homotopy groups, direct chain level calculation is seldom the method of choice. Rather, one uses chains as a tool for developing more sophisticated algebraic techniques, notably spectral sequences (J. Peter May 112)
- We give an illustration that both shows that chain level calculations are sometimes practicable even when there are many non-zero differentials to determine and indicates why one might not wish to attempt such calculations for really complicated spaces. (J. Peter May 112)
- Lemma. The degree of the antipodal map an : S n −→ S n is (−1)n+1 . (J. Peter May 112)
- We shall use cellular chains to compute the homology of RP n (J. Peter May 112)
- The nth integral homology group of such a manifold M is Z if M is orientable and zero if M is not orientable. The nth mod 2 homology group of M is Z2 whether or not M is orientable. (J. Peter May 113)
- This calculation well illustrates general facts about the homology of compact connected closed n-manifolds M (J. Peter May 113)
- A “generalized homology theory” E∗ is defined to be a system of functors Eq (X, A) and natural transformations ∂ : Eq (X, A) −→ Eq−1 (A) that satisfy all of our axioms except for the dimension axiom. (J. Peter May 115)
- following our proposal that E∗ (X, A) be taken as an alternative notation for H∗ (X, A; π). (J. Peter May 115)
- One of the themes of this chapter is the relationship between homology theories on pairs of spaces and reduced homology theories on based spaces. The latter are more convenient in most advanced work in algebraic topology (J. Peter May 115)
- For a based space X, we define the reduced homology of X to be ̃ Ẽq (X) = Eq (X, ∗). (J. Peter May 115)
- Since the basepoint is a retract of X, there results a direct sum decomposition E∗ (X) ∼ = ∼ = E ̃ Ẽ∗ (X) ⊕ E∗ (∗) (J. Peter May 115)
- the exactness axiom implies that there is a reduced long exact sequence · · · −→ E ̃ Ẽq (A) −→ E ̃ Ẽq (X) −→ Eq (X, A) ∂ ∂ − → E ̃ Ẽq−1 (A) −→ · · · . (J. Peter May 115)
- We can obtain the unreduced homology groups as special cases of the reduced ones. For an unbased space X, we define a based space X+ by adjoining a disjoint basepoint to X. By the additivity axiom, we see immediately that E∗ (X) = E ̃ Ẽ∗ (X+ ). (J. Peter May 115)
- In fact, the unreduced cone on a space Y coincides with the reduced cone on Y+ : the line through the disjoint basepoint is identified to the cone point when constructing the reduced cone on Y+ . Therefore the unreduced cofiber of an unbased map f coincides with the reduced cofiber of the based map f+ . (J. Peter May 116)
- The observant reader will have noticed that the unreduced suspension of X is not the reduced suspension on X+ . Rather, under either interpretation of suspension, Σ(X+ ) is homotopy equivalent to the wedge of Σ(X) and a circle. (J. Peter May 116)
- a special case of the reduced homology of spaces. Theorem. For any cofibration i : A −→ X, the quotient map q : (X, A) −→ (X/A, ∗) induces an isomorphism E∗ (X, A) −→ E∗ (X/A, ∗) = E ̃ Ẽ∗ (X/A). (J. Peter May 116)
- Recall that a basepoint ∗ ∈ X is nondegenerate if the inclusion {∗} −→ X is a cofibration (J. Peter May 117)
- This ensures that the inclusion of the line through the basepoint in the unreduced suspension of X is a cofibration, so that the map from the unreduced suspension to the suspension that collapses out the line through the basepoint is a homotopy equivalence. (J. Peter May 117)
- Theorem. For a nondegenerately based space X, there is a natural isomorphism Σ : E ̃ Ẽq (X) ∼ = ∼ = E ̃ Ẽq+1 (ΣX). (J. Peter May 117)
- Corollary. For any n and q, ̃ Ẽq (S n ) ∼ = ∼ = Eq−n (∗). (J. Peter May 117)
- In the study of generalized homology theories, it is most convenient to restrict attention to reduced homology theories defined on nondegenerately based spaces. (J. Peter May 118)
- Definition. A reduced homology theory ̃ Ẽ∗ consists of functors ̃ Ẽq from the homotopy category of nondegenerately based spaces to the category of Abelian groups that satisfy the following axioms (J. Peter May 118)
- For the additivity axiom, we note that the cofiber of a disjoint union of maps is the wedge of the cofibers of the given maps (J. Peter May 119)
- The first is the long exact sequence of a triple (X, A, B) of spaces B ⊂ A ⊂ X, which is just like its analogue for homotopy groups. Proposition. For a triple (X, A, B), the following sequence is exact: · · · −→ Eq (A, B) i i −→∗ Eq (X, B) j j∗ −→ Eq (X, A) ∂ ∂ − → Eq−1 (A, B) −→ · · · . Here i : (A, B) −→ (X, B) and j : (X, B) −→ (X, A) are inclusions and ∂ is the composite Eq (X, A) ∂ ∂ − → Eq−1 (A) −→ Eq−1 (A, B). (J. Peter May 120)
- Theorem (Mayer-Vietoris sequence). Let (X; A, B) be an excisive triad and set C = A ∩ B. The following sequence is exact: · · · −→ Eq (C) ψ − → Eq (A) ⊕ Eq (B) φ − → Eq (X) ∆ ∆ −→ Eq−1 (C) −→ · · · . (J. Peter May 120)
- Alternatively, one can use CW approximation. For a CW triad, there is a short exact sequence 0 −→ C∗ (C) −→ C∗ (A) ⊕ C∗ (B) −→ C∗ (X) −→ 0 whose associated long exact sequence is the Mayer-Vietoris sequence. (J. Peter May 121)
- We have seen that the compactness of spheres S n and cylinders S n × I implies that, for any choice of basepoint in X0 , the natural map colim π∗ (Xi ) −→ π∗ (X) is an isomorphism. We shall use the additivity and weak equivalence axioms and the Mayer-Vietoris sequence to prove the analogue for homology (J. Peter May 122)
- We record an algebraic description of the colimit of a sequence for use in the proof. Lemma. Let fi : Ai −→ Ai+1 be a sequence of homomorphisms of Abelian groups. Then there is a short exact sequence s a short 0 −→ i Ai ct seq − → i Ai β − → colim Ai −→ 0, where α(ai ) = ai − fi (ai ) for ai ∈ Ai and the restriction of β to Ai is the canonical map given by the definition of a colimit (J. Peter May 123)
- The proof makes use of a useful general construction called the “telescope” of the Xi , denoted tel Xi (J. Peter May 123)
- Inductively, let Y0 = X0 × {0} and suppose that we have constructed Yi ⊃ Xi × {i}. Define Yi+1 to be the double mapping cylinder Yi ∪ Mi+1 obtained by identifying (x, i) ∈ Yi with (x, i) ∈ Mi+1 for x ∈ Xi . Define tel Xi to be the union of the Yi , with the colimit topology (J. Peter May 123)
- Remark. There is a general theory of “homotopy colimits,” which are up to homotopy versions of colimits. The telescope is the homotopy colimit of a sequence. The double mapping cylinder that we used in approximating excisive triads by CW triads is the homotopy pushout of a diagram of the shape • ←− • −→ •. We implicitly used homotopy coequalizers in constructing CW approximations of spaces (J. Peter May 124)
- Definition. For based spaces X, define the Hurewicz homomorphism h : πn (X) −→ H ̃ H̃n (X) by h([f ]) = f∗ (in ). (J. Peter May 125)
- The dimension axiom implicitly fixes a generator i0 of ̃ H̃0 (S 0 ), and we choose generators in of ̃ H̃n (S n ) inductively by setting Σin = in+1 . (J. Peter May 125)
- Lemma. Let X be a wedge of n-spheres. Then h : πn (X) −→ H ̃ H̃n (X) is the Abelianization homomorphism if n = 1 and is an isomorphism if n > 1. (J. Peter May 126)
- Theorem (Hurewicz). Let X be any (n − 1)-connected based space. Then h : πn (X) −→ H ̃ H̃n (X) is the Abelianization homomorphism if n = 1 and is an isomorphism if n > 1. (J. Peter May 126)
- Here the groups on the left are defined in terms of homotopy groups and were used in our construction of cellular chains, while the groups on the right are those of our given homology theory. We use the groups on the right to construct cellular chains in our given theory, and we find that the isomorphism is compatible with differentials (J. Peter May 127)
- Thus let X be a CW complex. For each integer n, define Cn (X) = Hn (X n , X n−1 ) ∼ = ∼ = H ̃ H̃n (X n /X n−1 ). Define d : Cn (X) −→ Cn−1 (X) to be the composite Hn (X n , X n−1 ) ∂ ∂ − → Hn−1 (X n−1 ) −→ Hn−1 (X n−1 , X n−2 ). It is not hard to check that d ◦ d = 0. (J. Peter May 127)
- Theorem. There is a natural isomorphism H∗ (X, A) ∼ = ∼ = H∗ (C∗ (X, A)) under which the natural transformation ∂ agrees with the natural transformation induced by the connecting homomorphisms associated to the short exact sequences 0 −→ C∗ (A) −→ C∗ (X) −→ C∗ (X, A) −→ 0. (J. Peter May 127)
- By the dimension and additivity axioms, we know the homology of wedges of spheres (J. Peter May 128)
- (Hint: construct M (π, n) as the cofiber of a map between wedges of spheres.) The spaces M (π, n) are called Moore spaces. (J. Peter May 129)
- (Hint: start with M (π, n), using the Hurewicz theorem, and kill its higher homotopy groups.) The spaces K(π, n) are called Eilenberg-Mac Lane spaces (J. Peter May 129)
- The standard topological n-simplex is the subspace (J. Peter May 131)
- There are “face maps” (J. Peter May 131)
- degeneracy maps (J. Peter May 131)
- A map f : ∆n −→ X is called a singular n-simplex (J. Peter May 132)
- The singular homology of X is usually defined in terms of this chain complex: H∗ (X; π) = H∗ (C∗ (X) ⊗ π). (J. Peter May 132)
- We define a space ΓX, called the “geometric realization of the total singular complex of X,” (J. Peter May 132)
- We may define the geometric realization |K∗ | of general simplicial sets exactly as we defined the geometric realization ΓX = |S∗ X| of the total singular complex of a topological space. In fact, the total singular complex and geometric realization functors are adjoint (J. Peter May 134)
- One can define a simplicial object in any category C as a sequence of objects Kn of C connected by face and degeneracy maps in C that satisfy the commutation relations that we have displayed. Thus we have simplicial groups, simplicial Abelian groups, simplicial spaces, and so forth. We can think of simplicial sets as discrete simplicial spaces, and we then see that geometric realization generalizes directly to a functor | − | from the category S U of simplicial spaces to the category U of spaces (J. Peter May 135)
- We note one of the principal features of geometric realization (J. Peter May 135)
- The projections induce maps of simplicial spaces from X∗ × Y∗ to X∗ and Y∗ . On passage to geometric realization, these give the coordinates of a map |X∗ × Y∗ | −→ |X∗ | × |Y∗ |. It turns out that this map is always a homeomorphism (J. Peter May 135)
- Now restrict attention to simplicial sets K∗ and L∗ . Then the homeomorphism just specified is a map between CW complexes. However, it is not a cellular map; rather, it takes the n-skeleton of |K∗ × L∗ | to the 2n-skeleton of |K∗ | × |L∗ |. It is homotopic to a cellular map, no longer a homeomorphism, and there results a chain homotopy equivalence C∗ (|K∗ × L∗ |) −→ C∗ (|K∗ |) ⊗ C∗ (|L∗ |) (J. Peter May 135)
- In particular, for spaces X and Y , there is a natural chain homotopy equivalence from the singular chain complex C∗ (X × Y ) to the tensor product C∗ (X) ⊗ C∗ (Y ). (J. Peter May 135)
- The space BG is called the classifying space of G (J. Peter May 136)
- We may view Bn (G) as the orbit space En (G)/G. (J. Peter May 136)
- It is less intuitive, but true, that the space E(G) is contractible. By the long exact homotopy sequence, these facts imply that πq+1 (BG) ∼ = ∼ = πq (G) (J. Peter May 136)
- We illustrate these ideas by defining the “classifying spaces” and “universal bundles” associated to topological groups G (J. Peter May 136)
- Since geometric realization commutes with products (J. Peter May 137)
- This allows us to iterate the construction, setting B 0 (G) = G and B n (G) = B(B n−1 (G)) for n ≥ 1. Specializing to a discrete Abelian group π, we define K(π, n) = B n (π). (J. Peter May 137)
- t is called a “transfer homomorphism.” (J. Peter May 137)
- Indeed, if one focuses on singular chains, then one eschews chain level computations in principle as well as in practice. (J. Peter May 139)
- We here recall some classical results in homological algebra that explain how to calculate H∗ (X; π) from H∗ (X) ≡ H∗ (X; Z) and how to calculate H∗ (X × Y ) from H∗ (X) ⊗ H∗ (Y ). (J. Peter May 139)
- Recall that an R-module M is said to be flat if the functor M ⊗ N is exact (that is, preserves exact sequences in the variable N ). (J. Peter May 139)
- For a principal ideal domain (PID) R, the only torsion product is the first one, denoted Tor R 1 (M, N ) (J. Peter May 139)
- It can be computed by constructing a short exact sequence 0 −→ F1 −→ F0 −→ M −→ 0 and tensoring with N to obtain an exact seqence 0 −→ Tor R 1 (M, N ) −→ F1 ⊗ N −→ F0 ⊗ N −→ M ⊗ N −→ 0, where F1 and F0 are free R-modules. That is, we choose an epimorphism F0 −→ M and note that, since R is a PID, its kernel F1 is also free. (J. Peter May 139)
- Theorem (Universal coefficient). Let R be a P ID and let X be a flat chain complex over R. Then, for each n, there is a natural short exact sequence 0 −→ Hn (X) ⊗ M α α − → Hn (X ⊗ M ) β → Tor − R 1 (Hn−1 (X), M ) −→ 0. The sequence splits, so that Hn (X ⊗ M ) ∼ = ∼ = (Hn (X) ⊗ M ) ⊕ Tor R 1 (Hn−1 (X), M ), but the splitting is not natural. (J. Peter May 140)
- In Chapter 20 §3, we shall see an important class of examples in which the splitting is very far from being natural (J. Peter May 140)
- Corollary. If R is a field, then α : H∗ (X) ⊗ M −→ H∗ (X; M ) is a natural isomorphism. (J. Peter May 140)
- 2. The Künneth theorem The universal coefficient theorem in homology is a special case of the Künneth theorem. (J. Peter May 140)
- Theorem (Künneth). Let R be a P ID and let X be a flat chain complex and Y be any chain complex. Then, for each n, there is a natural short exact sequence Y be any c 0 −→ p+q=n Hp (X)⊗Hq (Y ) α α − → Hn (X⊗Y ) β , there is β −→ p+q=n−1 Tor R 1 (Hp (X), Hq (Y )) −→ 0. The sequence splits, so that Hn (X ⊗ Y ) ∼ = ∼ =( o t p+q=n Hp (X) ⊗ Hq (Y )) ⊕ ( p+q=n−1 Tor R 1 (Hp (X), Hq (Y ))), but the splitting is not natural. (J. Peter May 140)
- Corollary. If R is a field, then α : H∗ (X) ⊗ H∗ (Y ) −→ H∗ (X ⊗ Y ) is a natural isomorphism. (J. Peter May 140)
- Since all modules over a field are free and thus flat (J. Peter May 141)
- . We assume that the reader has seen Ext modules, which measure the failure of Hom to be an exact functor. For a PID R, the only Ext module is the first one, denoted Ext1 R R (M, N ). It can be computed by constructing a short exact sequence 0 −→ F1 −→ F0 −→ M −→ 0 and applying Hom to obtain an exact seqence 0 −→ Hom(M, N ) −→ Hom(F0 , N ) −→ Hom(F1 , N ) −→ Ext1 R R (M, N ) −→ 0, where F1 and F0 are free R-modules (J. Peter May 142)
- Theorem (Universal coefficient). Let R be a P ID and let X be a free chain complex over R. Then, for each n, there is a natural short exact sequence 0 −→ Ext1 R R (Hn−1 (X), M ) β − → H n (X; M ) α α − → Hom(Hn (X), M ) −→ 0. The sequence splits, so that H n (X; M ) ∼ = ∼ = Hom(Hn (X), M ) ⊕ Ext1 R R (Hn−1 (X), M ), but the splitting is not natural. (J. Peter May 142)
- Corollary. If R is a field, then α : H ∗ (X; M ) −→ Hom(H∗ (X), M ) is a natural isomorphism. (J. Peter May 142)
- Second, when R is a PID, a short exact sequence 0 −→ L ′ −→ L −→ L ′′ −→ 0 of R-modules gives rise to a six-term exact sequence (J. Peter May 143)
- For Rmodules L, M , and N , we have an adjunction Hom(L ⊗ M, N ) ∼ = ∼ = Hom(L, Hom(M, N )). (J. Peter May 144)
- We also have a natural homomorphism Hom(L, M ) ⊗ N −→ Hom(L, M ⊗ N ), and this is an isomorphism if either L or N is a finitely generated projective Rmodule (J. Peter May 144)
- Again, we have a natural map Hom(L, M ) ⊗ Hom(L ′ , M ′ ) −→ Hom(L ⊗ L ′ , M ⊗ M ′ ), which is an isomorphism if L and L′ are finitely generated and projective or if L is finitely generated and projective and M = R. (J. Peter May 144)
- For its flatness hypothesis, it is useful to remember that, for any Noetherian ring R, the dual Hom(F, R) of a free R-module is a flat R-module (J. Peter May 144)
- The essential new feature is the cup product structure that makes the cohomology of X with coefficients in a commutative ring R a commutative graded R-algebra. (J. Peter May 145)
- For general spaces X, we can use ΓX = |S∗ X| as a canonical CW approximation functor (J. Peter May 146)
- By our observations about cochain complexes (J. Peter May 147)
- This product makes H ∗ (X; R) into a graded unital, associative, and “commutative” R-algebra. Here commutativity is understood in the appropriate graded sense (J. Peter May 147)
- In both diagrams, τ (x ⊗ y) = (−1) pq y ⊗ x if deg x = p and deg y = q. The reason is that, on the topological level, t permutes p-cells past q-cells and, on the level of cellular chains, this involves the transposition S p+q = S p ∧ S q −→ S q ∧ S p = S p+q . We leave it as an exercise that this map has degree (−1)pq . It is this fact that forces the cup product to be commutative in the graded sense (J. Peter May 148)
- In principle, the way to compute cup products is to pass to cellular chains from a cellular approximation to the diagonal map ∆. The point is that ∆ fails to be cellular since it carries the n-skeleton of X to the 2n-skeleton of X × X. In practice, this does not work very well and more indirect means of computation must be used. (J. Peter May 148)
- 4. An example: RP n and the Borsuk-Ulam theorem (J. Peter May 148)
- We shall later use Poincaré duality to give a quick proof that the cohomology algebra H ∗ (RP n ; Z2 ) is a truncated polynomial algebra Z2 [x]/(xn+1 ), where deg x = 1. (J. Peter May 148)
- We use this fact together with covering space theory to prove a celebrated result known as the Borsuk-Ulam theorem. A map g : S m −→ S n is said to be antipodal if it takes pairs of antipodal points to pairs of antipodal points. It then induces a map f : RP m −→ RP n such that the following diagram commutes: (J. Peter May 149)
- Theorem. If m > n ≥ 1, then there exist no antipodal maps S m −→ S n . (J. Peter May 149)
- Theorem (Borsuk-Ulam). For any continuous map f : S n −→ Rn , there exists x ∈ S n such that f (x) = f (−x). (J. Peter May 149)
- Proof. Suppose for a contradiction that f (x) 6= f (−x) for all x. We could then define a continuous antipodal map g : S n −→ S n−1 by letting g(x) be the point at which the vector from 0 through f (x) − f (−x) intersects S n−1 . (J. Peter May 149)
- We give an outline of one of the most striking features of cohomology: the cohomology groups of a space X with coefficients in the homotopy groups of a space Y control the construction of homotopy classes of maps X −→ Y . As a matter of motivation, this helps explain why one is interested in general coefficient groups. (J. Peter May 150)
- Definition. Fix n ≥ 1. A connected space X is said to be n-simple if π1 (X) is Abelian and acts trivially on the homotopy groups πq (X) for q ≤ n; X is said to be simple if it is n-simple for all n. (J. Peter May 150)
- We ask when f can be extended to a map X n+1 −→ Y that restricts to the given map on A. (J. Peter May 150)
- Let f : X n −→ Y be a map (J. Peter May 150)
- If we compose the attaching maps S n → X of cells of X A with f , we obtain elements of πn (Y ). These elements specify a well defined “obstruction cocycle” cf ∈ C n+1 (X, A; πn (Y )). (J. Peter May 150)
- Clearly, by considering extensions cell by cell, f extends to X n+1 if and only if cf = 0 (J. Peter May 150)
- considering extensions cell by cell, f extends to X n+1 if and only if cf = 0. This is not a computable criterion. However, if we allow ourselves to modify f a little, then we can refine the criterion to a cohomological one that often is computable. (J. Peter May 150)
- Theorem. For f : X n −→ Y , the restriction of f to X n−1 extends to a map n+1 X → Y if and only if [cf ] = 0 in H n+1 (X, A; πn (Y )). (J. Peter May 150)
- Theorem. Given maps f, f ′ : X n → Y and a homotopy rel A of their restrictions to X n−1 , there is an obstruction class in H n (X, A; πn (Y )) that vanishes if and only if the restriction of the given homotopy to X n−2 extends to a homotopy f ≃ f ′ rel A. (J. Peter May 150)
- Theorem (Mayer-Vietoris sequence). Let (X; A, B) be an excisive triad and set C = A ∩ B. The following sequence is exact: · · · −→ E q−1 (C) ∆∗ ∆∗ −−→ E q (X) φ∗ −→ E q (A) ⊕ E q (B) ψ∗ −−→ E q (C) −→ · · · . Here, if i : C −→ A, j : C −→ B, k : A −→ X, and ℓ : B −→ X are the inclusions, then φ ∗ (χ) = (k ∗ (χ), ℓ ∗ (χ)) and ψ ∗ (α, β) = i ∗ (α) − j ∗ (β) and ∆∗ is the composite E q−1 (C) δ → δ → E − q (A, C) ∼ = ∼ = E q (X, B) −→ E q (X). (J. Peter May 155)
- Theorem (Relative Mayer-Vietoris sequence). The following sequence is exact: · · · −→ E q−1 (Y, C) ∆∗ ∆∗ −−→ E q (Y, X) φ∗ −→ E q (Y, A) ⊕ E q (Y, B) ψ∗ −−→ E q (Y, C) −→ · · · . Here, if i : (Y, C) −→ (Y, A), j : (Y, C) −→ (Y, B), k : (Y, A) −→ (Y, X), and ℓ : (Y, B) −→ (Y, X) are the inclusions, then φ ∗ (χ) = (k ∗ (χ), ℓ ∗ (χ)) and ψ ∗ (α, β) = i ∗ (α) − j ∗ (β) and ∆∗ is the composite E q−1 (Y, C) −→ E q−1 (A, C) ∼ = ∼ = E q−1 (X, B) δ δ − → E q (Y, X). (J. Peter May 155)
- We shall use the additivity and weak equivalence axioms and the MayerVietoris sequence to explain how to compute E ∗ (X). The answer is more subtle than in homology because, algebraically, limits are less well behaved than colimits: they are not exact functors from diagrams of Abelian groups to Abelian groups. (J. Peter May 156)
- That is, we may as well define lim 1 Ai to be the displayed cokernel. We then have the following result. Theorem. For each q, there is a natural short exact sequence 0 −→ lim 1 E q−1 (Xi ) −→ E q (X) π π − → lim E q (Xi ) −→ 0, where π is induced by the inclusions Xi −→ X. (J. Peter May 156)
- Rather than go into the general theory, we simply display how the “first right derived functor” lim 1 of an inverse sequence of Abelian groups can be computed (J. Peter May 156)
- We say that an inverse sequence fi : Ai+1 −→ Ai satisfies the Mittag-Leffler condition if, for each fixed i, there exists j ≥ i such that, for every k > j, the image of the composite Ak −→ Ai is equal to the image of the composite Aj −→ Ai . For example, this holds if all but finitely many of the fi are epimorphisms or if the Ai are all finite. As a matter of algebra, we have the following vanishing result. Lemma. If the inverse sequence fi : Ai+1 −→ Ai satisfies the Mittag-Leffler condition, then lim 1 Ai = 0. (J. Peter May 157)
- If we assume given a theory that satisfies the axioms, we see that the cochains with coefficients in π of a CW complex X can be redefined by C n (X; π) = H n (X n , X n−1 ; π), with differential d : C n (X; π) −→ C n+1 (X; π) the composite H n (X n , X n−1 ; π) −→ H n (X n ) δ δ − → H n+1 (X n+1 , X n ). That is, the following result holds. Theorem. C ∗ (X; π) as just defined is isomorphic to Hom(C∗ (X), π). (J. Peter May 157)
- Cup products are “unstable,” in the sense that they vanish on suspensions. This is an indication of how much more information they carry than the mere additive groups. The proof given by this sequence of exercises actually applies to any “multiplicative” cohomology theory, that is, any theory that has suitable cup products (J. Peter May 158)
- Commentary: Additively, cohomology groups are “stable,” in the sense that ̃ H̃ p (Y ) ∼ = ∼ = H ̃ H̃ p+1 (ΣY ). (J. Peter May 158)
- It is apparent that there is a kind of duality relating the construction of homology and cohomology. In its simplest form, this is reflected by the fact that evaluation of cochains on chains gives a natural homomorphism C p (X; π) ⊗ Cp (X; ρ) −→ π ⊗ ρ. This passes to homology and cohomology to give an evaluation pairing H p (X; π) ⊗ Hp (X; ρ) −→ π ⊗ ρ. (J. Peter May 159)
- When R is a field and the Hp (X; R) are finite dimensional vector spaces, the adjoint of this pairing is an isomorphism H p (X; R) ∼ = ∼ = HomR (Hp (X; R), R). (J. Peter May 159)
- That is, the cohomology groups of X are the vector space duals of the homology groups of X. (J. Peter May 159)
- We shall study manifolds without boundary in this chapter, turning to manifolds with boundary in the next. We do not assume that M is differentiable. It is known that M can be given the structure of a finite CW complex, and its homology and cohomology groups are therefore finitely generated. When M is differentiable, it is not hard to prove this using Morse theory, but it is a deep theorem in the general topological case (J. Peter May 159)
- If R is a field and M is “R-orientable,” then there is an “R-fundamental class” z ∈ Hn (M ; R). The composite of the cup product and evaluation on z gives a cup product pairing H p (M ; R) ⊗ H n−p (M ; R) −→ R. (J. Peter May 159)
- One version of the Poincaré duality theorem asserts that this pairing is nonsingular, so that its adjoint is an isomorphism H p (M ; R) = ∼ = HomR (H ∼ n−p (M ; R), R) ∼ = ∼ = Hn−p (M ; R). (J. Peter May 159)
- Theorem (Poincaré duality). Let M be a compact R-oriented n-manifold. Then, for an R-module π, there is an isomorphism D : H p (M ; π) −→ Hn−p (M ; π). (J. Peter May 160)
- Proposition. If M is a compact n-manifold, then an R-orientation of M determines and is determined by an R-fundamental class z ∈ Hn (M ; R). (J. Peter May 160)
- Corollary. Let Tp ⊂ H p (M ) be the torsion subgroup. The cup product pairing α ⊗ β −→ hαβ, zi induces a nonsingular pairing H p (M )/Tp ⊗ H n−p (M )/Tn−p −→ Z. (J. Peter May 160)
- Corollary. As a graded ring, H ∗ (CP n ) is the truncated polynomial algebra Z[α]/(αn+1 ), where deg α = 2. That is, H 2q (CP n ) is the free Abelian group with generator αq for 1 ≤ q ≤ n. Proof (J. Peter May 161)
- We shall see that an oriented manifold is R-oriented for any commutative ring R (J. Peter May 161)
- a field. Then α ⊗ β −→ hα ∪ β, zi defines a nonsingular pairing H p (M ; R) ⊗R H n−p (M ; R) −→ R. We shall see that every manifold is Z2 -oriented, and an argument exactly like that for CP n allows us to compute the cup products in H ∗ (RP n ; Z2 ). We used this information in our proof of the Borsuk-Ulam theorem (J. Peter May 161)
- Corollary. Let M be a connected compact R-oriented n-manifold, where R is (J. Peter May 161)
- Corollary. As a graded ring, H ∗ (RP n ; Z2 ) is the truncated polynomial algebra Z2 [α]/(α n+1 ), where deg α = 1. That is, α q is the non-zero element of H q (RP n ; Z2 ) for 1 ≤ q ≤ n. (J. Peter May 161)
- Using the evident natural map from the tensor product of homologies to the homology of a tensor product, we see that ∩ passes to homology to induce a pairing ∩ : H ∗ (X; π) ⊗R H∗ (X; R) −→ H∗ (X; π). (J. Peter May 162)
- Inspecting definitions, we see that, on elements, these observations prove the fundamental identity hα ∪ β, xi = hβ, α ∩ xi. (J. Peter May 163)
- For use in the proof of the Poincaré duality theorem, we observe that the cap product generalizes to relative cap products (J. Peter May 163)
- By excision, exactness, and homotopy invariance, we have isomorphisms Hi (M, M − x) = ∼ = Hi (U, U − x) ∼ = ∼ = H ∼ ̃ H̃i−1 (U − x) ∼ = ∼ = H ̃ H̃i−1 (S n−1 ). (J. Peter May 163)
- We think of Hn (M, M − x) as a free R-module on one generator, but the generator (which corresponds to a unit of the ring R) is unspecified. Intuitively, an R-orientation of M is a consistent choice of generators. Definition. An R-fundamental class of M at a subspace X is an element z ∈ Hn (M, M − X) such that, for each x ∈ X, the image of z under the map Hn (M, M − X) −→ Hn (M, M − x) induced by the inclusion (M, M − X) −→ (M, M − x) is a generator. If X = M , we refer to z ∈ Hn (M ) as a fundamental class of M . An R-orientation of M is an open cover {Ui } and R-fundamental classes zi of M at Ui such that if Ui ∩ Uj is non-empty, then zi and zj map to the same element of Hn (M, M − Ui ∩ Uj ). (J. Peter May 164)
- We say that M is R-orientable if it admits an R-orientation (J. Peter May 164)
- Theorem (Vanishing). Let M be an n-manifold. For any coefficient group π, Hi (M ; π) = 0 if i > n, and ̃ H̃n (M ; π) = 0 if M is connected and is not compact. (J. Peter May 164)
- We can use this together with Mayer-Vietoris sequences to construct R-fundamental classes at compact subspaces from R-orientations (J. Peter May 164)
- Theorem. Let K be a compact subset of M . Then, for any coefficient group π, Hi (M, M − K; π) = 0 if i > n, and an R-orientation of M determines an Rfundamental class of M at K. In particular, if M is compact, then an R-orientation of M determines an R-fundamental class of M (J. Peter May 164)
- Corollary. Let M be a connected compact n-manifold, n > 0. Then either M is not orientable and Hn (M ; Z) = 0 or M is orientable and the map Hn (M ; Z) −→ Hn (M, M − x; Z) ∼ = ∼ =Z is an isomorphism for every x ∈ M . (J. Peter May 165)
- As an aside, the corollary leads to a striking example of the failure of the naturality of the splitting in the universal coefficient theorem (J. Peter May 165)
- Thus the left and right vertical arrows are zero. If the splittings of the rows were natural, this would imply that the middle vertical arrow is also zero (J. Peter May 165)
- We begin with the general observation that homology is “compactly supported” in the sense of the following result. Lemma. For any space X and element x ∈ Hq (X), there is a compact subspace K of X and an element k ∈ Hq (K) that maps to x. (J. Peter May 166)
- the much more subtle statement that Hn (M ) = 0 if M is connected and is not compact. (J. Peter May 166)
- Remember that homology is a covariant functor with compact supports. Cohomology is a contravariant functor, and it does not have compact supports (J. Peter May 168)
- We would like to prove the Poincaré duality theorem by inductive comparisons of Mayer-Vietoris sequences, and the opposite variance of homology and cohomology makes it unclear how to proceed. To get around this, we introduce a variant of cohomology that does have compact supports and has enough covariant functoriality to allow us to proceed by comparisons of Mayer-Vietoris sequences (J. Peter May 168)
- Consider the set K of compact subspaces K of M . This set is directed under inclusion; to conform with our earlier discussion of colimits, we may view K as a category whose objects are the compact subspaces K and whose maps are the inclusions between them. We define H q cq (M ) = colim H q (M, M − K), where the colimit is taken with respect to the homomorphisms H q (M, M − K) −→ H q (M, M − L) (J. Peter May 168)
- Intuitively, thinking in terms of singular cohomology, its elements are represented by cocycles that vanish off some compact subspace (J. Peter May 168)
- A map f : M −→ N is said to be proper if f −1 (L) is compact in M when L is compact in N . This holds, for example, if f is the inclusion of a closed subspace (J. Peter May 169)
- Theorem (Poincaré duality). Let M be an R-oriented n-manifold. Then D : Hcp cp (M ) −→ Hn−p (M ) is an isomorphism. (J. Peter May 169)
- We shall prove a generalization to not necessarily compact manifolds. (J. Peter May 169)
- 6. The orientation cover There is an orientation cover of a manifold that helps illuminate the notion of orientability (J. Peter May 171)
- Proposition. Let M be a connected n-manifold. Then there is a 2-fold cover p : ̃ M̃ −→ M such that ̃ M̃ is connected if and only if M is not orientable (J. Peter May 171)
- Corollary. If M is simply connected, or if π1 (M ) contains no subgroup of index 2, then M is orientable. If M is orientable, then M admits exactly two orientations (J. Peter May 172)
- We can use homology with coefficients in a commutative ring R to construct an analogous R-orientation cover. It depends on the units of R. For example, if R = Z2 , then the R-orientation cover is the identity map of M since there is a unique unit in R. This reproves the obvious fact that any manifold is Z2 oriented. The evident ring homomorphism Z −→ R induces a natural homomorphism H∗ (X; Z) −→ H∗ (X; R), and we see immediately that an orientation of M induces an R-orientation of M for any R. (J. Peter May 172)
- The Poincaré duality theorem imposes strong constraints on the Euler characteristic of a manifold. It also leads to new invariants, most notably the index. (J. Peter May 173)
- Moreover, there is a relative version of Poincaré duality in the context of manifolds with boundary, and this leads to necessary algebraic conditions on the cohomology of a manifold that must be satisfied if it is to be a boundary. (J. Peter May 173)
- We shall later outline the theory of cobordism, which leads to necessary and sufficient algebraic conditions for a manifold to be a boundary. (J. Peter May 173)
- The Euler characteristic χ(X) of a space with finitely generated homology is defined by istic χ(X) χ(X) = i (−1)i rank Hi (X; Z). P The universal coefficient theorem implies that P heorem im χ(X) = i (−1)i dim Hi (X; F ) for any field of coefficients F (J. Peter May 173)
- We may take F = Z2 , and so dispense with the requirement that M be oriented. If n is odd, the summands of χ(M ) cancel in pairs, and we obtain the following conclusion. Proposition. If M is a compact manifold of odd dimension, then χ(M ) = 0. (J. Peter May 173)
- This pairing is nonsingular. Since α ∪ β = (−1)m β ∪ α, it is skew symmetric if m is odd and is symmetric if m is even. When m is odd, we obtain the following conclusion. Proposition. If M is a compact oriented n-manifold, where n ≡ 2 mod 4, then χ(M ) is even. (J. Peter May 174)
- Lemma. Let F be a field of characteristic 6= 2, V be a finite dimensional vector space over F , and φ : V × V −→ F be a nonsingular skew symmetric bilinear form. Then V has a basis {x1 , . . ., xr , y1 , . . ., yr } such that φ(xi , yi ) = 1 for 1 ≤ i ≤ r and φ(z, w) = 0 for all other pairs of basis elements (z, w). Therefore the dimension of V is even (J. Peter May 174)
- Lemma. Let V be a finite dimensional real vector space and φ : V × V −→ R be a nonsingular symmetric bilinear form. Define q(x) = φ(x, x). Then V has a basis {x1 , . . ., xr , y1 , . . ., ys } such that φ(z, w) = 0 for all pairs (z, w) of distinct basis elements, q(xi ) = 1 for 1 ≤ i ≤ r and q(yj ) = −1 for 1 ≤ j ≤ s. The number r − s is an invariant of φ, called the signature of (J. Peter May 174)
- Definition. Let M be a compact oriented n-manifold. If n = 4k, define the index of M , denoted I(M ), to be the signature of the cup product form H 2k (M ; R)⊗ H 2k (M ; R) −→ R. If n 6≡ 0 mod 4, define I(M ) = 0. (J. Peter May 174)
- Proposition. For any compact oriented n-manifold, χ(M ) ≡ I(M ) mod 2. (J. Peter May 174)
- The Euler characteristic and index are related by the following congruence. (J. Peter May 174)
- Lemma. If M and M ′ are compact oriented n-manifolds, then I(M ∐ M ′ ) = I(M ) + I(M ′ ), where M ∐ M ′ is given the evident orientation induced from those of M and M ′ . (J. Peter May 175)
- Lemma. Let M be a compact oriented m-manifold and N be a compact oriented n-manifold. Then I(M × N ) = I(M ) · I(N ), where M × N is given the orientation induced from those of M and N . (J. Peter May 175)
- Let Hn = {(x1 , . . ., xn )|xn ≥ 0} be the upper half-plane in Rn . Recall that an n-manifold with boundary is a Hausdorff space M having a countable basis of open sets such that every point of M has a neighborhood homeomorphic to an open subset of H n . A point x is an interior point if it has a neighborhood homeomorphic to an open subset of Hn − ∂Hn ∼ = Rn ; otherwise it is a boundary point. It is a fact called “invariance of domain” that if U and V are homeomorphic subspaces of Rn and U is open, then V is open. Therefore, a homeomorphism of an open subspace of Hn onto an open subspace of Hn carries boundary points to boundary points. (J. Peter May 176)
- We denote the boundary of an n-manifold M by ∂M . Thus M is a manifold without boundary if ∂M is empty; M is said to be closed if, in addition, it is compact. The space ∂M is an (n − 1)-manifold without boundary. (J. Peter May 176)
- It is a fundamental question in topology to determine which closed manifolds are boundaries (J. Peter May 176)
- we can ask whether or not a smooth (= differentiable) closed manifold is the boundary of a smooth manifold (with the induced smooth structure) (J. Peter May 176)
- Remember that χ(M ) = 0 if M is a closed manifold of odd dimension. (J. Peter May 176)
- Proposition. If M = ∂W , where W is a compact (2m + 1)-manifold, then χ(M ) = 2χ(W ). (J. Peter May 176)
- Corollary. If M = ∂W for a compact manifold W , then χ(M ) is even. (J. Peter May 176)
- For example, since χ(RP 2m ) = 1 and χ(CP n ) = n+ 1, this criterion shows that 2m RP and CP 2m cannot be boundaries. Notice that we have proved that these are not boundaries of topological manifolds, let alone of smooth ones. (J. Peter May 177)
- The index gives a more striking criterion: if a closed oriented 4k-manifold M is the boundary of a (topological) manifold, then I(M ) = 0. (J. Peter May 177)
- In the case of smooth manifolds, it can be seen in terms of inward-pointing unit vectors of the normal line bundle of the embedding ∂M −→ M . (J. Peter May 177)
- We let M be an n-manifold with boundary, n > 0, throughout this section, and we let R be a given commutative ring. We say that M is R-orientable (or orientable if R = Z) if its interior ̊ M̊ = M − ∂M is R-orientable; similarly, an R-orientation of M is an R-orientation of its interior (J. Peter May 177)
- Theorem (Topological collaring). There is an open neighborhood V of ∂M in M such that the identification ∂M = ∂M × {0} extends to a homeomorphism V ∼ = ∼ = ∂M × [0, 1). (J. Peter May 177)
- It follows that the inclusion M̊ M̊ −→ M is a homotopy equivalence and the inclusion ∂M −→ M is a cofibration (J. Peter May 177)
- Proposition. An R-orientation of M determines an R-orientation of ∂M . (J. Peter May 177)
- Proposition. If M is compact and R-oriented and z∂M ∈ Hn−1 (∂M ) is the fundamental class determined by the induced R-orientation on ∂M , then there is a unique element z ∈ Hn (M, ∂M ) such that ∂z = z∂M ; z is called the R-fundamental class determined by the R-orientation of M . (J. Peter May 178)
- Theorem (Relative Poincaré duality). Let M be a compact R-oriented nmanifold with R-fundamental class z ∈ Hn (M, ∂M ; R). Then, with coefficients taken in any R-module π, capping with z specifies duality isomorphisms (J. Peter May 178)
- Theorem. If M is the boundary of a compact oriented (4k + 1)-manifold, then I(M ) = 0. (J. Peter May 179)
- We first give an algebraic criterion for the vanishing of the signature of a form and then show that the cup product form on the middle dimensional cohomology of M satisfies the criterion. Lemma. Let W be a n-dimensional subspace of a 2n-dimensional real vector space V . Let φ : V × V −→ R be a nonsingular symmetric bilinear form such that φ : W × W −→ R is identically zero. Then the signature of φ is zero. (J. Peter May 179)
- We here give a homotopical way of constructing ordinary theories that makes no use of chains, whether cellular or singular. We also show how to construct cup and cap products homotopically. This representation of homology and cohomology in terms of Eilenberg-Mac Lane spaces is the starting point of the modern approach to homology and cohomology theory, and we shall indicate how theories that do not satisfy the dimension axiom can be represented (J. Peter May 183)
- indicate how theories that do not satisfy the dimension axiom can be represented. We shall also describe Postnikov systems, which give a way to approximate general (simple) spaces by weakly equivalent spaces built up out of Eilenberg-Mac Lane spaces. This is conceptually dual to the way that CW complexes allow the approximation of spaces by weakly equivalent spaces built up out of spheres (J. Peter May 183)
- Finally, we present the important notion of cohomology operations and relate them to the cohomology of Eilenberg-Mac Lane spaces. (J. Peter May 183)
- Recall that a reduced homology theory on based CW complexes is a sequence of functors E ̃ Ẽq from the homotopy category of based CW complexes to the category of Abelian groups. (J. Peter May 183)
- By a result of Milnor, if X has the homotopy type of a CW complex, then so does ΩX. By the Whitehead theorem, we therefore have a homotopy equivalence σ̃ : K(π, n) −→ ΩK(π, n + 1). (J. Peter May 183)
- This map is the adjoint of a map σ : ΣK(π, n) −→ K(π, n + 1). (J. Peter May 183)
- Theorem. For CW complexes X, Abelian groups π and integers n ≥ 0, there are natural isomorphisms ̃ H̃q (X; π) ∼ = ∼ colimn πq+n (X ∧ K(π, n)). = (J. Peter May 184)
- Definition. A prespectrum is a sequence of based spaces Tn , n ≥ 0, and based maps σ : ΣTn −→ Tn+1 . (J. Peter May 184)
- If X = S 0 , then X ∧ K(π, n) = K(π, n). (J. Peter May 184)
- The example at hand is the Eilenberg-Mac Lane prespectrum {K(π, n)}. Another example is the “suspension prespectrum” {Σn X} of a based space X; the required maps Σ(Σn X) −→ Σn+1 X are the evident identifications. When X = S 0 , this is called the sphere prespectrum. (J. Peter May 184)
- Theorem. Let {Tn } be a prespectrum such that Tn is (n − 1)-connected and of the homotopy type of a CW complex for each n. Define ̃ Ẽq (X) = colimn πq+n (X ∧ Tn ), where the colimit is taken over the maps πq+n (X ∧ Tn ) Σ Σ → πq+n+1 (Σ(X ∧ Tn )) − ∼ = ∼ = πq+n+1 (X ∧ ΣTn ) id ∧σ id ∧σ −−−→ πq+n+1 (X ∧ Tn+1 ). Then the functors ̃ Ẽq define a reduced homology theory on based CW complexes. (J. Peter May 184)
- We need some preliminaries to prove the ad Definition. Define the weak product i i Yi of a set of based spaces Yi to Definition. Def be the subspace of i Qw Yi consisting of those points all but finitely many of whose Q coordinates are basepoints (J. Peter May 184)
- Since passage to colimits preserves exact sequences (J. Peter May 184)
- Lemma. For a set of based spaces {Yi }, the canonical map i πq (Yi ) −→ πq ( i Yi ) i spaces {Yi }, the c πq (Yi ) −→ πq ( i i Yi ) is an isomorphism. (J. Peter May 185)
- Example. Applying the theorem to the sphere prespectrum, we find that the stable homotopy groups πqs qs (X) give the values of a reduced homology theory; it is called “stable homotopy theory (J. Peter May 185)
- Theorem. For CW complexes X, Abelian groups π, and integers n ≥ 0, there are natural isomorphisms ̃ H̃ n (X; π) ∼ = ∼ = [X, K(π, n)]. (J. Peter May 185)
- If Z has a multiplication φ : Z × Z −→ Z such that the basepoint ∗ of Z is a two-sided unit up to homotopy, so that Z is an “H-space,” then φ induces an “addition” [X, Z] × [X, Z] −→ [X, Z]. (J. Peter May 186)
- . We say that Z is “grouplike” if there is a map χ : Z −→ Z such that φ(id × χ)∆ : Z −→ Z is homotopic to the trivial map, and then χ∗ : [X, Z] −→ [X, Z] sends an element x ∈ [X, Z] to x−1 . (J. Peter May 186)
- . If Z is a grouplike homotopy associative and commutative H-space, then the functor [X, Z] takes values in Abelian groups. (J. Peter May 186)
- Actually, the existence of inverses can be deduced if Z is only “grouplike” in the weaker sense that π0 (X) is a group, but we shall not need the extra generality. Now consider the multiplication on a loop space ΩY given by composition of loops. Our proof that π1 (Y ) is a group and π2 (Y ) is an Abelian group amounts to a proof of the following result. Lemma. For any based space Y , ΩY is a grouplike homotopy associative Hspace and Ω 2 Y is a grouplike homotopy associative and commutative H-space. (J. Peter May 186)
- Definition. An Ω-prespectrum is a sequence of based spaces Tn and weak homotopy equivalences σ̃ : Tn −→ ΩTn+1 . (J. Peter May 186)
- It is a consequence of a general result called the Brown representability theorem that every reduced cohomology theory is represented in this fashion by an Ω-prespectrum. (J. Peter May 187)
- We can also construct cap products homotopically. To do so, it is convenient to bring function spaces into play, using the obvious isomorphisms [X, Y ] ∼ = ∼ = π0 F (X, Y ) and evaluation maps ε : F (X, Y ) ∧ X −→ Y. (J. Peter May 188)
- We have implicitly studied the represented functors k(X) = [X, Y ] by decomposing X into cells. This led in particular to the calculation of ordinary represented cohomology [X, K(π, n)] by means of cellular chains (J. Peter May 190)
- There is an Eckmann-Hilton dual way of studying [X, Y ] by decomposing Y into “cocells.” We briefly describe this decomposition of spaces into their “Postnikov systems” here (J. Peter May 190)
- This decomposition answers a natural question: how close are the homotopy groups of a CW This deco groups of a C Q type? Since n complex X to being a complete set of invariants for its homotopy K(πn (X), n) has the same homotopy groups as X but is generally not weakly homotopy equivalent to it, some added information is needed. If X is simple, it turns out that the homotopy groups together with an inductively defined sequence of cohomology classes give a complete set of invariants. (J. Peter May 190)
- Recall that a connected space X is said to be simple if π1 (X) is Abelian and acts trivially on πn (X) for n ≥ 2. A Postnikov system for a simple based space X consists of based spaces Xn together with based maps αn : X −→ Xn and pn+1 : Xn+1 −→ Xn , (J. Peter May 190)
- n ≥ 1, such that pn+1 ◦αn+1 = αn , X1 is an Eilenberg-Mac Lane space K(π1 (X), 1), pn+1 is the fibration induced from the path space fibration over an EilenbergMac Lane space K(πn+1 (X), n + 2) by a map k n+2 : Xn −→ K(πn+1 (X), n + 2), (J. Peter May 191)
- and αn induces an isomorphism πq (X) → πq (Xn ) for q ≤ n. It follows that πq (Xn ) = 0 for q > n. (J. Peter May 191)
- The system can be displayed diagrammatically as follows (J. Peter May 191)
- The maps αn induce a weak equivalence X → lim Xn , but the inverse limit generally will not have the homotopy type of a CW complex. The a CW complex. The maps αn induce a weak equivalence X → lim Xn , but the inverse limit generally will not have the homotopy type of a CW complex. The “k-invariants” k n+2 that specify the system are to be regarded as cohomology classes k n+2 ∈ H n+2 (Xn ; πn+1 (X)). (J. Peter May 191)
- By our definition of a Postnikov system, we must define Xn+1 to be the homotopy fiber of k n+2 . Thus its points are pairs (ω, x) consisting of a path ω : I → K(πn+1 (X), n+2) and a point x ∈ Xn such that ω(0) = ∗ and ω(1) = k n+2 (x). (J. Peter May 192)
- Lemma (Yoneda). There is a canonical bijection between natural transformations Φ : k −→ k ′ and elements φ ∈ k ′ (Z). (J. Peter May 192)
- Consider a “represented functor” k(X) = [X, Z] and another contravariant functor k ′ from the homotopy category of based CW complexes to the category of sets (J. Peter May 192)
- Corollary. There is a canonical bijection between natural transformations Φ : [−, Z] −→ [−, Z ′ ] and elements φ ∈ [Z, Z ′ ]. (J. Peter May 192)
- Theorem. Cohomology operations ̃ H̃ q (−; π) −→ H ̃ H̃ q+n (−; ρ) are in canonical bijective correspondence with elements of ̃ H̃ q+n (K(π, q); ρ). (J. Peter May 192)
- In general, cohomology operations are only natural transformations of setvalued functors. However, stable operations are necessarily homomorphisms of cohomology groups (J. Peter May 192)
- . To determine all cohomology operations, we need only compute the cohomology of all EilenbergMac Lane spaces (J. Peter May 193)
- We have described an explicit construction of these spaces as topological Abelian groups in Chapter 16 §5, and this construction leads to an inductive method of computation (J. Peter May 193)
- Theorem. For n ≥ 0, there are stable cohomology operations Sq n : H q (X; Z2 ) −→ H q+n (X; Z2 ), called the Steenrod operations. They satisfy the following properties (J. Peter May 193)
- There are also formulas, called the Adem relations, describing Sq i Sq j , as a linear combination of operations Sq i+j−k Sq k , 2k (J. Peter May 193)
- It turns out that the Steenrod operations generate all mod 2 cohomology operations. In fact, the identity map of K(Z2 , q) specifies a fundamental class ιq ∈ H q (K(Z2 , q); Z2 ), and the following theorem holds. Theorem. H ∗ (K(Z2 , q); Z2 ) is a polynomial algebra whose generators are certain iterates of Steenrod operations applied to the fundamental class ιq . Explicitly, writing Sq I = Sq i1 · · · Sq ij for a sequence of positive integers I = {i1 , . . ., ij }, the generators are the Sq I ιq for those sequences I such that ir ≥ 2ir+1 for 1 ≤ r < j and i1 < i2 + · · · + ij + q. (J. Peter May 193)
- The β are called Bockstein operations. (J. Peter May 193)
- For Abelian groups π and ρ, show that [K(π, n), K(ρ, n)] ∼ = ∼ = Hom(π, ρ). (J. Peter May 193)
- We shall require our open covers to be numerable, as can always be arranged when B is paracompact (J. Peter May 195)
- In the case of non-connected base spaces, the fibers over points in different components may have different dimension (J. Peter May 195)
- A map (g, f ) of vector bundles is an isomorphism if and only if f is a homeomorphism and g restricts to an isomorphism on each fiber. (J. Peter May 195)
- We say that two vector bundles over B are equivalent if they are isomorphic over B, so that there is an isomorphism (g, id) between them. We let En (B) denote the set of equivalence classes of n-plane bundles over B (J. Peter May 195)
- Thus we have a contravariant set-valued functor En (−) on spaces (J. Peter May 195)
- Vector bundles should be thought of as rather rigid geometric objects, and the equivalence relation between them preserves that rigidity. Nevertheless, equivalence classes of n-plane bundles can be classified homotopically. (J. Peter May 195)
- In turn, the starting point of the classification theorem is the observation that the functor En (−), like homology and cohomology, is homotopy invariant in the sense that it factors (J. Peter May 195)
- through the homotopy category hU . In less fancy language, this amounts to the following result. Proposition. The pullbacks of an n-plane bundle p : E −→ B along homotopic maps f0 , f1 : A −→ B are equivalent. (J. Peter May 196)
- It can be verified on general abstract nonsense grounds, using Brown’s representability theorem, that the functor En (−) is representable in the form [−, BO(n)] (J. Peter May 196)
- It is far more useful to have an explicit concrete construction of the relevant “classifying space” BO(n). (J. Peter May 196)
- We construct a particular n-plane bundle γn : En −→ BO(n), called the “universal n-plane bundle.” By pulling back γn along (homotopy classes of) maps f : B −→ BO(n), we obtain a natural transformation of functors [−, BO(n)] −→ En (−). We show that this natural transformation is a natural isomorphism of functors by showing how to construct a map (g, f ), unique up to homotopy, from any given n-plane bundle E over any space B to the universal n-plane bundle En ; it is in this sense that En is “universal.” (J. Peter May 196)
- Let Vn (R q ) be the Stiefel variety of orthonormal n-frames in R q . Its points are n-tuples of orthonormal vectors in Rq , and it is topologized as a subspace of (Rq )n or, equivalently, as a subspace of (S q−1 )n . It is a compact manifold. Let Gn (Rq ) be the Grassmann variety of n-planes in Rq . Its points are the n-dimensional subspaces of Rq . Sending an n-tuple of orthonormal vectors to the n-plane they span gives a surjective function Vn (Rq ) −→ Gn (Rq ), and we topologize Gn (Rq ) as a quotient space of Vn (Rq ). It too is a compact manifold (J. Peter May 196)
- We define the classifying space BO(n) to be Gn (R∞ ). (J. Peter May 196)
- Let Enq nq be the subbundle of the trivial bundle Gn (Rq ) × Rq whose points are the pairs (x, v) such that v is a vector in the plane x; denote the projection of Enq nq by γnq nq , so that γnq nq (x, v) = x. When n = 1, γ is called the “canonical line bundle” over RP q−1 . (J. Peter May 196)
- We may let q go to infinity. We let En = En∞ and let γn = γn∞ : En −→ BO(n) (J. Peter May 196)
- Theorem. The natural transformation Φ : [−, BO(n)] −→ En (−) obtained by sending the homotopy class of a map f : B −→ BO(n) to the equivalence class of the n-plane bundle f ∗ En is a natural isomorphism of functors. (J. Peter May 197)
- it is called the Gauss map of the tangent bundle of M (J. Peter May 197)
- Similarly, using the orthogonal complements of tangent planes, we obtain the Gauss map E(ν) −→ E q q−n of the q normal bundle ν of the embedding of M in R . (J. Peter May 197)
- Definition. Let k ∗ be a cohomology theory, such as H ∗ (−; π) for an Abelian group π. A characteristic class c of degree q for n-plane bundles is a natural assignment of a cohomology class c(ξ) ∈ k q (B) to bundles ξ with base space B. (J. Peter May 197)
- Thus, if (g, f ) is a map from a bundle ζ over A to a bundle ξ over B, so that ζ is equivalent to f ∗ ξ, then f ∗ c(ξ) = c(ζ). (J. Peter May 198)
- Since the functor En is represented by BO(n), the Yoneda lemma specializes to give the following result. Lemma. Evaluation on γn specifies a canonical bijection between characteristic classes of n-plane bundles and elements of k ∗ (BO(n)). (J. Peter May 198)
- classes of n-plane bundles and elements of k ∗ (BO(n)). The formal similarity to the definition of cohomology operations is obvious (J. Peter May 198)
- Moreover, the behavior of characteristic classes with respect to operations on bundles can be determined by calculating the maps on cohomology induced by maps between classifying spaces (J. Peter May 198)
- We are particularly interested in Whitney sums of bundles (J. Peter May 198)
- The internal sum, or Whitney sum, of two bundles over the same base space B is obtained by pulling back their external sum along the diagonal map of B (J. Peter May 198)
- For example, let ε denote the trivial line bundle over any space. We have the operation that sends an n-plane bundle ξ over B to the (n + 1)-plane bundle ξ ⊕ ε over B. There is a classifying map in : BO(n) −→ BO(n + 1) that is characterized up to homotopy by i∗n n (γn+1 ) = γn ⊕ ε. (J. Peter May 198)
- Theorem. For n-plane bundles ξ over base spaces B, n ≥ 0, there are characteristic classes wi (ξ) ∈ H i (B; Z2 ), i ≥ 0, called the Stiefel-Whitney classes. They satisfy and are uniquely characterized by the following axioms. (J. Peter May 199)
- Every mod 2 characteristic class for n-plane bundles can be written uniquely as a polynomial in the Stiefel-Whitney classes {w1 , . . ., wn }. (J. Peter May 199)
- The mod 2 cohomology H ∗ (BO(n); Z2 ) is the polynomial algebra Z2 [w1 , . . ., wn ]. (J. Peter May 199)
- P to consider as formal sums xi , deg xi = i guarantee that the sum isPfinite. a vector bundle ξ to be wi (ξ); (J. Peter May 199)
- Suppose that M immerses in Rq with normal bundle ν. Then τ (M ) ⊕ ν ∼ = ∼ = εq and we have the “Whitney duality formula” w(M ) ∪ w(ν) = 1, (J. Peter May 199)
- which shows how to calculate tangential Stiefel-Whitney classes in terms of normal Stiefel-Whitney classes, and conversely. This formula can be used to prove non-immersion results when we know w(M ) (J. Peter May 199)
- Since the normal bundle of the standard embedding S q −→ Rq+1 is trivial, w(S q ) = 1. A manifold is said to be parallelizable if its tangent bundle is trivial. For some manifolds M , we can show that M is not parallelizable by showing that one of its Stiefel-Whitney classes is non-zero, but this strategy fails for M = S q . (J. Peter May 200)
- We describe some standard computations in the cohomology of projective spaces that give less trivial examples. Write ζq for the canonical line bundle over RP q in this section (J. Peter May 200)
- For example, w(RP q ) = 1 if and only if q = 2k − 1 for some k (as the reader should check) and therefore RP q can be parallelizable only if q is of this form (J. Peter May 200)
- If R q+1 admits a bilinear product without zero divisors, then it is not hard to prove that τ (RP q ) ∼ = ∼ = Hom(ζq , ζ ⊥ q⊥ ) admits q linearly independent cross-sections and is therefore trivial. We conclude that Rq+1 can admit such a product only if q+1 = 2k for some k. The real numbers, complex numbers, quaternions, and Cayley numbers show that there is such a product for q + 1 = 1, 2, 4, and 8. As we shall explain in the next chapter, these are in fact the only q for which Rq+1 admits such a product (J. Peter May 200)
- While the calculation of w(RP q ) just given is quite special, there is a remarkable general recipe, called the “Wu formula,” for the computation of w(M ) in terms of Poincaré duality and the Steenrod operations in H ∗ (M ; Z2 ). In analogy with w(M ), general recipe, called the “Wu formula,” for the computation of w( Poincaré duality and the Steenrod operations in H ∗ (M ; Z2 ). In anal we define the total Steenrod square of an element x by Sq(x) = i Sq i (x). (J. Peter May 200)
- Theorem (Wu formula). Let M be a smooth closed n-manifold with funda(M ; Z2 ). Then the total Stiefel-Whitney class w(M ) is equal Theorem (Wu formula). Let M be a smooth closed n-manifold with funda mental class z ∈ HnP (M ; Z2 ). Then the total Stiefel-Whitney class w(M ) is equ to Sq(v), where v = vi ∈ H ∗∗ (M ; Z2 ) is the unique cohomology class such that hv ∪ x, zi = hSq(x), zi for all x ∈ H (M ; Z2 ). Thus, for k ≥ 0, vk ∪ x = Sq k (x) for all x ∈ H n−k (M ; Z2 ), ∗ and wk (M ) = vk i+j=k Sq i (vj ). (J. Peter May 201)
- The basic reason that such a formula holds is that the StiefelWhitney classes can be defined in terms of the Steenrod operations (J. Peter May 201)
- Since the tangent bundle of M depends on its smooth structure, this is rather surprising (J. Peter May 201)
- Characteristic classes determine important numerical invariants of manifolds, called their characteristic numbers. (J. Peter May 201)
- Definition. Let M be a smooth closed R-oriented n-manifold with fundamental class z ∈ Hn (M ; R). For a characteristic class c of degree n, define the tangential characteristic number c[M ] ∈ R by c[M ] = hc(τ (M )), zi. Similarly, define the normal characteristic number c[ν(M )] by c[ν(M )] = hc(ν(M )), zi, where ν(M ) is the normal bundle associated to an embedding of M in Rq for q sufficiently large. (These numbers are well defined because any two embeddings of M in Rq for large q are isotopic and have equivalent normal bundles.) (J. Peter May 201)
- arge q are isotopic and have equivalent normal bundles.) In particular, if ri are integers such that iri = n, then the monomial w r1 · · · wnrn nrn P is a characteristic class of degree n, and all mod 2 characteristic classes of degree n are linear combinations of these. Different manifolds can have the same Stiefel-Whitney numbers (J. Peter May 201)
- Lemma. If M is the boundary of a smooth compact (n + 1)-manifold W , then all tangential Stiefel-Whitney numbers of M are zero. (J. Peter May 201)
- Lemma. All tangential Stiefel-Whitney numbers of a smooth closed manifold M are zero if and only if all normal Stiefel-Whitney numbers of M are zero. (J. Peter May 201)
- Theorem (Thom). If M is a smooth closed n-manifold all of whose normal Stiefel-Whitney numbers are zero, then M is the boundary of a smooth (n + 1)manifold. (J. Peter May 202)
- Thus we need only compute the Stiefel-Whitney numbers of M to determine whether or not it is a boundary. By Wu’s formula, the computation only requires knowledge of the mod 2 cohomology of M , with its Steenrod operations (J. Peter May 202)
- There are several ways to construct the Stiefel-Whitney classes. The most illuminating one depends on a simple, but fundamentally important, construction on vector bundles, namely their “Thom spaces.” (J. Peter May 202)
- Definition. Let ξ : E −→ B be an n-plane bundle. Apply one-point compactification to each fiber of ξ to obtain a new bundle Sph(E) over B whose fibers are spheres S n with given basepoints, namely the points at ∞. These basepoints specify a cross-section B −→ Sph(E). Define the Thom space T ξ to be the quotient space T (ξ) = Sph(E)/B. That is, T (ξ) is obtained from E by applying fiberwise one-point compactification and then identifying all of the points at ∞ to a single basepoint (denoted ∞). (J. Peter May 202)
- Remark. If we give the bundle ξ a Euclidean metric and let D(E) and S(E) denote its unit disk bundle and unit sphere bundle, then there is an evident homeomorphism between T ξ and the quotient space D(E)/S(E) (J. Peter May 202)
- In turn, D(E)/S(E) is homotopy equivalent to the cofiber of the inclusion S(E) −→ D(E) and therefore to the cofiber of the projection S(E) −→ B. (J. Peter May 202)
- If the bundle ξ is trivial, so that E = B × Rn , then Sph(E) = B × S n . Quotienting out B amounts to the same thing as giving B a disjoint basepoint and then forming the smash product B+ ∧ S n . That is, in this case the Thom complex is Σn B+ . Therefore, for any cohomology theory k ∗ , k q (B) = k̃ k̃ q (B+ ) ∼ = ∼ = k̃ k̃ n+q (T ξ). (J. Peter May 202)
- which is called the “Thom diagonal. (J. Peter May 203)
- This should look very similar to the problem of patching local fundamental classes to obtain a global one; that is, it looks like a question of orientation (J. Peter May 203)
- Definition. Let ξ : E −→ B be an n-plane bundle. An R-orientation, or Thom class, of ξ is an element µ ∈ ̃ H̃ n (T ξ; R) such that, for every point b ∈ B, i∗b b (µ) is a generator of the free R-module ̃ H̃ n (Sbn bn ). (J. Peter May 203)
- We leave it as an instructive exercise to verify that an R-orientation of a closed n-manifold M determines and is determined by an R-orientation of its tangent bundle τ (M ). (J. Peter May 203)
- Theorem (Thom isomorphism theorem). Let µ ∈ H ̃ H̃ n (T ξ; R) be a Thom class for an n-plane bundle ξ : E −→ B. Define Φ : H q (B; R) −→ H ̃ H̃ n+q (T ξ; R) by Φ(x) = x ∪ µ. Then Φ is an isomorphism. (J. Peter May 203)
- However, much the best proof from the point of view of anyone seriously interested in algebraic topology is to apply the Serre spectral sequence of the bundle Sph(E). (J. Peter May 203)
- . Use of a field ensures that the cohomology of the relevant direct limits is the inverse limit of the cohomologies (J. Peter May 203)
- The Serre spectral sequence is a device for computing the cohomology of the total space E of a fibration from the cohomologies of its base B and fiber F . It measures the cohomological deviation of H ∗ (E) from H ∗ (B)⊗H ∗ (F ) (J. Peter May 203)
- Just as in orientation theory for manifolds, the question of orientability depends on the structure of the units of the ring R, and this leads to the following conclusion. Proposition. Every vector bundle admits a unique Z2 -orientation. This can be proved along with the Thom isomorphism theorem by a MayerVietoris argument (J. Peter May 204)
- First, taking the characteristic class point of view, we define the Stiefel-Whitney classes in terms of the Steenrod operations by setting wi (ξ) = Φ −1 Sq i Φ(1) = Φ −1 Sq i (J. Peter May 204)
- Here S(γ1 ) is the infinite sphere S ∞ , which is the universal cover of RP ∞ and is therefore contractible (J. Peter May 204)
- We indicate two constructions of the Stiefel-Whitney classes (J. Peter May 204)
- As we shall explain in §8, passage from topological groups to their classifying spaces is a product-preserving functor, at least up to homotopy. (J. Peter May 204)
- Indeed, up to homotopy, inner conjugation by an element of G induces the identity map on BG for any topological group G. (J. Peter May 204)
- . The subring H ∗ ((RP ∞ )n ; Z2 )Σn of elements invariant under the action is the polynomial algebra on the elementary symmetric functions σi , 1 ≤ i ≤ n, in the variables αi . (J. Peter May 205)
- By the Künneth theorem, we see that H ∗ ((RP ∞ ) n ; Z2 ) = ⊗ i i=1 H ∗ (RP ∞ ; Z2 ) = Z2 [α1 , . . ., αn ], (J. Peter May 205)
- The resulting map ω ∗ : H ∗ (BO(n); Z2 ) −→ H ∗ ((RP ∞ ) n ; Z2 ) Σn is a ring homomorphism between polynomial algebras on generators of the same degrees. It turns out to be a monomorphism and therefore an isomorphism (J. Peter May 205)
- One advantage of this approach is that, since we know the Steenrod operations on H ∗ (RP ∞ ; Z2 ) and can read them off on H ∗ ((RP ∞ ) n ; Z2 ) by the Cartan formula, it leads to a purely algebraic calculation of the Steenrod operations in H ∗ (BO(n); Z2 ). Explicitly, the following “Wu formula” holds: Sq i (wj ) = fol i t=0 lowing “Wu for j+t−i−1 t u wi−t wj+t . (J. Peter May 205)
- The proof of the classification theorem for complex n-plane bundles works in exactly the same way as for real n-plane bundles, using complex Grassmann varieties. (J. Peter May 205)
- In fact, the fundamental groups of the real Grassmann varieties are Z2 , and their universal covers are their orientation covers. (J. Peter May 205)
- These covers are the oriented Grassmann varieties ̃ G̃n (Rq ). (J. Peter May 205)
- We write BU (n) = Gn (C∞ ) and BSO(n) = G̃n (R∞ ), (J. Peter May 205)
- and we construct universal complex n-plane bundles γn : EUn −→ BU (n) and oriented n-plane bundles γ̃n : E ̃ Ẽn −→ BSO(n) as in the first section (J. Peter May 205)
- Let E Un (B) denote the set of equivalence classes of complex nplane bundles over B and let E Ẽ Ẽn (B) denote the set of equivalence classes of oriented real n-plane bundles over B; (J. Peter May 205)
- Theorem. The natural transformation Φ : [−, BU (n)] −→ E Un (−) obtained by sending the homotopy class of a map f : B −→ BU (n) to the equivalence class of the n-plane bundle f ∗ EUn is a natural isomorphism of functors. (J. Peter May 205)
- Theorem. The natural transformation Φ : [−, BSO(n)] −→ Ẽ Ẽn (−) obtained by sending the homotopy class of a map f : B −→ BSO(n) to the equivalence class of the oriented n-plane bundle f ∗ Ẽn is a natural isomorphism of functors. (J. Peter May 206)
- Lemma. Evaluation on γn specifies a canonical bijection between characteristic classes of complex n-plane bundles and elements of k ∗ (BU (n)). Lemma. Evaluation on γ̃n specifies a canonical bijection between characteristic classes of oriented n-plane bundles and elements of k ∗ (BSO(n)). (J. Peter May 206)
- Theorem. H ∗ (BSO(n); Z2 ) ∼ = Z2 [w2 , . . ., wn ]. (J. Peter May 206)
- The Thom space of a complex or oriented real vector bundle is the Thom space of its underlying real vector bundle. We obtain characteristic classes in cohomology with any coefficients by applying cohomology operations to Thom classes, but it is rarely the case that the resulting characteristic classes generate all characteristic classes: the cases H ∗ (BO(n); Z2 ) and H ∗ (BSO(n); Z2 ) are exceptional (J. Peter May 206)
- Theorem. For n ≥ 1, there are elements ci ∈ H 2i (BU (n); Z), i ≥ 0, called the Chern classes. They satisfy and are uniquely characterized by the following axioms. (1) c0 = 1 and ci = 0 if i > n. (2) c1 is the canonical generator of H 2 (BU (1); Z) when n = 1. (3) i∗n ∗n (ci ) = ci . (4) p∗m m,n ) = ci . (ci ) = j j=0 cj ⊗ ci−j . The integral cohomology H ∗ (BU (n); Z) is the polynomial algebra Z[c1 , . . ., cn ]. (J. Peter May 207)
- The reader deserves to be warned about a basic inconsistency in the literature. Remark. With the discussion above, c1 (γ n+1 1n ) is the canonical generator of H (CP n ; Z), where γ1n+1 2 1n is the canonical line bundle of lines in Cn+1 and points on the line. This is the standard convention in algebraic topology. In algebraic geometry, it is more usual to define Chern classes so that the first Chern class of the dual of γ n+1 1n is the canonical generator of H 2 (CP n ; Z). With this convention, the nth Chern class would be (−1)n cn . It is often unclear in the literature which convention is being followed. (J. Peter May 207)
- Turning to oriented real vector bundles, we define the Pontryagin and Euler classes as follows, taking cohomology with coefficients in any commutative ring R. Definition. Define the Pontryagin classes pi ∈ H 4i (BO(n); R) by pi = (−1) i c ∗ (c2i ), c ∗ : H 4i (BU (n); R) −→ H 4i (BO(n); R); also write pi for πn∗ (pi ) ∈ H 4i (BSO(n); R). (J. Peter May 207)
- Definition. Define the Euler class e(ξ) ∈ H n (B; R) of an R-oriented n-plane bundle ξ over the base space B by e(ξ) = Φ−1 µ2 , where µ ∈ H n (T ξ; R) is the Thom class. Giving the universal oriented n-plane bundle over BSO(n) the R-orientation induced by its integral orientation, this defines the Euler class e ∈ H n (BSO(n); R). (J. Peter May 207)
- The name “Euler class” is justified by the following classical result, which well illustrates the kind of information that characteristic numbers can encode. 1 Theorem. If M is a smooth closed oriented manifold, then the characteristic number e[M ] = he(τ (M )), zi ∈ Z is the Euler characteristic of M . (J. Peter May 208)
- The presence of 2-torsion makes the description of the integral cohomology rings of BO(n) and BSO(n) quite complicated, and these rings are almost never used in applications. Rather, one uses the mod 2 cohomology rings and the following description of the cohomology rings that result by elimination of 2-torsion. (J. Peter May 208)
- Theorem. Take coefficients in a ring R in which 2 is a unit. Then H ∗ (BO(2n)) ∼ = ∼ H = ∗ (BO(2n + 1)) ∼ = ∼ = H ∗ (BSO(2n + 1)) ∼ = ∼ = R[p1 , . . ., pn ] and H ∗ (BSO(2n)) ∼ = ∼ = R[p1 , . . ., pn−1 , e], with e 2 = pn . (J. Peter May 208)
- Consider bundles ξ : Y −→ B with fiber G. For spaces U in a numerable open cover O of B, there are homeomorphisms φ : U ×G −→ p−1 (U ) such that p◦φ = π1 . We say that Y is a principal G-bundle if Y has a free right action by G, B is the orbit space Y /G, ξ is the quotient map, and the φ are maps of right G-spaces. We say that ξ : Y −→ B is a universal principal G-bundle if Y is a contractible space. In particular, for any topological group G whose identity element is a nondegenerate basepoint, such as any Lie group G, the map p : EG −→ BG constructed in Chapter 16 §5 is a universal principal G-bundle (J. Peter May 208)
- Observe that the long exact sequence of homotopy groups of a universal principal G-bundle gives isomorphisms πq (BG) ∼ = ∼ = πq−1 (G) for q ≥ 1. (J. Peter May 208)
- There is a classification theorem for principal G-bundles. Let PG(B) denote the set of equivalence classes of principal G-bundles over B, where two principal G-bundles over B are equivalent if there is a G-homeomorphism over B between them. Via pullback of bundles, this is a contravariant set-valued functor on the homotopy category of spaces. Theorem. Let γ : Y −→ Y /G be any universal principal G-bundle. The natural transformation Φ : [−, Y /G] −→ PG(−) obtained by sending the homotopy class of a map f : B −→ Y /G to the equivalence class of the principal G-bundle f ∗ Y is a natural isomorphism of functors. (J. Peter May 209)
- Here an action is effective if gf = f for every f ∈ F implies g = e. (J. Peter May 209)
- For a principal Gbundle Y , let G act on Y × F by g(y, f ) = (yg −1 , gf ) and let Y ×G F be the orbit space (Y × F )/G. With the correct formal definition of a fiber bundle with group G and fiber F , every such fiber bundle p : E −→ B is equivalent to one of the form Y ×G F −→ Y /G ∼ = ∼ = B for some principal G-bundle Y over B (J. Peter May 209)
- In fact, the “associated principal G-bundle” Y can be constructed as the function space of all maps ψ : F −→ E such that ψ is an admissible homeomorphism onto some fiber Fb = p −1 (b). (J. Peter May 209)
- We conclude that, for any F , PG(B) is naturally isomorphic to the set of equivalence classes of bundles with group G and fiber F over B. Fiber bundles with group O(n) and fiber Rn are real n-plane bundles, fiber bundles with group U (n) and fiber C n are complex n-plane bundles, and fiber bundles with group SO(n) and fiber Rn are oriented real n-plane bundles. Thus the classification theorems of the previous sections could all be rederived as special cases of the general classification theorem for principal G-bundles stated in this section (J. Peter May 209)
- The fact that it is a generalized cohomology theory is a consequence of the Bott periodicity theorem, which is one of the most important and influential theorems in all of topology (J. Peter May 211)
- we explain how the Adams operations in K-theory allow a quick solution to the “Hopf invariant one problem.” (J. Peter May 211)
- One implication is the purely algebraic theorem that the only possible dimensions of a real (not necessarily associative) division algebra are 1, 2, 4, and 8 (J. Peter May 211)
- We shall only discuss complex Ktheory, although there is a precisely analogous construction of real K-theory KO (J. Peter May 211)
- From the point of view of algebraic topology, real K-theory is a substantially more powerful invariant, but complex K-theory is usually more relevant to applications in other fields. (J. Peter May 211)
- We consider the set V ect(X) of equivalence classes of vector bundles over a space X. (J. Peter May 211)
- . The set V ect(X) forms an Abelian monoid (= semi-group) under Whitney sum, and it forms a semi-ring with multiplication given by the (internal) tensor product of vector bundles over X. (J. Peter May 211)
- There is a standard construction, called the Grothendieck construction, of an Abelian group G(M ) associated to an Abelian monoid M : one takes the quotient of the free Abelian group generated by the elements of M by the subgroup generated by the set of elements of the form m + n − m ⊕ n, where ⊕ is the sum in M . The evident morphism of Abelian monoids i : M −→ G(M ) is universal: for any homomorphism of monoids f : M −→ G, where G is an Abelian group, there is a unique homomorphism of groups f ˜ f˜ : G(M ) −→ G such that f ˜ f˜ ◦ i = f (J. Peter May 211)
- If M is a semi-ring, then its multiplication induces a multiplication on G(M ) such that G(M ) is a ring, called the Grothendieck ring of M . If the semi-ring M is commutative, then the ring G(M ) is commutative (J. Peter May 211)
- Definition. The K-theory of X, denoted K(X), is the Grothendieck ring of the semi-ring V ect(X). An element of K(X) is called a virtual bundle over X. We write [ξ] for the element of K(X) determined by a vector bundle ξ. (J. Peter May 211)
- we have the function d : V ect(X) −→ Z that sends a vector bundle to the dimension of its restriction to the component of the basepoint ∗. Since d is a homomorphism of semi-rings, it induces a dimension function d : K(X) −→ Z, which is a homomorphism of rings. Since d is an isomorphism when X is a point, d can be identified with the induced map K(X) −→ K(∗). Definition. The reduced K-theory ̃ K̃(X) of a based space X is the kernel of d : K(X) −→ Z. It is an ideal of K(X) and thus a ring without identity. Clearly K(X) ∼ = ∼ = K ̃ K̃(X) × Z. (J. Peter May 212)
- We say that bundles ζ and ξ are stably equivalent if, for a sufficiently large q, the bundles ζ ⊕ εq−m and ξ ⊕ εq−n are equivalent, where m = d(ζ) and n = d(ξ). (J. Peter May 212)
- E U (X) ∼ = ∼ = [X+ , BU ]. (J. Peter May 212)
- Let E U (X) be the set of stable equivalence classes of vector bundles over X (J. Peter May 212)
- Proposition. If ξ : E −→ X is a vector bundle over X, then there is a bundle η over X such that ξ ⊕ η is equivalent to εq for some q. (J. Peter May 212)
- The space ΓE of sections of E is a vector space under fiberwise addition and scalar multiplication (J. Peter May 212)
- The resulting short exact sequence of vector bundles, like any other short exact sequence of vector bundles, splits as a direct sum (J. Peter May 212)
- Corollary. Every virtual bundle over X can be written in the form [ξ] − q for some bundle ξ and non-negative integer q. (J. Peter May 212)
- Corollary. There is a natural isomorphism E U (X) −→ K ̃ K̃(X). (J. Peter May 212)
- Corollary. Give Z the discrete topology. For compact spaces X, there is a natural isomorphism K(X) ∼ = ∼ = [X+ , BU × Z]. (J. Peter May 212)
- Proposition. The space BU × Z is a ring space up to homotopy. That is, there are additive and multiplicative H-space structures on BU × Z such that the associativity, commutativity, and distributivity diagrams required of a ring commute up to homotopy. (J. Peter May 213)
- The study of ring spaces such as this is a relatively new, and quite deep, part of algebraic topology. However, the reader should feel reasonably comfortable with the additive H-space structure on BU . (J. Peter May 213)
- Theorem. K(S 2 ) is generated as a ring by [H] subject to the single relation ([H] − 1)2 = 0. Therefore, as Abelian groups, K(S 2 ) is free on the basis {1, [H]} and ̃ K̃(S 2 ) is free on the basis {1 − [H]}. (J. Peter May 214)
- Conversely, an isomorphism f from the trivial bundle over S 1 to itself gives a way to glue together the trivial bundles over D and D′ to reconstruct a bundle over S 2 . Say that two such “clutching functions” f are equivalent if the bundles they give rise to are equivalent. A careful analysis of the form of the possible clutching functions f leads to a canonical example in each equivalence class and (J. Peter May 214)
- Theorem (Bott periodicity). For compact spaces X, α : K(X) ⊗ K(S 2 ) −→ K(X × S 2 ) is an isomorphism. (J. Peter May 214)
- The following useful observation applies to any representable functor, not just K-theory. Lemma. For nondegenerately based spaces X and Y , the projections of X × Y on X and on Y and the quotient map X×Y −→ X∧Y induce a natural isomorphism ̃ K̃(X ∧ Y ) ⊕ K ̃ K̃(X) ⊕ K ̃ K̃(Y ) ∼ = ∼ = K ̃ K̃(X × Y ), and ̃ K̃(X ∧ Y ) is the kernel of the map ̃ K̃(X × Y ) −→ ̃ K̃(X) ⊕ K ̃ K̃(Y ) induced by the inclusions of X and Y in X × Y . (J. Peter May 214)
- Theorem (Bott periodicity). For nondegenerately based compact spaces X, β : ̃ K̃(X) ⊗ K ̃ K̃(S 2 ) −→ K ̃ K̃(X ∧ S 2 ) = K ̃ K̃(Σ 2 X) is an isomorphism. (J. Peter May 215)
- Since ̃ K̃(S 2 ) ∼ = Z with generator b, the theorem implies that multiplication by the “Bott element” b specifies an isomorphism [X, BU × Z] = ∼ = K ∼ ̃ K̃(X) −→ K ̃ K̃(Σ 2 X) ∼ = ∼ = [X, Ω 2 (BU × Z)] (J. Peter May 215)
- Bott’s map β can also be proved to be a homotopy equivalence using only basic algebraic topology. Since BU and ΩSU are simply connected spaces of the homotopy types of CW complexes, a relative version of the Hurewicz theorem called the Whitehead theorem shows that β will be a weak equivalence and therefore a homotopy equivalence if it induces an isomorphism on integral homology. (J. Peter May 215)
- A purely algebraic dualization argument proves that, as a ring, H∗ (BU ) ∼ = ∼ Z[γi |i ≥ 1], = where γi is the image of a generator of H2i (CP ∞ ) under the map induced by the inclusion of CP ∞ = BU (1) in BU (J. Peter May 216)
- In any case, it should now be clear that we have a periodic Ω-prespectrum and therefore a generalized cohomology theory represented by it. Definition. The K-theory Ω-prespectrum KU has spaces KU2i = BU ×Z and KU2i+1 = U for all i ≥ 0. The structure maps are given by the canonical homotopy equivalence U ≃ ΩBU = Ω(BU × Z) and the Bott equivalence BU × Z ≃ ΩU . (J. Peter May 216)
- We have a resulting reduced cohomology theory on based spaces such that ̃ K̃ 2i (X) = K̃(X) and K̃ 2i+1 (X) = K̃(ΣX) for all integers i. This theory has products that are induced by tensor products of bundles over compact spaces and that are induced by suitable maps φ : KUi ∧ KUj −→ KUi+j in general, just as for the cup product in ordinary cohomology. It is standard to view this simply as a Z2 -graded theory with groups ̃ K̃ 0 (X) and K̃ 1 (X). (J. Peter May 216)
- Definition. Let E0 be the zero section of E. Define the projective bundle π : P (E) −→ X by letting the non-zero complex numbers act on E − E0 by scalar multiplication on fibers and taking the orbit space under this action. Equivalently, the fiber π −1 (x) ⊂ P (E) is the complex projective space of lines through the origin in the fiber ξ −1 (x) ⊂ E. Define the canonical line bundle L(E) over P (E) to be the subbundle of the pullback π ∗ E of ξ along π whose points are the pairs consisting of a line in a fiber of E and a point on that line. Let Q(E) be the quotient bundle π ∗ E/L(E) and let H(E) denote the dual of L(E). (J. Peter May 216)
- Regard K(P (E)) as a K(X)algebra via π ∗ : K(X) −→ K(P (E)). Theorem (Bott periodicity). Let L be a line bundle over X and let H = H(L ⊕ ε). Then the K(X)-algebra K(P (L ⊕ ε)) is generated by the single element [H] subject to the single relation ([H] − 1)([L][H] − 1) = 0. (J. Peter May 216)
- It will be based on a generalization to projective bundles of the calculation of H ∗ (CP n ). The proofs of both results are intertwined with the proof of the following “splitting principle,” which allows the deduction of explicit formulas about general bundles from formulas about sums of line bundles. (J. Peter May 217)
- Theorem (Splitting principle). There is a compact space F (E) and a map p : F (E) −→ X such that p∗ E is a sum of line bundles over F (E) and both p∗ : H ∗ (X; Z) −→ H ∗ (F (E); Z) and p∗ : K(X) −→ K(F (E)) are monomorphisms. (J. Peter May 217)
- Lemma (Splitting lemma). Both π ∗ : H ∗ (X; Z) −→ H ∗ (P (E); Z) and π ∗ : K(X) −→ K(P (E)) are monomorphisms. (J. Peter May 217)
- Theorem. Let x = c1 (L(E)) ∈ H 2 (P (E); Z). Then H ∗ (P (E); Z) is the free Theorem. Let x = c1 (L(E)) ∈ H 2 (P (E); Z). Then H ∗ (P (E); Z) is the free H (X; Z)-module on the basis 1, x, . . ., xn−1 , and the Chern classes of ξ are char∗ acterized by c0 (ξ) = 1 and the formula e fo n k=0 (−1) k ck (E)x n−k = 0. (J. Peter May 217)
- One first a line bundle E, L(E) = E and c1 (E) = c′1 (E) by the definition of x. One first verifies by direct calculation that if E = L1 ⊕ · · · ⊕ Ln is a sum of line bundles, then ′ 1≤k≤n (x − c1 (Lk )) = 0. This implies that ck (E) is the kth elementary symmetric 1≤k≤n ′ (x − c1 (Lk )) = 0. This implies that ck k (E) is the kth elementary symmetric Q polynomial in the c1 (Lk ). By the Whitney sum formula for the Chern classes, this implies that c′k k (E) = ck (E) in this case (J. Peter May 217)
- The following analogue in K-theory of the previous theorem holds. Observe that, since they are continuous operations on complex vector spaces, the exterior powers λk can be applied fiberwise to give natural operations on vector bundles. Theorem. Let H = H(E). Then K(P (E)) is the free K(X)-module on the Theorem. Let H = H(E). Then K(P (E)) is the fre basis 1, [H], . . ., [H]n−1 , and the following formula holds: d t n k=0 (−1) k [H] k [λ k E] = 0. (J. Peter May 218)
- Projective bundles are closely related to Thom spaces (J. Peter May 218)
- Theorem (Thom isomorphism theorem). Define Φ : K(X) −→ K ̃ K̃(T (ξ)) by Φ(x) = x · λE . Then Φ is an isomorphism. (J. Peter May 219)
- We have seen above that ordinary cohomology and K-theory enjoy similar properties. The splitting theorem implies a direct connection between (J. Peter May 219)
- The example we are interested in is the “Chern character,” which gives rise to (J. Peter May 219)
- Example. Taking R = Q, define the Chern character ch(E) ∈ H ∗∗ (X; Q) by ch(E) = f ˆ mple. Taking R = Q, define the fˆ(E), where f (t) = et = ti /i!. (J. Peter May 219)
- For line bundles L and L′ , we have c1 (L ⊗ L′ ) = c1 (L) + c1 (L′ ). One way to see this is to recall that BU (1) ≃ K(Z, 2) and that line bundles are classified by their Chern classes regarded as elements of [X+ , BU (1)] ∼ = ∼ = H 2 (X; Z). (J. Peter May 219)
- Lemma. The Chern character specifies a ring homomorphism ch : K(X) −→ H ∗∗ (X; Q). (J. Peter May 220)
- Lemma. For n ≥ 1, the Chern character maps ̃ K̃(S 2n ) isomorphically onto the image of H 2n (S 2n ; Z) in H 2n (S 2n ; Q). Therefore cn : K̃(S 2n ) −→ H 2n (S 2n ; Z) is a monomorphism with cokernel Z(n−1)! . (J. Peter May 220)
- Together with some of the facts given in Chapter 23 §7, this has a remarkable application to the study of almost complex structures on spheres. Recall that a smooth manifold of even dimension admits an almost complex structure if its tangent bundle is the underlying real vector bundle of a complex bundle. (J. Peter May 220)
- Theorem. S 2 and S 6 are the only spheres that admit an almost complex structure. (J. Peter May 220)
- It is classical that S 2 and S 6 admit almost complex structures and that S 4 does not. (J. Peter May 220)
- We then have the following basic result, which actually holds for general compact spaces X provided that we replace singular cohomology by Č Čech cohomology. Theorem. For any finite based CW complex X, ch induces an isomorphism ̃ K̃ ∗ (X) ⊗ Q −→ H ̃ H̃ ∗∗ (X; Q). (J. Peter May 220)
- Visibly, this is a morphism of monoids, Λ(ξ ⊕ η) = Λ(ξ)Λ(η). It therefore extend to a homomorphism of groups Λ : K(X) −→ G, (J. Peter May 221)
- There are natural operations in K-theory, called the Adams operations, that are somewhat analogous to the Steenrod operations in mod 2 cohomology. In fact, the analogy can be given content by establishing a precise relationship between the Adams and Steenrod operations, but (J. Peter May 221)
- Theorem. For each non-zero integer k, there is a natural homomorphism of rings ψ k : K(X) −→ K(X). These operations satisfy the following properties. (1) ψ 1 = id and ψ −1 is induced by complex conjugation of bundles. (2) ψ k ψ ℓ = ψ kℓ = ψ ℓ ψ k . (3) ψ p (x) ≡ xp mod p for any prime p. (4) ψ k (ξ) = ξ k if ξ is a line bundle. (5) ψ k (x) = k n x if x ∈ K ̃ K̃(S 2n ). (J. Peter May 221)
- Recall that the subring of symmetric polynomials in the polynomial algebra Z[x1 , . . ., xn ] is the polynomial algebra Z[σ1 , . . ., σn ], where σi = x1 x2 · · · xi + · · · is the ith elementary symmetric function. (J. Peter May 221)
- Remark. The observant reader will have noticed that, by analogy with the definition of the Stiefel-Whitney classes, we can define characteristic classes in Ktheory by use of the Adams operations and the Thom isomorphism, setting ρk (E) = Φ−1 ψ k Φ(1) for n-plane bundles E (J. Peter May 223)
- We give one of the most beautiful and impressive illustrations of the philosophy described in the first chapter. We define a numerical invariant, called the “Hopf invariant,” of maps f : S 2n−1 −→ S n and show that it can only rarely take the value one. (J. Peter May 223)
- We then indicate several problems whose solution can be reduced to the question of when such maps f take the value one. Adams’ original solution to the Hopf invariant one problem used secondary cohomology operations in ordinary cohomology and was a critical starting point of modern algebraic topology. The later realization that a problem that required secondary operations in ordinary cohomology could be solved much more simply using primary operations in Ktheory had a profound impact on the further development of the subject. (J. Peter May 223)
- Definition. Let X be the cofiber of a based map f : S 2n−1 −→ S n , where n ≥ 2. Then X is a CW complex with a single vertex, a single n-cell i, and a single 2n-cell j. (J. Peter May 223)
- H ̃ H̃ ∗ (X) is free Abelian on generators x = [i] and y = [j]. Define an integer h(f ), the Hopf invariant of f , by x 2 = h(f )y (J. Peter May 223)
- Theorem. If h(f ) = ±1, then n = 2, 4, or 8. (J. Peter May 224)
- Theorem. If S n−1 is an H-space, then n = 1, 2, 4, or 8 (J. Peter May 224)
- The determination of which spheres are H-spaces has the following implications. Theorem. Let ω : Rn × Rn −→ Rn be a map with a two-sided identity element e 6= 0 and no zero divisors. Then n = 1, 2, 4, or 8. (J. Peter May 226)
- Note that ω need not be bilinear, just continuous. Also, it need not have a strict unit; all that is required is that e be a two-sided unit up to homotopy for the restriction of ω to Rn − {0}. Theorem. If S n is parallelizable, then n = 0, 1, 3, or 7. (J. Peter May 226)
- suppose that S n is parallelizable, so that its tangent bundle τ is trivial. (J. Peter May 226)
- Cobordism theories were introduced shortly after K-theory, (J. Peter May 227)
- We shall describe the cobordism of smooth closed manifolds, but this is in fact a particularly elementary example. Other examples include smooth closed manifolds with extra structure on their stable normal bundles: orientation, complex structure, Spin structure, or symplectic structure for example. All of these except the symplectic case have been computed completely. (J. Peter May 227)
- The area is pervaded by insights from algebraic topology that are quite mysterious geometrically. For example, the complex cobordism groups turn out to be concentrated in even degrees: every smooth closed manifold of odd dimension with a complex structure on its stable normal bundle is the boundary of a compact manifold (with compatible bundle information). However, there is no geometric understanding of why this should be the case. The analogue with “complex” replaced by “symplectic” is false. (J. Peter May 227)
- We consider the problem of classifying smooth closed n-manifolds M . One’s first thought is to try to classify them up to diffeomorphism, but that problem is in principle unsolvable. Thom’s discovery that one can classify such manifolds up to the weaker equivalence relation of “cobordism (J. Peter May 227)
- We say that two smooth closed nmanifolds M and N are cobordant if there is a smooth compact manifold W whose boundary is the disjoint union of M and N , ∂W = M ∐ N . We write Nn for the set of cobordism classes of smooth closed n-manifolds. It is convenient to allow the empty set ∅ as an n-manifold for every n. Disjoint union gives an addition on the set Nn . (J. Peter May 227)
- Nn is a vector space over Z2 . Cartesian product of manifolds defines a multiplication Nm × Nn −→ Nm+n . This operation is bilinear, associative, and commutative, and the zero dimensional manifold with a single point provides an identity element. We conclude that N∗ is a graded Z2 -algebra. (J. Peter May 227)
- Theorem (Thom). N∗ is a polynomial algebra over Z2 on generators ui of dimension i for i > 1 and not of the form 2 r − 1. (J. Peter May 227)
- As already stated in our discussion of Stiefel-Whitney numbers, it follows from the proof of the theorem that a manifold is a boundary if and only if its normal Stiefel-Whitney numbers are zero (J. Peter May 228)
- Theorem. Two smooth closed n-manifolds are cobordant if and only if their normal Stiefel-Whitney numbers, or equivalently their tangential Stiefel-Whitney numbers, are equal. (J. Peter May 228)
- Explicit generators ui are known. Write [M ] for the cobordism class of a manifold M . Then we can take u2i = [RP 2i ] (J. Peter May 228)
- The strategy for the proof of Thom’s theorem is to describe Nn as a homotopy group of a certain Thom space. The homotopy group is a stable one, and it turns out to be computable by the methods of generalized homology theory (J. Peter May 228)
- and we have the following translation of our problem in manifold theory to a problem in homotopy theory (J. Peter May 228)
- shall sketch the proof in the next section, where we shall also explain the ring structure on π∗ (T O) that makes it a Z2 -algebra. (J. Peter May 228)
- Theorem (Thom). For sufficiently large q, Nn is isomorphic to πn+q (T O(q)). Therefore Nn ∼ = ∼ = πn (T O). Moreover, N∗ and π∗ (T O) are isomorphic as Z2 -algebras. (J. Peter May 228)
- Given a smooth closed n-manifold M , we may embed it in Rn+q for q sufficiently large, and we let ν be the normal bundle of the embedding. (By the Whitney embedding theorem, q = n suffices (J. Peter May 228)
- Embed M as the zero section of the total space E(ν). Then a standard result in differential topology known as the tubular neighborhood theorem implies that the identity map of M extends to an embedding of E(ν) onto an open neighborhood U of M in R n+q . (J. Peter May 228)
- The “PontryaginThom construction” associates a map t : S n+q −→ T (ν) to our tubular neighborhood U (J. Peter May 229)
- The Thom space was tailor made to allow this construction (J. Peter May 229)
- By an implication of Sard’s theorem known as the transversality theorem (J. Peter May 229)
- . In fact, T O is a commutative and associative ring prespectrum in the sense of the following definition. Definition. Let T be a prespectrum. Then T is a ring prespectrum if there are maps η : S 0 −→ T0 and φm,n : Tm ∧ Tn −→ Tm+n such that the following diagrams are homotopy commutative (J. Peter May 230)
- For example, the Eilenberg-Mac Lane Ω-prespectrum of a commutative ring R is an associative and commutative ring prespectrum (J. Peter May 230)
- 3. It is denoted HR or sometimes, by abuse, K(R, 0). Similarly, the K-theory Ω-prespectrum is an associative and commutative ring prespectrum. The sphere prespectrum, whose nth space is S n , is another example (J. Peter May 231)
- Lemma. If T is an associative ring prespectrum, then π∗ (T ) is a graded ring. If T is commutative, then π∗ (T ) is commutative in the graded sense. (J. Peter May 231)
- Calculation of the homotopy groups π∗ (T O) proceeds by first computing the homology groups H∗ (T O; Z2 ) and then showing that the stable Hurewicz homomorphism maps π∗ (T O) monomorphically onto an identifiable part of H∗ (T O; Z2 ). (J. Peter May 231)
- Just as we defined the homotopy groups of a prespectrum T by the formula πn (T ) = colim πn+q (Tq ), (J. Peter May 231)
- we define the homology and cohomology groups of T with respect to a homology theory k∗ and cohomology theory k ∗ on spaces by the formulas kn (T ) = colim k̃ k̃n+q (Tq ), (J. Peter May 231)
- In fact, this definition of cohomology is inappropriate in general, differing from the appropriate definition by a lim 1 error term. However, the definition is correct when k ∗ is ordinary cohomology with coefficients in a field R and each H̃ n+q (Tq ; R) is a finite dimensional vector space over R. (J. Peter May 232)
- Observe that there is no cup product in H ∗ (T ; R): the maps in the limit system factor through the reduced cohomologies of suspensions, in which cup products are identically zero (J. Peter May 232)
- The Hurewicz homomorphisms πn+q (Tq ) −→ ̃ H̃n+q (Tq ; Z) pass to colimits to give the stable Hurewicz homomorphism h : πn (T ) −→ Hn (T ; Z). (J. Peter May 232)
- If T is an associative and commutative ring prespectrum, then h : π∗ (T ) −→ H∗ (T ; R) is a map of graded commutative rings. (J. Peter May 232)
- Recall that we have Thom isomorphisms Φq : H n (BO(q)) −→ H ̃ H̃ n+q (T O(q)) obtained by cupping with the Thom class µq ∈ H ̃ H̃ q (T O(q)). (J. Peter May 232)
- We therefore obtain a “stable Thom isomorphism” Φ : H n (BO) −→ H n (T O) on passage to limits. We have dual homology Thom isomorphisms Φn : H ̃ H̃n+q (T O(q)) −→ Hn (BO(q)) (J. Peter May 232)
- Theorem. H∗ (BO) is the polynomial algebra Z2 [bi |i ≥ 1]. Let ai ∈ Hi (T O) be the element characterized by Φ(ai ) = bi . Corollary. H∗ (T O) is the polynomial algebra Z2 [ai |i ≥ 1]. (J. Peter May 233)
- Recall from Chapter 23 §6 that we have a homotopy equivalence j : RP ∞ −→ T O(1). (J. Peter May 233)
- Since the Steenrod operations are stable and natural, they pass to limits to define natural operations Sq i : H n (T ) −→ H n+i (T ) for i ≥ 0 and prespectra T . (J. Peter May 234)
- The homology and cohomology of T O are built up from π∗ (T O) and Steenrod operations (J. Peter May 234)
- Definition. The mod 2 Steenrod algebra A is the quotient of the free associative Z2 -algebra generated by elements Sq i , i ≥ 1, by the ideal generated by the Adem relations (which are stated in Chapter 22 §5) (J. Peter May 234)
- Lemma. For spaces X, H ∗ (X) has a natural A-module structure. Lemma. For prespectra T , H ∗ (T ) has a natural A-module structure (J. Peter May 234)
- The elements of A are stable mod 2 cohomology operations, and our description of the cohomology of K(Z2 , q)s in Chapter 22 §5 implies that A is in fact the algebra of all stable mod 2 cohomology operations (J. Peter May 234)
- Recall that HZ2 denotes the EilenbergMac Lane Ω-prespectrum {K(Z2 , q)}. (J. Peter May 234)
- Lemma. As a vector space, A is isomorphic to H ∗ (HZ2 ). (J. Peter May 234)
- Theorem. A has a basis consisting of the operations Sq I = Sq i1 · · · Sq ij , where I runs over the sequences {i1 , . . ., ij } of positive integers such that ir ≥ 2ir+1 for 1 ≤ r < j. (J. Peter May 234)
- What is still more important to us is that A not only has the composition product A ⊗ A −→ A, it also has a coproduct ψ : A −→ A ⊗ A. (J. Peter May 234)
- Algebraic structures like this, with compatible products and coproducts, are called “Hopf algebras.” (J. Peter May 234)
- Theorem. Let N∗ be the algebra defined abstractly by N∗ = Z2 [ui |i > 1 and i 6= 2 r − 1], where deg ui = i. Define a homomorphism of algebras f : H∗ (T O) −→ N∗ by f (ai ) = homomorphism of algebras f : H∗ (T ui if i is not of the form 2r − 1 0 if i = 2r − 1. Then the composite g : H∗ (T O) γ − → A∗ ⊗ H∗ (T O) id ⊗f −−−→ A∗ ⊗ N∗ (J. Peter May 236)
- is an isomorphism of both A-comodules and Z2 -algebras. (J. Peter May 236)
- Now consider the Hurewicz homomorphism h : π∗ (T ) −→ H∗ (T ) of a prespectrum T . (J. Peter May 236)
- Theorem. h : π∗ (T O) −→ H∗ (T O) is a monomorphism and g◦h maps π∗ (T O) isomorphically onto N∗ . (J. Peter May 236)
- We shall prove that a smooth closed n-manifold M is a boundary if and only if all of its normal Stiefel-Whitney numbers are zero. (J. Peter May 236)
- Polynomials in the StiefelWhitney classes are elements of H ∗ (BO) (J. Peter May 236)
- follows that cobordant manifolds have the same normal Stiefel-Whitney numbers. (J. Peter May 236)
- -Whitney numbers. The assignment of Stiefel-Whitney numbers to corbordism classes of n-manifolds specifies a homomorphism # : H n (BO) ⊗ Nn −→ Z2 (J. Peter May 236)
- To say that all normal Stiefel-Whitney numbers of M are zero is to say that w#[M ] = 0 for all w ∈ H n (BO) (J. Peter May 237)
- this implies that [M ] = 0 and thus that M is a boundary. (J. Peter May 237)
- We think of prespectra as “stable objects” that have associated homotopy, homology, and cohomology groups. Imagine that we have a good category of stable objects, analogous to the category of based spaces, that is equipped with all of the constructions that we have on based spaces: wedges (= coproducts), colimits, products, limits, suspensions, loops, homotopies, cofiber sequences, fiber sequences, smash products, function objects, and so forth. Let us call the stable objects in our imagined category “spectra” and call the category of such objects S . We have in mind an analogy with the notions of presheaf and sheaf. (J. Peter May 238)
- The homology and cohomology groups of Σ∞ X are the (reduced) homology and cohomology groups of X; the homotopy groups of Σ ∞ X are the stable homotopy groups of X. (J. Peter May 238)
- Continuing our thought exercise, we can form the homotopy category hS of spectra and can define homotopy groups in terms of homotopy classes of maps from sphere spectra to spectra (J. Peter May 239)
- Reflection on the periodic nature of K-theory suggests that we should define sphere spectra of negative dimension and define homotopy groups πq (X) for all integers q (J. Peter May 239)
- That is, we develop a theory of CW spectra using sphere spectra as the domains of attaching maps (J. Peter May 239)
- The Whitehead and cellular approximation theorems hold, and every spectrum X admits a CW approximation ΓX −→ X. We define the set [X, Y ] of morphisms X −→ Y in h̄ h̄S to be the set of homotopy classes of maps ΓX −→ ΓY . This is a stable category in the sense that the functor Σ : h̄ h̄S −→ h h̄ h̄S is an equivalence of categories. More explicitly, the natural maps X −→ ΩΣX and ΣΩX −→ X are isomorphisms in h h̄ h̄S . (J. Peter May 239)
- In particular, up to isomorphism, every object in the category h̄ h̄S is a suspension, hence a double suspension (J. Peter May 239)
- This implies that each [X, Y ] is an Abelian group and composition is bilinear. (J. Peter May 239)
- so that cofiber sequences and fiber sequences are equivalent. Therefore cofiber sequences give rise to long exact sequences of homotopy groups. (J. Peter May 239)
- The homotopy groups of wedges and products of spectra are given by topy π∗ ( i roups of Xi ) = i edges and product π∗ (Xi ) and π∗ ( i of spectr Xi ) = i π∗ (Xi ). (J. Peter May 239)
- Therefore, if only Theref map i e, if only Q Xi −→ fi i nitely many πq (Xi ) a Xi is an isomorphism (J. Peter May 239)
- A spectrum E represents a homology theory E∗ and a cohomology theory E ∗ specified in terms of smash products and function spectra by Eq (X) = πq (X ∧ E) and E q (X) = π−q F (X, E) ∼ = ∼ = [X, Σ q E]. (J. Peter May 239)
- Moreover, every homology or cohomology theory on h h̄ h̄S is so represented by some spectrum E. (J. Peter May 239)
- Hπ for the “Eilenberg-Mac Lane spectrum” that represents ordinary cohomology with coefficients in (J. Peter May 239)
- Its only non-zero homotopy group is π0 (Hπ) = π, and the Hurewicz homomorphism maps this group isomorphically onto H0 (Hπ; Z). When π = Z2 , the natural map H0 (HZ2 ; Z) −→ H0 (HZ2 ; Z2 ) is also an isomorphism (J. Peter May 239)
- Returning to our motivating example, we write M O for the “Thom spectrum” that arises from the Thom prespectrum T O. The reader may sympathize with a student who claimed that M O stands for “Mythical Object.” (J. Peter May 239)
- However, these early constructions were far more primitive than our outline suggests. While they gave a satisfactory stable category, the underlying category of spectra did not have products, limits, and function objects, and its smash product was not associative, commutative, or unital. In fact, a fully satisfactory category of spectra was not constructed until 1995. (J. Peter May 240)
- Definition. A spectrum E is a prespectrum such that the adjoints σ̃ : En −→ ΩEn+1 of the structure maps σ : ΣEn −→ En+1 are homeomorphisms. A map f : T −→ T ′ of prespectra is a sequence of maps fn : Tn −→ Tn′ such that σ ′ n′ ◦ Σfn = fn+1 ◦ σn for all n. A map f : E −→ E ′ of spectra is a map between E and E ′ regarded as prespectra. (J. Peter May 240)
- We define wedges and colimits of spectra by first performing the construction on the prespectrum level and then applying the functor L. If we start with spectra and construct products or limits spacewise, then the result is again a spectrum; that is, limits of spectra are the limits of their underlying prespectra. Thus the category S is complete and cocomplete. (J. Peter May 240)
- We now have cylinders E ∧ I+ and thus can define homotopies between maps of spectra. Similarly we have cones CE = E ∧ I (where I has basepoint 1), suspensions ΣE = E ∧ S 1 , path spectra F (I, E) (where I has (J. Peter May 240)
- Similarly, we define the smash product T ∧X and function prespectrum F (X, T ) of a based space X and a prespectrum T spacewise. For a spectrum E, we define E ∧ X by applying L to the prespectrum level construction; the prespectrum F (X, E) is already a spectrum. (J. Peter May 240)
- basepoint 0), and loop spectra ΩE = F (S 1 , E). The development of cofiber and fiber sequences proceeds exactly as for based spaces (J. Peter May 241)
- The essential point is that homotopy and homology commute with colimits. (J. Peter May 241)
- It is not true that cohomology converts colimits to limits in general, because of lim 1 error terms, and this is one reason that our definition of the cohomology of prespectra via limits is inappropriate except under restrictions that guarantee the vanishing of lim 1 terms (J. Peter May 241)
- We define QX = ∪Ωq Σq X, and we find that the nth space of Σ∞ X is QΣn X. It should be apparent that the homotopy groups of the space QX are the stable homotopy groups of X. (J. Peter May 241)
- For example, the homotopy groups of the K-theory spectrum are Z for every even integer and zero for every odd integer (J. Peter May 241)
- Thus, if we have a prespectrum T whose invariants we are interested in, such as an Eilenberg-Mac Lane Ω-prespectrum or the K-theory Ω-prespectrum, then we can construct a spectrum LKT that has the same invariants (J. Peter May 241)
- The real work involves the smash product of spectra (J. Peter May 241)
- there is also considerable payoff in explicit concrete calculations, as the computation of π∗ (M O) (J. Peter May 241)
- The subject in its earlier days was blessed with some of the finest expositors of mathematics, for example Steenrod, Serre, Milnor, and Adams (J. Peter May 243)
- Two introductions to algebraic topology starting from de Rham cohomology: R. Bott and L.W. Tu. Differential forms in algebraic topology. Springer-Verlag. 1982. I. Madsen and J. Tornehave. From calculus to cohomology. de Rham cohomology and characteristic classes. Cambridge University Press. 1997. (J. Peter May 244)
- The classic reference on Morse theory, with an exposition of the Bott periodicity theorem: J. Milnor. Morse theory. Annals of Math. Studies No. 51. Princeton University Press. 1963. (J. Peter May 244)
- Two good basic references on equivariant algebraic topology, classically called the theory of transformation groups (see also §§16, 21 below): G. Bredon. Introduction to compact transformation groups. Academic Press. 1972. T. tom Dieck. Transformation groups. Walter de Gruyter. 1987. A more advanced book, a precursor to much recent work in the area: T. tom Dieck. Transformation groups and representation theory. Lecture Notes in Mathematics Vol. 766. Springer-Verlag. 1979 (J. Peter May 245)
- Two classical treatments and a good modern treatment of homological algebra: H. Cartan and S. Eilenberg. Homological algebra. Princeton University Press. 1956. S. MacLane. Homology. Springer-Verlag. 1963. C.A. Weibel. An introduction to homological algebra. Cambridge University Press. 1994. (J. Peter May 245)
- P.G. Goerss and J.F. Jardine. Simplicial homotopy theory. Birkhäuser. To appear. (J. Peter May 245)
- Two classic papers of Serre: J.-P. Serre. Homologie singuliére des espaces fibrés. Applications. Annals of Math. (2)54(1951), 425–505. J.-P. Serre. Groupes d’homotopie et classes de groupes abéliens. Annals of Math. (2)58(1953), 198–232. A nice exposition of some basic homotopy theory and of Serre’s work: S.-T. Hu. Homotopy theory. Academic Press. 1959. (J. Peter May 245)
- There are other important spectral sequences in the context of fibrations, mainly due to Eilenberg and Moore. Three references: S. Eilenberg and J.C. Moore. Homology and fibrations, I. Comm. Math. Helv. 40(1966), 199–236. (J. Peter May 245)
- L. Smith. Homological algebra and the Eilenberg-Moore spectral sequences. Trans. Amer. Math. Soc. 129(1967), 58–93. V.K.A.M. Gugenheim and J.P. May. On the theory and applications of differential torsion products. Memoirs Amer. Math. Soc. No. 142. 1974. There is a useful guidebook to spectral sequences: J. McCleary. User’s guide to spectral sequences. Publish or Perish. 1985. (J. Peter May 246)
- A compendium of the work of Steenrod and others on the construction and analysis of the Steenrod operations: N.E. Steenrod and D.B.A. Epstein. Cohomology operations. Annals of Math. Studies No. 50. Princeton University Press. 1962. (J. Peter May 246)
- A general treatment of Steenrod-like operations: J.P. May. A general algebraic approach to Steenrod operations. In Lecture Notes in Mathematics Vol. 168, 153–231. Springer-Verlag. 1970. A nice book on mod 2 Steenrod operations and the Adams spectral sequence: R. Mosher and M. Tangora. Cohomology operations and applications in homotopy theory. Harper and Row. 1968. (J. Peter May 246)
- A classic and a more recent standard treatment that includes K-theory: N.E. Steenrod. Topology of fibre bundles. Princeton University Press. 1951. Fifth printing, 1965. D. Husemoller. Fibre bundles. Springer-Verlag. 1966. Third edition, 1994. A general treatment of classification theorems for bundles and fibrations: J.P. May. Classifying spaces and fibrations. Memoirs Amer. Math. Soc. No. 155. 1975. (J. Peter May 246)
- A good reference for the basic calculations of characteristic classes: A. Borel. Topology of Lie groups and characteristic classes. Bull. Amer. Math. Soc. 61(1955), 297–432. (J. Peter May 246)
- Two proofs of the Bott periodicity theorem that only use standard techniques of algebraic topology, starting from characteristic class calculations: H. Cartan et al. Périodicité des groupes d’homotopie stables des groupes classiques, d’après Bott. Séminaire Henri Cartan, 1959/60. Ecole Normale Supérieure. Paris. (J. Peter May 246)
- E. Dyer and R.K. Lashof. A topological proof of the Bott periodicity theorems. Ann. Mat. Pure Appl. (4)54(1961), 231–254. (J. Peter May 247)
- Two classical lecture notes on K-theory: R. Bott. Lectures on K(X). W.A. Benjamin. 1969. This includes a reprint of perhaps the most accessible proof of the complex case of the Bott periodicity theorem, namely: M.F. Atiyah and R. Bott. On the periodicity theorem for complex vector bundles. Acta Math. 112(1994), 229–247. (J. Peter May 247)
- J.F. Adams. Vector fields on spheres. Annals of Math. 75(1962), 603–632. (J. Peter May 247)
- The basic source for the structure theory of (connected) Hopf algebras: J. Milnor and J.C. Moore. On the structure of Hopf algebras. Annals of Math. 81(1965), 211–264. (J. Peter May 247)
- The classic analysis of the structure of the Steenrod algebra as a Hopf algebra: J. Milnor. The Steenrod algebra and its dual. Annals of Math. 67(1958), 150–171. (J. Peter May 247)
- Two classic papers of Adams; the first constructs the Adams spectral sequence relating the Steenrod algebra to stable homotopy groups and the second uses secondary cohomology operations to solve the Hopf invariant one problem: J.F. Adams. On the structure and applications of the Steenrod algebra. Comm. Math. Helv. 32(1958), 180–214. (J. Peter May 247)
- J.F. Adams. On the non-existence of elements of Hopf invariant one. Annals of Math. 72(1960), 20–104. (J. Peter May 248)
- The beautiful classic paper of Thom is still highly recommended: R. Thom. Quelques propriétés globals des variétés différentiables. Comm. Math. Helv. 28(1954), 17–86. Thom computed unoriented cobordism. Oriented and complex cobordism came later. In simplest form, the calculations use the Adams spectral sequence: J. Milnor. On the cobordism ring Ω∗ and a complex analogue. Amer. J. Math. 82(1960), 505–521. (J. Peter May 248)
- Two classical references, the second of which also gives detailed information about complex cobordism that is of fundamental importance to the subject. G.W. Whitehead. Generalized homology theories. Trans. Amer. Math. Soc. 102(1962), 227–283. J.F. Adams. Stable homotopy and generalised homology. Chicago Lectures in Mathematics. University of Chicago Press. 1974. Reprinted in 1995. (J. Peter May 248)
- Foundations for equivariant stable homotopy theory are established in: L.G. Lewis, Jr., J.P. May, and M.Steinberger (with contributions by J.E. McClure). Equivariant stable homotopy theory. Lecture Notes in Mathematics Vol. 1213. Springer-Verlag. 1986. (J. Peter May 248)
- . Localization and completion; rational homotopy theory Since the early 1970s, it has been standard practice in algebraic topology to localize and complete topological spaces, and not just their algebraic invariants, at sets of primes and then to study the subject one prime at a time, or rationally. Two of the basic original references are: D. Sullivan. The genetics of homotopy theory and the Adams conjecture. Annals of Math. 100(1974), 1–79. A.K. Bousfield and D.M. Kan. Homotopy limits, completions, and localizations. Lecture Notes in Mathematics Vol. 304. Springer-Verlag. 1972. A more accessible introduction to localization and a readable recent paper on completion are: P. Hilton, G. Mislin, and J. Roitberg. Localization of nilpotent groups and spaces. North-Holland. 1975. (J. Peter May 249)
- When spaces are rationalized, there is a completely algebraic description of the result. The main original reference and a more accessible source are: D. Sullivan. Infinitesimal computations in topology. Publ. Math. IHES 47(1978), 269–332. A.K. Bousfield and V.K.A.M. Gugenheim. On PL de Rham theory and rational homotopy type. Memoirs Amer. Math. Soc. No. 179. 1976 (J. Peter May 249)
- Another area well established by the mid-1970s. The following book is a delightful read, with capsule introductions of many topics other than infinite loop space theory, a very pleasant starting place for learning modern algebraic topology: J.F. Adams. Infinite loop spaces. Annals of Math. Studies No. 90. Princeton University Press. 1978 (J. Peter May 249)
- The following survey article is less easy going, but gives an indication of the applications to high dimensional geometric topology and to algebraic K-theory: J.P. May. Infinite loop space theory. Bull. Amer. Math. Soc. 83(1977), 456–494. (J. Peter May 249)
- Adams’ book cited in §16 gives a spectral sequence for the computation of stable homotopy groups in terms of generalized cohomology theories. Starting from complex cobordism and related theories, its use has been central to two waves of major developments in stable homotopy theory (J. Peter May 250)
- A good exposition for the first wave: D.C. Ravenel. Complex cobordism and stable homotopy groups of spheres. Academic Press. 1986. (J. Peter May 250)
- The essential original paper and a very nice survey article on the second wave: E. Devinatz, M.J. Hopkins, and J.H. Smith. Nilpotence and stable homotopy theory. Annals of Math. 128(1988), 207–242. (J. Peter May 250)
- The cited Proceedings contain good introductory survey articles on several other topics in algebraic topology. A larger scale exposition of the second wave is: D.C. Ravenel. Nilpotence and periodicity in stable homotopy theory. Annals of Math. Studies No. 128. Princeton University Press. 1992. (J. Peter May 250)
- There is a leap from the level of this introductory book to that of the most recent work in the subject. One recent book that helps fill the gap is: P. Selick. Introduction to homotopy theory. Fields Institute Monographs No. 9. American Mathematical Society. 1997 (J. Peter May 250)
- There is a recent expository book for the reader who would like to jump right in and see the current state of algebraic topology; although it focuses on equivariant theory, it contains introductions and discussions of many non-equivariant topics: J.P. May et al. Equivariant homotopy and cohomology theory. NSF-CBMS Regional Conference Monograph. 1996. (J. Peter May 250)
- For the reader of the last section of this book whose appetite has been whetted for more stable homotopy theory, there is an expository article that motivates and explains the properties that a satisfactory category of spectra should have: J.P. May. Stable algebraic topology and stable topological algebra. Bulletin London Math. Soc. 30(1998), 225–234 (J. Peter May 250)

Last Annotation: 04/11/2019

- Lift an exact sequence to a complex (Paolo Aluffi 670)

- “object of interest:the probien of studyiug functions quetient is just the solution10 the natural wiversal probien of studyiug functions 10 othe sets withidentical behavioron “equivalent” eluents. This i the primary 10 othe objective (Paolo Aluffi 630)
- This partly because the derived approach to an abstract study of cohomology. This partly because the derived category ofa abelian categoryisnot an abeliancategory and simple notions such ax kernel, cokernel, exact sequences are notavailable in D(A). (Paolo Aluffi 631)
- On the other hand, going past these difficulties, one finds that enough structure remains to do much homological algebra: objects in D(A) have cohomology, and there are ‘distinguished triangles’(cf. §3.4) sbutracting exact sequences and giving (Paolo Aluffi 631)
- . The derived category is a triangulated category, like the more manageable homotopic category K(A) that we will soon define (Paolo Aluffi 632)
- after constructions that determine cochain complexes ‘up to quask-isomorphism’. However, quasi-isomorphisms appear hard to deal with directly. Thus, we look for ‘more mauageable notions that may work as an effective replacement (Paolo Aluffi 633)
- Definition 4.8. A homotopy h between two morphisms of cochain complexes. of BL — M* is a collection of morphisms BLM such that Vi Bal mdighoht + KY od, ‘Wesaythat o®is homotopic to 3* and write a® ~ 3 there is a homotopy between ‘Wesaytha a* and 8°. (Paolo Aluffi 633)
- Definition 4.9. A morphism a* : L* — M® is a homotopy equivalence if there x morphism 4° M* —+ L* such that a* 8" ~ 1» and 3*+a* ~ 1s. The complexes L*, M* are said to be homotopy equivalent if there i a homotopy equivalence plexes L*, por (Paolo Aluffi 634)
- Corollary 4.11. Homotopy equivalent complezes have isomorphic cohomology (Paolo Aluffi 634)
- ss we will do in §5.3.is precisely that quasi-isomorphisi nnd howotopy equivalence are equivalent notions for (bounded) complexes of ijective or projective modules (Paolo Aluffi 635)
- an additive functor does not preserve quasi-isoworphisin (while an ercise 4.15): an add ezact functordoe). (Paolo Aluffi 635)
- meut with theorem status becauwe itis at the root of The contentof Theorem 4.14 s that any mechanism ‘asociating to a mathematical object a cochain complex determined up to homotopy ‘asociating to a mathematical object a cochain complex determined up to homotopy il give ise to a slew of interesting invariants: apply your favorite additive functor to any such complex, take cobomology, and ‘Theorem 4.14 guarantees that the ret wil be independent of the specific chosen complex (Paolo Aluffi 636)
- Theorem 4.14. Let & : A — Bb an addito functor Semen ho abelian cate gories. If L*, M* are homotopy equivalent complexes in C(A), then the cohomology gories. If complces EW). EE) ere hamaghi (Paolo Aluffi 636)
- applications. This i» the moral of Theorem 4.14: homotopy equivalent complexes have the same cobormology for & much better reason than complexes linked by & have the same cobormology for & much better reason than complexes linked by & Quaskisomorphism. If our general aim is to understand What t means to make Quaskisomorphism. If our general aim is to understand What t means to make all quasi-somorphisms invertible (that i, understand the derived category D(A), all quasi-somorphisms invertible (that i, understand the derived category D(A), we may begin hy making homotopy equivalences invertible. This produces a new we may begin hy making homotopy equivalences invertible. This produces a ne category, the houotopic category’of complexes, that we approach in this section. (Paolo Aluffi 638)
- ‘Wealoexaive the privileged positionofbouudedcomplexes ofprojective and ‘Wealoexaive the privileged positionofbouudedcomplexes ofprojective and injective objects regarding homotopy: for these complexes, quas-bomorplisms are ecesarily homotopy equivalences (Paolo Aluffi 638)
- 5.1. Homotopic maps are identified in the derived category. To this homotopiccategoryis 8 necewarytop on theway fomC(A) to D(A), (Paolo Aluffi 638)
- Lemma 5.1 tells us that every functor transforming quasiisomorp! isomorphisms must factorthrough the category obtained from C(A) by identifying isomorphisms must factorthrough the category obtained from C( together homotopic morphisms. It is time to definethis category. (Paolo Aluffi 640)
- .2. Definition ofthe homotopic category of complexes. Definition 5.2. Let A be an abelian category. The homotopy category K(A) of Definition 5.2. Let A be an abelian category. The homotopy category K(A) of cochain complezes in A is the category whose objects are the cochain complexes in cochain complezes in A is the category whose objects are the co A (thais, the same objects of C(A)) and whose morphisms are Homa (L*, M*) := Home) (L*, M*)/ ~ (Paolo Aluffi 640)
- However,note that, in general, the homotopic category is mot abelian. Indeed, homotopic maps do not have the same kernel or cokernel in general,so defining homotopic maps do not have the same kernel or cokernel in general,so defining these notions becomes problematic. As we already mentioned in $3.4, K(A) is a these notions becomes problematic. As we already mentioned in $3.4, K(A) is a friangulate category; the ‘distinguished triangles’ are the triangles arising from friangulate category; t the cones of morphisms, (Paolo Aluffi 640)
- Recall (Defition VITLG.1)that au Ramodule M is ‘projectiveif thefunctor Homa(M,) is exact and it i “nective if Homp(_,M) i exact. We can adopt Homa(M,) is exact and it i “nective these defiitions in any abelian category: (Paolo Aluffi 641)
- Example 5.6. The category offinite abelian groups is abelian (surprise, surprise), but contains o noatvial projective or njctve objects (Exercise 3). . (Paolo Aluffi 642)
- Definition 5.7. An abelian category A has enoughpoectves fo every object A in A there exis projective object P in A aad an epimorphism P— 4 The in A there exis projective object P in A aad an epimorphism P— 4 The category bs caught or evry ahs AinAthen ivi category bs caught or e nA aad a nouomorphisn A>—Q. (Paolo Aluffi 642)
- These definitionsare not ew tothe reader, ines we rn scrous thei VIL: in particular,we have lady abserved tha, for every commutative ring F, A-Mod in particular,we have lady abserved tha, for every commutative ring F, A-Mod has enough prjectives (ths i no challenging. free modes and thir direct su has enough prjectives (ths i no challenging. free modes and thir direct su mands are projective, Proposition VIILE.4) nd nugh inectives (thisis challeng. ig; we verified in Corollary VIILS.12). (Paolo Aluffi 642)
- shows thatwe shoud ot expect an abelcategory A to have An important cas in which ane can show that thee az enongh injectives is the category of sheaves of abelian groups over a topological vpace; this iy a key step in the definitionof sheaf cohomology as ‘derived functor (Paolo Aluffi 642)
- In general, categories of sheaves do not have enough projectives. Onthe other hand, the category offinitely generated abelian groups has enough projectives but not. the category offinitely generated abelia enough (indo, no nontrivial injective. (Paolo Aluffi 642)
- A counlicated way of say that a couplex. N* iu C(A) is exact s to asert that the identity map idy» aad the trivial map 0 induce the same morphism in cohomology, as they would ifthey were homotopic to each ober, It is however easy to homology, as they would ifthey were homotopic to each ober, It is however easy t construct examples ofexact complexes for which the identity is not homotopic to 0. construct examples ofexact complexes for which the identity is not homotopic to 0. As the reader will vey (Exercise 5.11), this hs to do with whether the complex As the reader will vey (Exercise 5.11), this hs to do with whether the complex splits or nok;in general, complex N° is said to be ‘split exactif dysis homosplits or nok;in general, complex N° is said to be ‘split exactif dysis homotopic to 0, Kepiog in mind that projective or injective objects ‘cause’sequences topic to 0 to split, it (Paolo Aluffi 643)
- Corollary 5.13 is the first manifestation ofthe principle captured more fully Corollary 5.13 is the first manifestation ofthe principle captured more fully by Theoreun 5.9: we have just verified that,for stably bounded complexes of by Theoreun 5.9: we have just verified that,for stably bounded complexes of projectivesor injective, ‘quasi-somorpbic to’ thesame as ‘homotopyequivalent proje 00. (Paolo Aluffi 645)
- Another consequence of Lemna 5.11 is the following remark, showing that quasi-omorphisms aze ‘no zero-divsors’ up to homotopy, with respect to morquasi-omorphisms aze ‘no zero-divso phians from complexes of projective. (Paolo Aluffi 645)
- Definition 6.1. Let A be an object of an abelian category A. A projective resolution of A i a quasibomorphism P* — «(A), where P* ix a complex in C=°(P). An inectio resolution of A isa quas-omorphism «(4) = Q°, where Q* is & An inectio r complex in C()). (Paolo Aluffi 651)
- Remark 6.2. The terminclogy is potentially confusing, sinc it hints that the resolutions themseles way be projectiveinjective as objects of the abeliancateresolutions themseles way be projectiveinjective as objects of the abel sory C(A),orof it bounded vavitions. Thi is ot the case (Brercie 63 (Paolo Aluffi 651)
- Now consider the category of homotopy classes of quasi-isomorphisms with target (A) (cf. Remark 4.5): this is ‘homotopic category of resolutions’ of A. We din tht prsciv Roclutlom, 1 hey ext, an baton. thi exegry (Analogously, injective resolutions are final in the homotopic category of quasiisomorphism with source «(4).) (Paolo Aluffi 651)
- Proposition 8.4. Any two projective (resp.,inje of an abelian category A are homotopy equislent. (Paolo Aluffi 653)
- The nioral we extract fromthese considerations is thatif an abelian category A has enough (say) projective, then we can ssociate with each object A of A an has enough (say) projective, then we can ssociate with each object A of A an object of KS¥(P). determined up to homotopy. This can fact be donefunctorially. object of KS¥(P). determined up to homotopy. This can fact be donefunctorially. in the sense that morphs in A can be lifted to morphisus of corresponding in the sense that m projective resolutions (Paolo Aluffi 653)
- In practice, this says that we can use this subcategory of K(P) as a replacementfor theorigina abelian category A. We are very close o achieving our gonlof mentfor theorigina abelian category A. We are very close o achieving our gonlof identifying the ‘essential nature’ ofcohomology: since homotopy equivalent com plexes hae the same cohomology (by Theorem 4.14) and additive functors preserve plexes hae the same cohomology (by Theorem 4.14) and additive functors preserv howotopy equivalence, objects ofK-(P) are ideal carrier ofcohomology invariants: howotopy equivalence, objects ofK-(P) are ideal carrier ofcohomology invariants: ‘while applying au adlitive functor to obiects of A in general destroys cohonmolog‘while applying au adlitive functor to obiects of A in general destroys cohonmological information, applying ‘the same functor’ to a corresponding obiect in K~(P) preserves that information. (Paolo Aluffi 653)
- Theorem 0.7. Let A be an abelian category with eno Functor K(F) = D(A) i an equivalence of caegoris. ITA has enough inectivs, then the functor K*(1) = D¥(A) is an equivalence ITA has of cutegoris (Paolo Aluffi 657)
- By Proposition 5.4,the functor C(A) — D(A)factors through the homotopi category; this of course holds for the bounded versious of these categories as well cate Ths (Paolo Aluffi 657)
- This, however, raises a question: if — i a functor and we are serious about replacing A with ts connterpart(s) in D(A), then there shonld be a way to about replacing A with ts connterpart(s) in D(A), then there shonld be a way to ‘reuterpret? fn this new coutext: cco i some natural way a ‘derived functor’ D(A) — D™(8). Oe would hope tht this fuctor should carry at least as tuuch D(A) — D™(8). Oe would hope tht this fu Information as # and satisfy bette propesties (Paolo Aluffi 663)
- eral. Why shonld they? For example, ‘tensor’ does not preserve projectives® (Ex ercise 7.1). Assuming that B hes enough projectives, the naturel way to ‘fix’ this proble is to apply a corresponding functor Pg constructed us in §6.3, msociating. proble is to apply a corresponding funct with each complex a projective resolution (Paolo Aluffi 664)
- Fore that easi iht-xact, and 2 Bacau i fe fads 0 anosher functorsee Gorllry VIL25. Functor tha are Jfedicats Go right-ezot onctors do preserve prolectives Gorllry V Bxercin 556. (Paolo Aluffi 664)
- ts, the subcategories whase objects have The paint is that there i no reason why cohomology concentrated in degree 0. The paint is that there i no reason why applying LS to a complex whose coboroology i concentrated in degree 0 should applying LS to a complex whose coboroology i concentrated in degree 0 sho yield a comple: with the sae property;unos 5 bs very special to begin vith. “This may be viewed as a misace. On tbe contraryit sone ofthe ma values “This may be viewed as a misace. On tbe contraryit sone ofthe ma values of deriving categories and functors. Recasting a additive functors A — B at of deriving categories and functors. Recasting a additive functors A — B at the level ofderived categorie, one ge acoms to interesting new invariants even the level ofderived categorie, one ge acoms to interesting new invariant whet wartug rou(the equivalent copy in the derived category of) A itl. (Paolo Aluffi 665)
- The moral of Example 7.2 i that deriving esact functors is relatively sraightforward, aud we cauot expect to lear anything new about an exact functor by forward, a deriving it (Paolo Aluffi 665)
- d we cauot expect to lear anything new about an exact functor by We will mostly interested in deriving functors that are not exact on the nose but preserve certain amountof exactness (Paolo Aluffi 665)
- property: There is a natural transformation LF 0 Bp Foo), « Jor cveryfunctor@ : K~(P(A)) — K~(P(8)) and every natural transformation Jor cveryfunctor@ : K~(P(A)) — K~(P(8)) and every natural transformation 7:40ns BooK(), there is a unique (sp to natural isomorphism) natural 7:40ns BooK(), there is a unique (sp to natural isomorphism) natural transformation~» L¥ inducing factorization ofy: 90 Pa ~s LF0 Pa transformat PaoK(®) (Paolo Aluffi 666)
- projectives. Let # : A — B be an additive functor. The i-th left-derived functor projectives. Let # : A — B be an addit Li of & in the functor A — B given by. L# =H ol 0 Paoin. For an object M in A, the complex in C(B)with Li(M) in degree —i and with For an object M in A, the complex in C(B)wit vanishing differentiali denoted by LyF(A). (Paolo Aluffi 667)
- Example 7.6. Every Remodule N determines a functor _@p N: M ++ M @N (see §VIIL2.2). The left-derived functor of _@N is denoted _ dp N and acts (see §VIIL2.2). The left-derived functor of _@N is denoted _ dp N and acts D(RtMod) — D-(R-Mod). The i-th leht-derived functor of _ 7 , viewed as a D(RtMod) — D-(R-Mod). The i-th leht-derived functor of _ 7 , viewed as a functor R-Mod — R-Mod, is Tof{(_@N):indeed,theconstruction ofTor(M,N) functor R-Mod — R-Mod, is Tof{(_@N):indeed,theconstruction ofTor(M,N) given in §VIIL24 matches precisely the ‘concrete’ interpretation of the ith left given in §VIIL24 matches pre derive functor given above. (Paolo Aluffi 668)
- “Alo note that we coud define the Ext functors x functors to Ab on say abelian category with enough injectives and/or projectives: any abeliancategory abelian category with enough injec he lack Hom Ranctors® to Ah (Paolo Aluffi 669)
- Long exactsequence of derived functors. The most remarkable property of the functors Tor; and Ext’ mentioned in Chapter VIII is probably that they epithe Jock of xacinesof © Hom, respectively, In th etme tha thy agree epithe Jock of xacinesof © Hom, respectively, In th etme ith the foncons dee 0 theyilonesse sequences (Paolo Aluffi 669)
- 0 Ls iN —0 in A dnc Tom exact sequen of dev functor’ the sequen for Toand in A dnc Tom exact sequen of dev functor Ext encountered i Chapter VIwillbe pricla cas (Paolo Aluffi 669)
- From a more sophisticated perspective, what happens is that derived functors in pang n $42, these tiagle play the rol of exact sequence nthe homotopic tit the vertices of a ‘distinguished triangle’ in the derived category: as we mentioned. in pang n $42, these tiagle play the rol of exact sequence nthe homotopic 228 rived iegorio which dono happen ob abn. Disinguibedtrngles 228 rived iegorio which dono happen ob abn. Disinguibedtrngles iw te 1 long exact sence,mh the sue way 4 dexact eens of complexes in the abelian case explored in §3.3. (Paolo Aluffi 669)
- on back fo th dinsy category ofsomplexs . The key poi threo the Glowing; assuming that A has enough projectives and that ough projectives and that 0— Ls i —N—0 is an exact snquence in A,can wo aeange forprojective solutions of L,M,N to form an exact vequence in CA)? (Paolo Aluffi 669)
- ‘Yes. This i often called the ‘horseshoe lemma’,ater the shape of the main diagram appearing in its proof. (Paolo Aluffi 670)
- of projective rvolutious. orally, we would like to say that the functor 3 assigning to every object of A a projective resolution in K(A) is “exact; but as K(A) is not but as K(A) is abel 1a 7.8 gets as close as posible to such (Paolo Aluffi 671)
- Lemma 7.11.Let A be an abelian category, and et © 0s Los Mrs Pr —r0 be an ezact sequence of complezes in A, where P* is projective for alli. Let F : A —B be any additive functor of abelian categories. Then the sequence obtained by applying # to (*). 0 FLY) — FM) — FP) —0, is exact. (Paolo Aluffi 673)
- Note that we are not asking .% to be exact in any sense. (Paolo Aluffi 673)
- quence of thelong exact cohomology sequence. Theorem 7.12. Let : A — B be an additive functor of abelian categories, and. Theorem 7.12. Let : A — B be an additive functo assume A has enough projectives. Every ezactsequence (Paolo Aluffi 673)
- in A inducesa long ezactsequence (Paolo Aluffi 674)
- 5. Relating #, L,#, RF. The reader may have woticed that © was derived Lo the let 10 obtain Tor. while Homwas derived Lo the right to obtain Ext. The symmetry mandating this choice lies inthe fact that i rght-exuet,while Hou is symmetry mandating this choice lies inthe fact that i rght-exuet,while Hou is left-exact: any additive functor can be derived to the left orto the ight (i the presence of enough projectives,rsp. enough injective), and the derived fusctors will it long exact sequences as proven in Theorem 7.12;but only functors satisfy measure of exactness can be recovered directly fromthei derived versions (Paolo Aluffi 675)
- Proposition 7.13. Let # : A — B be a right-ezac 0 fori <0. and LF is naturally isomorphic to F. (Paolo Aluffi 675)
- additive functor. Then LF (Paolo Aluffi 675)
- Going back to Theorem 7.12, we see that if # is right-ezact, then the teil end of the loug exact sequence of lef-derived functors for cousists of au application of fae. Thus, the situation in this case i the following: starting from short of fae. Th exact sequence 0— L— M—N—0 in A, we apply to obtain an exact sequence FL) = F(M) — FV) —0 in which we lost ‘the 0 on the lft’.The long exact sequence saves the day, continuing. the new sequence fuko an exact complex: (Paolo Aluffi 676)
- That is, Liweasures the extent to which higher LiF giv further measures ofthis failure (Paolo Aluffi 676)
- which & fails to be lefexact, and the (Paolo Aluffi 676)
- does not satisfy some exactness property. If # ix exact on the nose,then both LoS and ROS agree with F (up to natural isomorphisin); (Paolo Aluffi 677)
- How do we extract ‘cohomologicalinvariants’ fom a group GY Consider the category G-Mod of abelian groups endowed with a left-C-action, equivalently, the category G-Mod of abelian groups endowed with a left-C-action, equivalently, the category of Iet-Z{G}module, where ZIG] is the group ring bricly encountered in §IIL14. Objects of G-Mod may be called G-modules. (Paolo Aluffi 677)
- For a G-modle M, MC denotes the et of elements that are fi actionof G: these ae reasonably called the invariants of the action (Paolo Aluffi 677)
- are fixe uner the (Paolo Aluffi 677)
- re fixe uner the action. Note that MCis an abelian group carrying a trivial actionof G; is clear that setting M — MC defines a covariant functor © G-Mod — Ab. Both G-Mod and Ab are abelian categories, and it takes a woment to realize that © is a left-exact are abelian categories, and it takes a woment to realize that © is a left-exact functor: the reader should cither check this directly or do Exercise 7.16 aad then remember Claim VIIL1.19. The reader should in fact contemplate why this functor remember Claim VIIL1.19. The reader should in fact contemplate why this functor not right-xact:if G acts trivially & coset. m] of quotient M/L,there is no reason a priori why G should sct trivially on a representative m. (Paolo Aluffi 677)
- ‘The i-th right.derived functor of is deuoted H¥(G,_). Therefore, HO(G, M)= MC,aud for every short exact sequence of G-uodules (Paolo Aluffi 677)
- Example 7.14. Take G = Z. Then Z(G] is the ring Z[z, of Laurent polynomils. As Zfr,z~1)/(1 ) & Z, (Paolo Aluffi 678)
- ‘Example 7.15 (Finite cyclic groups). Let G = Cy be a cyclic group of order ms then Z[G] = Z{z/(2" — 1). (Paolo Aluffi 678)
- ‘There i a standard free resolutionof Z over a groupring Z[G], which leads to a coucrete description of group cohomology (Paolo Aluffi 678)
- Example 7.17. Let G be the Galois group of finite Galois extension k C F. Then G acts on the maltiplicative group F* of F, we can view F* as a Gmodule (Paolo Aluffi 679)
- Claim 7.18. H’(G,F*)=0. (Paolo Aluffi 679)
- (Clit 7.18 i igaiican: it goes under the aime of Hilbrt’s theorem 90, bocause in the case fn which G is the Galois group of a finite cychic extension, ft cause in the case fn which G is the Galois group o Cecovers precisely the classical at with this name. (Paolo Aluffi 680)
- (Gust as the elements of MC are called invariants, elements of Mg are called eoinvariants of the action). (Paolo Aluffi 680)
- The upshot is that in order to verify that a given collection of functors agrees with the derived functors of & given (say) let-exact hctor,itsuffices to verifythat they formacobomological functor aud tha theeffcosbilty condition bods. This they formacobomological functor aud tha theeffcosbilty condition bod in ed, for example,to obtain concrete realizations ofsheaf cobomology. (Paolo Aluffi 683)
- At this point we know how to construct lef-derived functors Li : A — B of in two words, Li(M)is computed by applying to a projective resolution of M in two words, Li(M)is computed by applying aud taking (co}homology ofthe resulting complex. (Paolo Aluffi 684)
- This says that projective objects are ‘acyclic’with respect to lefl-derived funtors: (Paolo Aluffi 684)
- Defition 8.1. Let # be an additive functor, An (srt. efdeived fonctors) if LF(M) = 0 for § #0. (Paolo Aluffi 684)
- An object M of A is acyclic Defition 8.1. Let # be an additive functor, (srt. efdeived fonctors) if LF(M) = 0 for § # (Paolo Aluffi 684)
- Example 8.2. Let R be a commutative ring. Recall (Definition VIIL2.13) that an Bode M is flat If© M is exact, or equivalently (by the symmetey of ©) if Bode M i M@n_is fat, (Paolo Aluffi 684)
- flat If© M is exact, or equivalently (by the symmetey of ©) if Flat modules ace acyclic with respect t tensor products, in very M@n_is fat, Flat mo strong sense: if Ns fat (Paolo Aluffi 684)
- then Tor(M, ¥) = 0 for i # 0 and all M; betier—proe it anew. Further, ine ( (Paolo Aluffi 684)
- a fat wiodule is acyclic for every functor defined aud all modules N. Thus, a fat wiodule is acyclic for every functor defined by _@p N, for all modules N. . (Paolo Aluffi 684)
- Here is the punch line. In §VIILG.4 we bad claimed that flaresolutions could be sed in place of fre otprojective resolutions,in order to compute Tor. We are Bog to verfy that Facyclic resolutions suffice in order to compule LiF. (Paolo Aluffi 684)
- Theorem 8.3 raises an interesting possibilty: since S-seyclic objcts suffice ia order to compute the derived functors of 5 (at lens, hen i right-exact)the order to compute the derived functors of 5 (at lens, hen i right-exact)the ceader con imaginethat there may be situations in which A docs not have enough ceader con imaginethat there may be situations in which A docs not have enough projecives, and yet lef-derved functors of » functor may be deedbecarse A projecives, and yet lef-derved functors of » functor may be deedbecars bas cuough -acyclc objects (in @ suitable eave). This i indood the case. (Paolo Aluffi 686)
- There i an alternative viewpoint on the questionaddressed by Theorens 8.3, 4° be an -acyclic rescltionof an object M, aud let Fy be projective Let 4° be an -acyclic rescltionof an object M, aud let Fy be projective ceolution. We will apply C(#) and place the resulting complexes as sides of an ceolut array: (Paolo Aluffi 686)
- gives the same result as taking cohomology oftherightmost column. Might there Botbe a way to ‘interpolate’between these two cohomologies, by cleverly fil ing in the dotted portion of this diagram? Double complezes may be wed to this effec (Paolo Aluffi 686)
- Definition 8.6. The total complez TC(M)" ofa double complex M** is defined by setting TC(M)" i= @,0, M", with diffcrentils (writen stenographically) do dy +4, (Paolo Aluffi 690)
- Claim 8.11. The total complex of MY" is the mapping cone of TC(a)*: TC(My)* MC(TC(a))". (Paolo Aluffi 696)
- auaple 7.6. Theoret 8.13 wakes good to show that the Tor functors could be on our old promise (aso made in §VITL2.4) to show that the compted by resolving the secon factor rather than the fr. (Paolo Aluffi 698)
- the localization of C may be defined by. of the ‘factions’in the tell of fractions;the localization of C may be defined by. taking the same objects as ia C and setting morphisms in the new category to be. compositions of roofs’, up to suitable equivalence relation (Paolo Aluffi 703)
- ses of) morphisms ofcocbain complexes, modulo suitable equivNote that the construction is applied to K(A), rather thau the lence relation. Note that the construction is applied to K(A), rather thau the simplerminded C(A): forone thing, one might ts wel start, rom K(A),since the functor C(A) — D(A) will have to factor through the homotopic category (Propofunctor C(A) — D(A) will have to factor through the homotopic category (Proposition 5.4). In any case it just so happens that, unfortunately, quasisomorphisms sition 5.4). In any case it just so happens that, unfortunately, quasisomorphisms do notform a localizing class of morphisms in C(A), while their homotapy classes do notform a localizi ore localziog K(A). (Paolo Aluffi 704)
- notion of triangulated category. The same applies fact to homotopic categories of complexes. We have poiated out in §5.2that these categories should not be expected to he (aad indeed are not)abelian, ht. they preserve enongh structure expected to he (aad indeed are not)abelian, ht. they preserve to make sense of, for example,long ext cohomology sequences (Paolo Aluffi 705)
- Theesential Ingredientsofatriangulated (additive) category are ‘translation Theesential Ingredientsofatriangulated (additive) category are ‘translation functor, which the cas of K(A)or D(A) srealized the shit functor L* + L{1)% functor, which the ca bein (Paolo Aluffi 705)
- There is more, including an fufasmous octahedral aziom (so named since a popular way to state It invokes an octahedral diagram). (Paolo Aluffi 705)
- That i, the third vertex” of a triangle with an assigned side a® L* —» M* is the ‘mapping cone of a*, and the unlabeled sides are the natural cochain morphisms ‘mapping cone of a*, and the unlabeled sides are M® = MC(a)*, MC(a)e — LILJ* studied in §4.1. (Paolo Aluffi 706)
- The category K(A) i triangulated: the translation functor is the basic shift of complex,and distinguished triangles are those isomorphic to (Paolo Aluffi 706)
- Applying the cohomology functor to a distinguished triangle in K(A) yields an exact triangle: i (Paolo Aluffi 706)
- . In general, cohomalogicalfunctor on » triangulated obtained in Proposition 4.1. In general, cohomalogicalfunctor on » triangulated category bs an additive functor to an abelian category, mapping distinguished trianeles to cxact triangles and hence inducing long exact sequences. (Paolo Aluffi 707)
- cohomology The functors 5 0 cohomological functor on K(A);this will not surprise onreader. The functors How(A,) and How(_, A)(to Ab)are cobomological functors ou every trisngulated How(A,) and How(_, A)(t category,for every object A. (Paolo Aluffi 707)
- Tiomorphisn in the derived category Is & les stringent notion than in the homotopic category, % there are ‘more’ distinguished triangles in D(A)than in homotopic category, K(A). Forexample, if opr pe Eo 0 hort exact sequence,a above, there i iu geueral wo distinguished triangle or or ZN Noh in K(A) (for any choice of +): indeed, N* need not he homotopically equivalent in K(A) (for any choice of +): indeed, N* need to MC(a)*. But wich tringles do exist in D(A)! (Paolo Aluffi 707)
- Given that we were thinking about triangles a moment ago, the following construction may be helpful in appreciating the notion of spectral vequeace. Let A Fes be an exact trisagle (Paolo Aluffi 709)
- Claim 9.1. The new triangle a 7 7 ry is again exact. (Paolo Aluffi 709)
- The reader will bave uo difculty proving this statement. (Bxercie 9.7). “The datum ofan exact triangle a above i called an czact couple which we find “The datum ofan exact triangle a above i called an czact couple which we find confusing since triangle hs three vertices (bu it is true thatonly two objects are confusing since triangle hs three vertices (bu it is true thatonly two objects are involved here);the new exact triaagle (couple) obtained ia Claim 9.1 ithe derived involved here);the new exact triaagle (couple) obtained ia Claim 9.1 ithe derived ‘couple, which we find even more confusing since there is no derived category or ‘couple, which w functor in sight. (Paolo Aluffi 709)
- Definition 0.2. A spectral sequence {(Ei,d:)},=1.2... 1 a sequence of objects Ey and morplisnis d, : E — Ey in an abelian category, such that dyed; 0 and Eig 2 kerd,/imd, A Thu, exact couple are way to produce spectral seuences. There sa ese Thu, exact couple are way to produce spectral s in which we can tu the crank ‘nftly many mes’ (Paolo Aluffi 710)
- that E; 2 Z;/B;, and let Ziyy kerd;, Bigs = imdy; define Zi41, Biss 85 the comepondio subobfecta of. Then o Zr o Zinn o Zr o Zinn Eng ® Br realizing Ey as a subquotient®*of E), and BC CBCBuC CZnCZLC Ch B-0, Define! Bu=UB Zuoi=()ZEw “This “ulinate’ subquotent E. i the limit of the spectral sequence; it i conmon “This “ulinate’ subquotent E. i the limit of the spectral sequence; it i conm to ay that the spectralsequence E, abu to Eee. By definition, if d, = drs to ay that the spectralsequence E, abu to Eee. By definition, if d, = drs +220, thet Zuo = Z, and Bu = By, 50 that Ev Ey; n this case we sa that +220, thet Zuo = Z, and the sequence collapses at Ey. (Paolo Aluffi 710)
- orking category R-Mod of modu invoking the FreydMitchell theorem) (Paolo Aluffi 710)
- too uncommon (Paolo Aluffi 710)
- Where is my double complex? We promised that spectral sequences could be used to ‘compute’ the cohomology of the total complex of a double complex. We will now see how a double complex gives rise to au exact couple and hence to a spectral sequence. This is a particular case of a useful mechanism producing an spectral sequence. This is a pa exact sequence from a filtration. A ‘descending titration’of M consists of a vequence of subobjects: M202 Ma 2 Mui2 (Paolo Aluffi 711)
- tons of G. If M is an object of an abelian category, a tration as above determines an associated graded object Ma pay wr) i= DuMm, ( (Paolo Aluffi 711)
- “The filtration on T* also determines tration on the cohomology of T%: we can take H*(T”)m to be the image in A(T“) of H(T3). Thus, we sso hove can take H*(T”)m to be the image in A(T“) of H(T3). Thus, we sso hove a graded object gr, H(T”). The relation between H(gri() and gr, H*(T“) is a graded object gr, H(T”). The relation between H(gri() and gr subi: this relation is what spectral sequences will help us understand. (Paolo Aluffi 712)
- The monomorphisms T5,,, € T3, define 8 monomorphism @,,T5, — @pTae ‘The monomorphisms T5,,, € T3, define 8 monomorphi of which gr3(T) is the cokernel: we have an exact sequence [C} 0— Dn Tn — Dp Tr — BT) —0 (Paolo Aluffi 712)
- Definition 9.3. The spectral sequence ofthe double complez M** (with respect to, the vertical filtration) is the spectral sequence determined by this exact couple. (Paolo Aluffi 713)
- s, ‘turning the crank’ moves the gr from inside the cohomology The limit ,E,, does not quite compute the cohomology of the 10 outside of It. The limit ,E,, does not quite compute the cohomology of the total complex, as we glibly announced In §8.3, butit computes the graded object total complex, as we glibly announced In §8.3, butit computes the graded object determined by filtration on the cohomology of the total complex, and this i good enough for many applications. (Paolo Aluffi 713)
- Theorem9.5. Let M** be a double complex in an abelian category. Assume that ‘Theorem9.5. Let M** be a double complex in an abelian category. Assume that MY fori <0, j <0, and let T* be the total complez of M**. Then, with notation as above, there czists a spectral sequence {(uE;udi)such that Er HUT), JE gr, HHT). (Paolo Aluffi 713)
- It follows that the sequence collapses tral sequence are 0 by degree considerations. It follows that the sequence collapses at,Ey,and by Theorem 9.5 this says that g,(H*(T))is isomorphic to the cohoat,Ey,and by Theorem 9.5 this says that g,(H*(T))is isomorphic to the cohoology of (*). In this case weclearlyhavegr, (H¥(T*)) & H*(T). so theconclusion ology of (*). In this case weclearlyhavegr, (H¥(T*)) & H*(T). so theconclusion reproduces the corresponding statementin Theorem 8.12. This what is meant reproduces the corresponding statementin Theorem 8.12. This by the incantation by an immediatespectral sequence argument’ (Paolo Aluffi 715)
- OFcourse, spectral sequences ae not limited to these simple applications. The Grothendieck spectral sequence ‘computes’ the derived functor of the composition Grothendieck spectral sequence ‘computes’ the derived functor of the composition of two functors: for example, if + A — B and # : B — C are two rightexact of two functors: for example, if + A — B and # : B — C are two rightexact functors and sends projectivs to projectives,then thre is a spectral sequence functors and sends projectivs to projectives,then thre is a spectral sequenc whose E; term callcts the compasitons ,#oL, ad that abuts tLy(#0.5) (Paolo Aluffi 717)
- For instance,if f : R — § is a homomorphism of commutative rings (so that S may be viewed as an R-module), A is an R-module, and B is an S-module (and hence an R-module,via f), then there is a ‘change-of-ring spectral sequence’ TorS(Tork(4,5), B) = Tork(A,B). In topology,the Serre spectral sequence can TorS(Tork(4,5), B) = Tork(A,B). In topology,the Serre spectral sequence can be use compute the homology of a bration in terms ofthe homology ofth base and of the fiber. (Paolo Aluffi 717)
- ctral sequences: the st of mathematicians X such spectrl sequence includes (but is not limited to) Adams, Atiyah, Barratt,Bloch, Bockstein, Bousfield, Cartan, Connes,Eilenberg, Federer, Frélicher, Green, Grothendieck, Hirzebruch, Hochschild, Hodge, Hurewicz, Federer, Frélicher, Green, Grothendieck, Hirzebruch, Hochschild, Hodge, Hurewicz, van Kampen, Kan, Kunaeth, Lerey, Lichtenbaum, Lyndon, May, Miller, Milnor, Moore, Novikov, de Rham, Quillen, Rothenberg, Serre, Steenrod, (Paolo Aluffi 717)

Last Annotation: 04/11/2019

- Cohomotopy groups? (J. P. May & K. Ponto 49)
- Key application of localization (J. P. May & K. Ponto 112)
- Write out the diagram for this, it’s like a section (J. P. May & K. Ponto 122)
- Example of proof in group theory using topological lift (J. P. May & K. Ponto 187)
- Big idea: replace zero in exact sequences with contractible spaces (J. P. May & K. Ponto 293)
- Interpret resolutions as cofibrant replacements (J. P. May & K. Ponto 303)
- Tor and Ext as homotopy groups of spheres (J. P. May & K. Ponto 305)
- Big results!! (J. P. May & K. Ponto 337)
- Proof of Bott Periodicity (J. P. May & K. Ponto 348)
- Spectral sequence of a filtered complex (J. P. May & K. Ponto 376)

- This material might include the fundamental group, covering spaces, ordinary homology and cohomology in its singular, cellular, axiomatic, and represented versions, higher homotopy groups and the Hurewicz theorem, basic homotopy theory including fibrations and cofibrations, Poincaré duality for manifolds and manifolds with boundary. The rudiments should also include a reasonable amount of categorical language and at least enough homological algebra for the universal coefficient and Künneth theorems (J. P. May & K. Ponto 7)
- . What next? Possibly K-theory, which is treated in [3] and, briefly, [89], and some idea of cobordism theory [34, 89]. (J. P. May & K. Ponto 7)
- A partial list of areas a student should learn is given in the suggestions for further reading of [89], and a helpful guide to further development of the subject (with few proofs) has been given by Selick [120]. (J. P. May & K. Ponto 7)
- the fundamental importance of the material is nowhere greater than in the theory of localization and completion of topological spaces 1 (J. P. May & K. Ponto 7)
- . Experts know that these constructions can be found in such basic 1970’s references as [20, 60, 129]. (J. P. May & K. Ponto 7)
- The notion of completion is particularly important because it relates directly to mod p cohomology, which is the invariant that algebraic topologists most frequently compute (J. P. May & K. Ponto 7)
- some standard topics, such as fibration and cofibration sequences, Postnikov towers, and homotopy limits and colimits, (J. P. May & K. Ponto 7)
- The only other preliminary that we require and that cannot be found in most of the books cited above is the Serre spectral sequence. There are several accessible sources for that, such as [32, 76, 95, 120], but to help make this book more self-contained, we shall give a concise primer on spectral sequences in Chapter 24 (J. P. May & K. Ponto 8)
- By far the longer of these parts is an introduction to model category theory. This material can easily be overemphasized, to the detriment of concrete results and the nuances needed to prove them. For example, its use would in no way simplify anything in the first half of the book. However, its use allows us to complete the first half by giving a conceptual construction and characterization of localizations and completions of general, not necessarily nilpotent, spaces. (J. P. May & K. Ponto 8)
- Anybody interested in any of these fields needs to know model category theory. It plays a role in homotopical algebra analogous to the role played by category theory in mathematics (J. P. May & K. Ponto 8)
- It develops the basic theory of bialgebras and Hopf algebras. (J. P. May & K. Ponto 8)
- The most important of these is Bousfield localization, which we shall construct model theoretically (J. P. May & K. Ponto 9)
- we can dualize the proof of the first theorem below (as given for example in [89]) to prove the second. Theorem 0.0.1. A weak homotopy equivalence e : Y −→ Z between CW complexes is a homotopy equivalence. Theorem 0.0.2. An integral homology isomorphism e : Y −→ Z between simple spaces is a weak homotopy equivalence. (J. P. May & K. Ponto 10)
- The argument is based on the dualization of cell complexes to cocell complexes, of which Postnikov towers are examples, and of the Homotopy Extension and Lifting Property (HELP) to co-HELP. Once this dualization is understood, it becomes almost transparent how one can construct and study the localizations and completions of nilpotent spaces simply by inductively localizing or completing their Postnikov towers one cocell at a time (J. P. May & K. Ponto 10)
- For conceptual clarity, we offer a slight reformulation of the original definition of a model category that focuses on weak factorization systems (WFS’s): a model category consists of a subcategory of weak equivalences together with a pair of related WFS’ (J. P. May & K. Ponto 10)
- We argue that classical algebraic topology, over at least the mathematical lifetime of the first author, has implicitly worked in the m-model structure (J. P. May & K. Ponto 11)
- Colocalization is dual to Bousfield localization, and this brings us to another reason for our introduction to model category theory, namely a perceived need for as simple and accessible an approach to Bousfield localization as possible. This is such a centrally important tool in modern algebraic topology (and algebraic geometry) that every student should see it (J. P. May & K. Ponto 11)
- In algebraic topology, algebras are always graded and often connected, meaning that they are zero in negative degrees and the ground field in degree zero. (J. P. May & K. Ponto 12)
- Under this assumption, bialgebras automatically have antipodes, so that there is no distinction between Hopf algebras and bialgebras. (J. P. May & K. Ponto 12)
- In Chapter 22, we explain the Hopf algebra proof of Thom’s calculation of the real cobordism ring and describe how the method applies to other unoriented cobordism theories. We also give the elementary calculational proof of complex Bott periodicity (J. P. May & K. Ponto 12)
- Chapter 9, the structure theory for rational Hopf algebras is used to describe the category of rational H-spaces and to explain how this information is used to study H-spaces in general. In Chapter 22, we explain the Hopf algebra proof of Thom’s calculation of the real cobordism ring and describe how the method applies to other (J. P. May & K. Ponto 12)
- . It is by now a standard convention in algebraic topology that spaces mean compactly generated spaces (J. P. May & K. Ponto 13)
- we implicitly restrict to spaces of the homotopy types of CW complexes whenever we talk about passage to homotopy. (J. P. May & K. Ponto 13)
- This allows us to define the homotopy category HoU simply by identifying homotopic maps; it is equivalent to the homotopy category of all spaces in U , not necessarily CW homotopy types, that is formed by formally inverting the weak homotopy equivalences. (J. P. May & K. Ponto 13)
- meaning that the inclusion ∗ −→ X is a cofibration (J. P. May & K. Ponto 13)
- The conventions mean that, when passing to homotopy categories, we implicitly approximate all spaces by weakly homotopy equivalent CW complexes, as we can do by [89, §10.3]. In particular, when we use Postnikov towers and pass to limits, which are not of the homotopy types of CW complexes, we shall implicitly approximate them by CW complexes (J. P. May & K. Ponto 13)
- The smash product X ∧ Y of based spaces X and Y is the quotient of the product X × Y by the wedge (or one-point union) X ∨ Y . We have adjunction homeomorphisms F (X ∧ Y, Z) ∼ = ∼ = F (X, F (Y, Z)) and consequent bijections [X ∧ Y, Z] ∼ = ∼ = [X, F (Y, Z)]. (J. P. May & K. Ponto 14)
- Let ZT denote the ring of integers localized at T , that is, the subring of Q consisting of rationals expressible as fractions k/ℓ, where ℓ is a product of primes not in T . We let Z[T −1 ] denote the subring of fractions k/ℓ, where ℓ is a product of primes in T . In particular, Z[p −1 ] has only p inverted. Let Z(p) denote the ring of integers localized at the prime ideal (p) or, equivalently, at the singleton set {p}. (J. P. May & K. Ponto 14)
- Let Zp denote the ring of p-adic integers. Illogically, but to avoid conflict of notation, we write Ẑ ẐT for the product over p ∈ T of the rings Zp . We then write Q Q̂ Q̂T for the ring Ẑ ẐT ⊗Q; when T = {p}, this is the ring of p-adic numbers. (J. P. May & K. Ponto 14)
- We write A(p) and ̂ Âp for the localization at p and the p-adic completion of an abelian group A (J. P. May & K. Ponto 15)
- In both the algebraic and topological literature, the ring Zp is sometimes denoted Z Ẑ Ẑp ; we would prefer that notation as a matter of logic, but the notation Zp has by now become quite standard. (J. P. May & K. Ponto 15)
- With k = Fp , such graded fields appear naturally in algebraic topology as the coefficients of certain generalized cohomology theories, called Morava Ktheories, and their homological algebra works exactly as for any other field, a fact that has real calculational applications (J. P. May & K. Ponto 15)
- consider a Laurent series algebra k[x, x −1 ] over a field k, where x has positive even degree. To an algebraic topologist, this is a perfectly good graded field: every non-zero element is a unit. To an algebraist, it is not (J. P. May & K. Ponto 15)
- That is, algebraic topologists do not usually allow the addition of elements of different degrees (J. P. May & K. Ponto 15)
- In categorical language, the category Ab∗ of graded abelian groups is a symmetric monoidal category under ⊗, meaning that ⊗ is unital (with unit Z concentrated in degree 0), associative, and commutative up to coherent natural isomorphisms (J. P. May & K. Ponto 15)
- For an algebraic topologist, the commutativity isomorphism γ : A ⊗ B −→ B ⊗ A is specified by γ(a ⊗ b) = (−1) pq b ⊗ a where deg(a) = p and deg(b) = q. A graded k-algebra with product φ is commutative if φ ◦ γ = φ; elementwise, this means that ab = (−1)pq ba (J. P. May & K. Ponto 15)
- but in algebraic topology this notion of commutativity is and always has been the default (at least since the early 1960’s). (J. P. May & K. Ponto 16)
- To an algebraic topologist, a polynomial algebra k[x] where x has odd degree is not a commutative k-algebra unless k has characteristic 2. The homology H∗ (ΩS n ; k), n even, is an example of such a non-commutative algebra. (J. P. May & K. Ponto 16)
- We shall make constant use of the theory of fibration and cofibration sequences (J. P. May & K. Ponto 19)
- a map i is a (based) cofibration if there is a lift λ in all such diagrams in which p is the map p0 : F (I+ , Y ) −→ Y given by evaluation at 0 for some space Y . This is a restatement of the homotopy extension property, or HEP (J. P. May & K. Ponto 19)
- ], a map p is a (based) fibration if there is a lift λ in all such diagrams in which i is the inclusion i0 : Y −→ Y ∧ I+ of the base of the cylinder. This is the covering homotopy property, or CHP (J. P. May & K. Ponto 19)
- One can think of model category theory as, in part, a codification of the notion of duality, called Eckmann-Hilton duality, that is displayed in the definitions of cofibrations and fibrations (J. P. May & K. Ponto 20)
- Lemma 1.1.4. Let i : A −→ X be a cofibration and Y be a space. Then the induced map i ∗ : F (X, Y ) −→ F (A, Y ) is a fibration and the fiber over the basepoint is F (X/A, Y (J. P. May & K. Ponto 20)
- how to use the defining lifting properties to construct new cofibrations and fibrations from given ones (J. P. May & K. Ponto 20)
- The homotopy cofiber Cf of f is the pushout Y ∪f CX of f and i0 : X −→ CX. Here the cone CX is X ∧ I, (J. P. May & K. Ponto 21)
- The homotopy fiber F f of f is the pullback X ×f P Y of f and p1 : P Y −→ Y . Here the path space P Y is F (I, Y ) (J. P. May & K. Ponto 21)
- When f : X −→ Y is a cofibration, the cofiber is canonically equivalent to the quotient Y /X (J. P. May & K. Ponto 21)
- when the given based map is a fibration, in which case the actual fiber f −1 (∗) and the homotopy fiber are canonically equivalent (J. P. May & K. Ponto 21)
- Lemma 1.1.6. Let f : X −→ Y be a map and Z be a space. Then the homotopy fiber F f ∗ of the induced map of function spaces f ∗ : F (Y, Z) −→ F (X, Z) is homeomorphic to F (Cf, Z), where Cf is the homotopy cofiber of f . (J. P. May & K. Ponto 21)
- The fiber F f ∗ is F (Y, Z) ×F (X,Z) P F (X, Z (J. P. May & K. Ponto 22)
- Clearly P F (X, Z) is homeomorphic to F (CX, Z) (J. P. May & K. Ponto 22)
- Since the functor F (−, Z) converts pushouts to pullbacks, (J. P. May & K. Ponto 22)
- the left square commutes up to homotopy (J. P. May & K. Ponto 22)
- the cofiber sequence construction gives a functor from the category of maps and commutative squares to the category of sequences of spaces and commutative ladders between them. (J. P. May & K. Ponto 22)
- Addendum 1.2.2. If X = X ′ , α is the identity map, the left square commutes, and β is a cofibration, then the canonical map γ : Cf −→ Cf ′ is a cofibration (J. P. May & K. Ponto 22)
- Homotopies between maps of unbased spaces, or homotopies between based maps that are not required to satisfy ht (∗) = ∗, are often called free homotopies (J. P. May & K. Ponto 25)
- Recall that a homotopy h : X × I −→ X is said to be a deformation if h0 is the identity map of X. (J. P. May & K. Ponto 25)
- The reduced mapping cylinder M i is obtained from M + i by collapsing the line {∗} × I to a point (J. P. May & K. Ponto 27)
- It is clear that two based maps that are in the same orbit under the action of π1 (Y, ∗) are freely homotopic. (J. P. May & K. Ponto 28)
- An H-space is a based space Y with a product µ : Y × Y −→ Y , written x · y or by juxtaposition, whose basepoint is a two-sided unit up to based homotopy. (J. P. May & K. Ponto 29)
- That is, the maps y 7→ ∗ · y and y 7→ y · ∗ are both homotopic to the identity map. (J. P. May & K. Ponto 29)
- Proposition 1.4.3. For an H-space Y , the action of π1 (Y, ∗) on [X, Y ] is trivial and therefore [X, Y ] ∼ = ∼ = [X, Y ]free . (J. P. May & K. Ponto 29)
- Definition 1.4.4. Take X = S n in Definition 1.4.1. The definition then specializes to define an action of the group π1 (Y, ∗) on the group πn (Y, ∗). When n = 1, this is the conjugation action of π1 (Y, ∗) on itself. A (connected) space Y is simple if π1 (Y, ∗) is abelian and acts trivially on πn (Y, ∗) for all n ≥ 2. (J. P. May & K. Ponto 29)
- Corollary 1.4.5. Any H-space is a simple space. (J. P. May & K. Ponto 29)
- As observed there, the sets [X, Y ] are groups if X is a suspension or Y is a loop space and are abelian groups if X is a double suspension or Y is a double loop space. However, there is additional structure at the ends of these sequences that will play a role in our work. (J. P. May & K. Ponto 29)
- This is of greatest interest when Z = S 0 . Since [−, −] refers to based homotopy classes, [S 0 , X] = π0 (X) and the sequence becomes π1 (X) f ∗ / π1 (Y ) ι ∗ / π0 (F f ) p ∗ / π0 (X) f ∗ / π0 (Y ). (J. P. May & K. Ponto 30)
- Observe that a fibration p : E −→ B need not be surjective,3 but either every point or no point of each component of B is in the image of E. (J. P. May & K. Ponto 31)
- This specializes to give a group homomorphism π1 (B, b) −→ π0 (Aut(Fb )). Here Aut(Fb ) is the topological monoid of (unbased) homotopy equivalences of Fb . We think of π1 (B, b) as acting “up to homotopy” on the space Fb , meaning that an element β ∈ π1 (B, b) determines a well-defined homotopy class of homotopy equivalences Fb −→ Fb (J. P. May & K. Ponto 31)
- (If p is a covering space, the action is by homeomorphisms, as in [89, p. 29].) As we shall use later, it follows that π1 (B) acts on the homology and cohomology groups of Fb . (J. P. May & K. Ponto 31)
- Intuitively, homotopy colimits are constructed from ordinary categorical colimits by gluing in cylinders so as to give domains for homotopies that allow us to replace equalities between maps that appear in the specification of ordinary colimits by homotopies between maps (J. P. May & K. Ponto 35)
- Since homotopies between given maps are not unique, not even up to homotopy, homotopy colimits give weak colimits in the homotopy category, in the sense that they satisfy the existence but not the uniqueness property of ordinary colimits (J. P. May & K. Ponto 35)
- We shall spell out the relevant algebraic property quite precisely for homotopy pushouts (or double mapping cylinders), homotopy coequalizers (or mapping tori), and sequential homotopy colimits (or telescopes) (J. P. May & K. Ponto 35)
- Definition 2.1.1. The homotopy pushout (or double mapping cylinder) M (f, g) of a pair of maps f : A −→ X and g : A −→ Y is the pushout written in alternative notations as (J. P. May & K. Ponto 35)
- X ∪f (A ∧ I+ ) ∪g Y. (J. P. May & K. Ponto 35)
- The following result is often called the “gluing lemma”. (J. P. May & K. Ponto 37)
- Corollary 2.1.4. If f is a cofibration and g is any map, then the natural quotient map M (f, g) −→ X ∪A Y is a homotopy equivalence. (J. P. May & K. Ponto 37)
- Definition 2.1.5. The homotopy coequalizer (or mapping torus) T (f, f ′ ) of a pair of maps f, f ′ : X −→ Y is the homotopy pushout of (f, f ′ ) : X ∨ X −→ Y and ∇ : X ∨ X −→ X. (J. P. May & K. Ponto 37)
- Since cofibrations must be inclusions (J. P. May & K. Ponto 38)
- The equalizer E(α, β) of functions α, β : S −→ U is {s|α(s) = β(s)} ⊂ S, as we see by checking the universal property ([89, p. 16]). Equivalently, it is the pullback of (α, β) : S −→ U × U and ∆ : U −→ U × U . (J. P. May & K. Ponto 38)
- Definition 2.1.7. The homotopy colimit (or mapping telescope) tel Xi of a sequence of maps fi : Xi −→ Xi+1 is the homotopy coequalizer of the identity map of Y = ∨i Xi and ∨i fi : Y −→ Y . It is homeomorphic to the union of mapping cylinders described in [89, p. 113] (J. P. May & K. Ponto 38)
- The definition of lim1 Gi for an inverse sequence of abelian groups is given, for example, in [89, p. 146]. It generalizes to give a definition for not necessarily abelian groups. However, the result is only a set in general, not a group. Definition 2.1.8. Let γi : Gi+1 −→ Gi , i ≥ 0, be homomorphisms of groups. Define a right action of the group G = ×i Gi on the set S = ×i Gi by (si ) ∗ (gi ) = (g −1 si γi (gi+1 )). The set of orbits of S under this action is called lim1 Gi (J. P. May & K. Ponto 38)
- Observe that lim Gi is the set of elements of G that fix the element (1) ∈ S whose coordinates are the identity elements of the Gi . Equivalently, lim Gi is the equalizer of the identity map of S and ×i γi : S −→ S. (J. P. May & K. Ponto 38)
- The following “ladder lemma” is analogous to the gluing lemma above. Lemma 2.1.10. Assume given a commutative diagram X0 f0 / α0 X1 / α1 ··· / Xi fi / αi Xi+1 / αi+1 ··· X ′ 0′ f0′ 0′ / X′ 1 1 ··· / X′ i i fi′ i′ ′ / Xi i+1 / ··· in which the fi and f ′ i′ are cofibrations. If the maps αi are homotopy equivalences, then so is their colimit colim Xi −→ colim X ′ i′ . (J. P. May & K. Ponto 39)
- Corollary 2.1.11. If the maps fi : Xi −→ Xi+1 are cofibrations, then the natural quotient map tel Xi −→ colim Xi is a homotopy equivalence. (J. P. May & K. Ponto 39)
- Definition 2.1.12. Let X be a (based) CW complex with n-skeleton X n . A map f : X −→ Z is called a phantom map if the restriction of f to X n is null homotopic for all n. (J. P. May & K. Ponto 39)
- Warning 2.1.13. This is the original use of the term “phantom map”, but the name is also used in some, but by no means all, of the more recent literature for maps f : X −→ Z such f ◦ g is null homotopic for all maps g : W −→ X, where W is a finite CW complex (J. P. May & K. Ponto 39)
- Of course, the two notions agree when X has finite skeleta (J. P. May & K. Ponto 39)
- The name comes from the fact that, with either definition, a phantom map f : X −→ Z induces the zero map on all homotopy, homology, and cohomology groups, since these invariants depend only on skeleta, and in fact only on composites f ◦ g, where g has finite domain (J. P. May & K. Ponto 39)
- As a concrete example, this result applies to show that [CP ∞ , S 3 ] contains an uncountable divisible subgroup (J. P. May & K. Ponto 40)
- Definition 2.2.1. The homotopy pullback (or double mapping path fibration) N (f, g) of a pair of maps f : X −→ A and g : Y −→ A is the pullback written in alternative notations as (J. P. May & K. Ponto 40)
- X ×f F (I+ , A) ×g Y. (J. P. May & K. Ponto 40)
- N (f, g) is the subspace of X×F (I+ , A)× Y that consists of those points (x, ω, y) such that ω(0) = f (x) and ω(1) = g(y) (J. P. May & K. Ponto 40)
- Corollary 2.2.3. Let f : X −→ A and g : Y −→ A be maps between connected spaces. There is a long exact sequence · · · −→ πn+1 (A) −→ πn N (f, g) (p0 ,p1 )∗ −−−−−→ πn (X) × πn (Y ) f ∗ −g f∗ −g∗ −−−−→ πn (A) −→ · · · −→ π1 (A) −→ π0 N (f, g) −→ ∗. (J. P. May & K. Ponto 41)
- Corollary 2.2.5. If f is a fibration and g is any map, then the natural injection X ×A Y −→ N (f, g) is a homotopy equivalence. (J. P. May & K. Ponto 41)
- Definition 2.2.8. The homotopy limit (or mapping microscope) 1 mic Xi of a sequence of maps fi : Xi+1 −→ Xi is the homotopy equalizer of the identity map of Y = ×i Xi and ×i fi : Y −→ Y . (J. P. May & K. Ponto 42)
- Lemma 2.2.10. Assume given a commutative diagram ··· / Xi+1 fi αi+1 Xi / αi ··· / X1 f0 / α1 X0 α0 ··· / Xi+1 fi Xi / ··· / X1 f0 / X0 in which the fi and f ′ i′ are fibrations. If the αi are homotopy equivalences, then so is their limit lim Xi −→ lim X ′ i′ . (J. P. May & K. Ponto 42)
- Proposition 2.2.11. If the maps fi : Xi+1 −→ Xi are fibrations, then the natural injection lim Xi −→ mic Xi is a homotopy equivalence (J. P. May & K. Ponto 43)
- The lim1 terms that appear in Propositions 2.1.9 and 2.2.9 are an essential, but inconvenient, part of algebraic topology. In practice, they are of little significance in most concrete applications, the principle reason being that they generally vanish on passage either to rationalization or to completion at any prime p, (J. P. May & K. Ponto 43)
- but only the Mittag–Leffler condition for the vanishing of lim 1 Gi will be relevant (J. P. May & K. Ponto 43)
- We consider a sequence of homomorphisms γi : Gi+1 −→ Gi . For j > i, let γ = γi γi+1 · · · γj−1 : Gj −→ Gi and let Gj = im γ . We say that the sequence {Gi , γi } satisfies the Mittag–Leffler condition if for each i there exists j(i) such that Gki i = G j(i) i for all k > j(i). That is, these sequences of images eventually stabilize. For example, this condition clearly holds if each γi is an epimorphism or if each Gi is a finite group (J. P. May & K. Ponto 43)
- The main conclusion is that either the Mittag–Leffler condition holds and lim1 Gi = 0 or, under further hypotheses that usually hold in the situations encountered in algebraic topology, the Mittag–Leffler condition fails and lim1 Gi is uncountable (J. P. May & K. Ponto 43)
- Since the functor [−, Z] converts finite wedges to finite products and thus to finite direct sums when it takes values in abelian groups (J. P. May & K. Ponto 46)
- Lemma 2.4.2. With q = 2, CP ∞ satisfies the hypotheses of Lemma 2.4.1 (J. P. May & K. Ponto 46)
- But the first Steenrod operation P 1 (see e.g. [94, 126]) satisfies P 1 x 6= 0, which by naturality contradicts P 1 (i) = 0. (J. P. May & K. Ponto 46)
- Corollary 2.4.3. There are uncountably many phantom maps CP ∞ −→ S 3 . (J. P. May & K. Ponto 46)
- It was observed in [89, p. 113], that homology commutes with sequential colimits of inclusions. The same holds more generally for suitably well-behaved filtered colimits (J. P. May & K. Ponto 47)
- Definition 2.5.1. A small category D is filtered if (i) For any two objects d and d ′ , there is an object e which admits morphisms d −→ e and d′ −→ e. (ii) For any two morphisms α, β : d −→ e, there is a morphism γ : e −→ f such that γα = γβ. (J. P. May & K. Ponto 47)
- Definition 2.5.2. A cardinal is an ordinal that is minimal among those of the same cardinality. A cardinal λ is regular if for every set I of cardinality less than λ and every set {Si |i ∈ I} of sets Si , each of cardinality less than λ, the cardinality of the union of the Si is less than λ. (J. P. May & K. Ponto 47)
- Lemma 2.5.7. A q-equivalence f : X −→ Y induces isomorphisms on homology and cohomology groups in dimensions less than q. (J. P. May & K. Ponto 48)
- Proof. Using mapping cylinders, we can replace f by a cofibration [89, p. 42]. Since weak equivalences induce isomorphisms on homology and cohomology, relative CW approximation [89, p. 76] and cellular approximation of maps [89, p. 74] show that we can replace X and Y by CW complexes with the same q-skeleton and can replace f by a cellular map that is the identity on the q-skeleton. Since in dimensions less than q the cellular chains and thus the homology and cohomology groups of a CW complex depend only on its q-skeleton, the conclusion follows. (J. P. May & K. Ponto 49)
- Proposition 2.5.9. Let X = lim Xn , where {Xn } is a convergent tower of fibrations. Then the canonical maps induce isomorphisms π∗ (X) ∼ = ∼ = lim π∗ (Xn ), H∗ (X) ∼ = ∼ = lim H∗ (Xn ), and H ∗ (X) ∼ = ∼ = colim H ∗ (Xn ) (J. P. May & K. Ponto 49)
- We may replace X by mic Xn . The inverse systems of homotopy groups satisfy the Mittag-Leffler condition, so that the lim1 error terms are trivial, and the isomorphism on homotopy groups follows (J. P. May & K. Ponto 49)
- we advertise an observation about cohomology with coefficients in a profinite abelian group (J. P. May & K. Ponto 49)
- We claim first that Hom(A, B) ∼ = ∼ = lim Hom(A, Bd ) and the derived functors lim n Hom(A, Bd ) are zero for all n > (J. P. May & K. Ponto 49)
- By Roos [116, Thm 3], there is a spectral sequence which converges from E = limp limq Hom(Ai , Bd ) to the derived functors lim n of the system of groups {Hom(Ai , Bd )} indexed on I × D op . (J. P. May & K. Ponto 49)
- We claim next that Ext(A, B) ∼ = ∼ lim Ext(A, Bd ) = (J. P. May & K. Ponto 50)
- We also want the reader to come away with the idea that cohomology classes, elements of ̃ H̃ n (X; π), are interchangable with (based) homotopy classes of maps, elements of [X, K(π, n)]. (J. P. May & K. Ponto 51)
- CW complexes (J. P. May & K. Ponto 51)
- Postnikov towers (J. P. May & K. Ponto 51)
- chapter with a feeling that these are such closely dual notions that there is really no reason to be more comfortable with one than the other (J. P. May & K. Ponto 51)
- A group is nilpotent if it has a central series that terminates after finitely many steps. It is equivalent that either its lower central series or its upper central series terminates after finitely many steps (J. P. May & K. Ponto 51)
- Call a group G a π-group if it has a (left) action of the group π as automorphisms of G. This means that we are given a homomorphism from π to the automorphism group of G (J. P. May & K. Ponto 52)
- We are thinking of π as π1 (X) and G as πn (X) for a space X. (J. P. May & K. Ponto 52)
- On the other hand, when G is abelian, a π-group is just a module over the group ring Z[π]. (J. P. May & K. Ponto 52)
- Recall that a connected space X is simple if π1 X is abelian and acts trivially on πn X. (J. P. May & K. Ponto 53)
- Clearly simple spaces and, in particular, simply connected spaces, are nilpotent. Connected H-spaces are simple and are therefore nilpotent. While it might seem preferable to restrict attention to simple or simply connected spaces, nilpotent spaces have significantly better closure properties under various operations (J. P. May & K. Ponto 53)
- the component F (X, Y )f of f in F (X, Y ) is nilpotent. This space is generally not simple even when X and Y are simply connected. (J. P. May & K. Ponto 53)
- Theorem 3.2.1. A connected space X is simple if and only if it admits a Postnikov tower of principal fibrations (J. P. May & K. Ponto 53)
- We can always construct maps αn : X −→ Xn such that αn induces an isomorphism on πi for i ≤ n and πi Xn = 0 for i > n just by attaching cells inductively to kill the homotopy groups of X in dimension greater than n. (J. P. May & K. Ponto 53)
- When X is simple, and only then, we can arrange further that Xn+1 is the homotopy fiber of a “k-invariant” k n+2 : Xn −→ K(πn+1 X, n + 2). This is what it means for X to have a “Postnikov tower of principal fibrations”. The name comes from the fact that Xn+1 is then the pullback along k n+2 of the path space fibration over K(πn+1 X, n + 2). Of course, the fiber of the resulting map pn+1 : Xn+1 −→ Xn is an Eilenberg-MacLane space K(πn+1 X, n + 1). (J. P. May & K. Ponto 53)
- One then constructs k n+2 by killing the higher homotopy groups of the cofiber Cαn and defines Xn+1 to be the fiber of k n+2 . However, the proof there is not complete since the check requires a slightly strengthened version of homotopy excision or the relative Hurewicz theorem (J. P. May & K. Ponto 54)
- We think of the maps Xn+1 −→ Xn as giving a decreasing filtration of X, and of course the fiber over the basepoint of this map is ΩKn . (J. P. May & K. Ponto 54)
- thinking of the inclusions of skeleta X n −→ X n+1 of a CW complex X as giving it an increasing filtration, and of course the quotient space X n+1 /X n is a wedge of suspensions ΣS n . (J. P. May & K. Ponto 55)
- Since right adjoints, such as P (−) or, more generally, F (X, −), preserve all categorical limits and since limits, such as pullbacks and sequential limits, commute with other limits, we find easily that products, pullbacks, and sequential limits of K -towers are again K -towers. (J. P. May & K. Ponto 55)
- model category theory focuses on cell complexes rather than CW complexes, which in fact play no role in that theory. It often applies to categories in which cell complexes can be defined just as in Definition 3.3.3, but there is no useful notion of a CW complex because the cellular approximation theorem [89, p. 74] fails (J. P. May & K. Ponto 56)
- The essential idea is just to apply the representability of cohomology, ̃ H̃ n (X; A) ∼ = ∼ = [X, K(A, n)], which is dual to the representability of homotopy groups, πn X = [S n , X], (J. P. May & K. Ponto 56)
- As we have noted, our towers Z = lim Zn are rarely of the homotopy types of CW complexes; it is for this reason that weak homotopy type rather than homotopy type appears in the following statement (J. P. May & K. Ponto 57)
- Theorem 3.3.9 (Dual Whitehead (second form)). If ξ : X −→ Y is an A cohomology isomorphism between connected spaces and Z is an A -tower, then ξ ∗ : [Y, Z] −→ [X, Z] is a bijection (J. P. May & K. Ponto 57)
- Proposition 3.3.11. Let R be a PID. Then f : X −→ Y is an R-homology isomorphism if and only if it is an R-cohomology isomorphism. (J. P. May & K. Ponto 58)
- Definition 3.3.10. Let R be a commutative ring and f : X −→ Y be a map. (J. P. May & K. Ponto 58)
- The following key result will make clear exactly where actions of the fundamental group and nilpotency of group actions enter into the theory of Postnikov towers. (J. P. May & K. Ponto 58)
- , we shall go very slowly through the following proof since it gives our first application of the Serre spectral sequence and a very explicit example of how one uses the representability of cohomology, (3.4.1) ̃ H̃ n (X; A) ∼ = ∼ [X, K(A, n)], = to obtain homotopical information. (J. P. May & K. Ponto 58)
- We regard K(A, n) as a name for any space whose only non-vanishing homotopy group is πn (K(A, n)) = A. With our standing CW homotopy type hypothesis, any two such spaces are homotopy equivalent. (J. P. May & K. Ponto 58)
- application of πn induces a bijection from the homotopy classes of maps K(A, n) −→ K(A, n) to Hom(A, A). One way to see that is to quote the Hurewicz and universal coefficient theorems [ (J. P. May & K. Ponto 58)
- addition in the cohomology group on the left is induced by the loop space multiplication on K(A, n) = ΩK(A, n + 1) (J. P. May & K. Ponto 58)
- Lemma 3.4.2. Let f : X −→ Y be a map of connected based spaces whose (homotopy) fiber F f is an Eilenberg-Mac Lane space K(A, n) for some abelian group A and n ≥ 1. (J. P. May & K. Ponto 59)
- There is a map k : Y −→ K(A, n + 1) and an equivalence ξ : X −→ F k such that the following diagram commutes, where π is the canonical fibration with (actual) fiber K(A, n) = ΩK(A, n + 1) (J. P. May & K. Ponto 59)
- There is a map k : Y −→ K(A, n + 1) and an equivalence λ : N f −→ F k such that the following diagram commutes, where π is as in (i) and ρ : N f → Y is the canonical fibration with (actual) fiber F f . (J. P. May & K. Ponto 59)
- Therefore the diagram displays equivalences showing that the sequence K(A, n) π /X f /Y k / K(A, n + 1) is equivalent to the fiber sequence generated by the map k (J. P. May & K. Ponto 59)
- Definition 3.5.1. A Postnikov A -tower is an A -tower X = lim Xi (see Definition 3.3.1) such that each Ki is a K(Ai , ni + 1) with Ai ∈ A , ni+1 ≥ ni ≥ 1, and only finitely many ni = n for each n ≥ 1. A map ψ : X −→ Y between Postnikov A -towers is cocellular if it is the limit of maps ψi : Xi −→ Yi (J. P. May & K. Ponto 62)
- Note that a product of Eilenberg-Mac Lane spaces Πj K(Aj , j) is an EilenbergMac Lane space K(Πj Aj , j). When A is closed under products, this makes it especially reasonable to use a single cocell at each stage of the filtration. (J. P. May & K. Ponto 62)
- Recall too that we are working in HoT , where spaces are implicitly replaced by CW approximations or, equivalently, where all weak equivalences are formally inverted (J. P. May & K. Ponto 63)
- Definition 3.5.3. An A -cocellular approximation of a space X is a weak equivalence from X to a Postnikov A -tower. (J. P. May & K. Ponto 63)
- The definition should be viewed as giving a kind of dual to CW approximation (J. P. May & K. Ponto 63)
- Just as CW approximation is the basis for the cellular construction of the homology and cohomology of general spaces, cocellular approximation is the basis for the cocellular construction of localizations and completions of general nilpotent spaces (J. P. May & K. Ponto 63)
- Theorem 3.5.4. Let X be an A -nilpotent space. (i) There is a Postnikov A -tower P (X) and a weak equivalence ξX : X −→ P (X); that is, ξ is a cocellular approximation of X. (J. P. May & K. Ponto 63)
- The spaces Yn,j and maps X −→ Yn,j are constructed by attaching (n + 1)-cells to X to kill the subgroup Gn,j of πn (X), using maps S n −→ X that represent generators of Gn,j as attaching maps, and then attaching higher dimensional cells to kill the homotopy groups in dimensions greater than (J. P. May & K. Ponto 64)
- Recall that an abelian group B is said to be T -local if it admits a structure of ZT -module, necessarily unique. It is equivalent that the multiplication map q : B −→ B be an isomorphism for all primes q not in T . (J. P. May & K. Ponto 81)
- We can define φ explicitly by setting AT = A⊗ ZT (J. P. May & K. Ponto 82)
- Lemma 5.1.2. Localization is an exact functor from abelian groups to ZT modules. (J. P. May & K. Ponto 82)
- The induced map φ∗ : H∗ (A; ZT ) −→ H∗ (AT ; ZT ) is an isomorphism for all abelian groups A. If B is T -local, then the homomorphism ̃ H̃∗ (B; Z) −→ H ̃ H̃∗ (B; ZT ) induced by the homomorphism Z −→ ZT is an isomorphism (J. P. May & K. Ponto 82)
- Any module over a PID R is the filtered colimit of its finitely generated submodules, and any finitely generated R-module is a finite direct sum of cyclic R-modules (J. P. May & K. Ponto 82)
- The localization functor commutes with colimits since it is a left adjoint (J. P. May & K. Ponto 82)
- homology of a filtered colimit of abelian groups is the colimit of their homologies (J. P. May & K. Ponto 82)
- one can use the standard simplicial construction of classifying spaces ([89, p. 126]) to give a construction of K(A, 1)’ (J. P. May & K. Ponto 82)
- The circle S 1 is a K(Z, 1). Our first example of a localized space is ST1 T1 , which not surprisingly turns out to be K(ZT , 1). (J. P. May & K. Ponto 83)
- Finally, consider A = Z, so that AT = ZT . The circle S 1 is a K(Z, 1). Our first example of a localized space is ST1 T1 , which not surprisingly turns out to be K(ZT , 1). AT can be constructed as the colimit of copies of Z together with the maps induced by multiplication by the primes not in T . For example, if we order the primes qi not in T by size and define rn inductively by r1 = q1 and rn = rn−1 q1 · · · qn = q1n . . . qn , then ZT is the colimit over n of the maps rn : Z −→ Z. We can realize these maps on π1 (S 1 ) by using the rn th power map S 1 −→ S 1 . Using the telescope construction ([89, p. 113]) to convert these multiplication maps into inclusions and passing to colimits, we obtain a space K(ZT , 1); the van Kampen theorem gives that the colimit has fundamental group ZT , and the higher homotopy groups are zero because a map from S n into the colimit has image in a finite stage of the telescope, which is equivalent to S 1 . The commutation of homology with colimits gives that the only non-zero reduced integral homology group of K(ZT , 1) is its first, which is ZT . (J. P. May & K. Ponto 83)
- In fact, as we shall use heavily in §5.4, for any groups G and H, not necessarily abelian, passage to fundamental groups induces a bijection (5.1.5) [K(G, 1), K(H, 1)] ∼ = ∼ Hom(G, H). = (J. P. May & K. Ponto 83)
- One way to see this is to observe that the classifying space functor from groups to Eilenberg–Mac Lane spaces (e.g. [89, p. 126]) gives an inverse bijection to π1 , but it can also be verified directly from the elementary construction of K(G, 1)’s that is obtained by realizing π1 as the fundamental group of a space X ([89, p. 35]) and then killing the higher homotopy groups of X. (J. P. May & K. Ponto 83)
- Proposition 5.2.7. A homomorphism φ : A −→ B of abelian groups is an algebraic localization at T if and only if the map, unique up to homotopy, φ : K(A, 1) −→ K(B, 1) that realizes φ on π1 is a topological localization at T . (J. P. May & K. Ponto 85)
- In view of Theorem 3.5.4, we may assume without loss of generality that X is a Postnikov tower lim Xi constructed from maps ki : Xi −→ K(Ai , ni +1), (J. P. May & K. Ponto 86)
- The given exact sequence implies that the homotopy fiber of the evident map K(G, 1) −→ K(G′′ , 1) is a K(G′ , 1) (J. P. May & K. Ponto 91)
- Recall that the lower central series of G is defined by Γ1 (G) = G and, inductively, Γj+1 (G) = [G, Γj (G)]. (J. P. May & K. Ponto 93)
- Recall that localization commutes with finite products. It does not commute with infinite products in general, but we have the following observation. Lemma 5.5.6. If Gi is a T -local group for all elements of an indexing set I, then e i∈I Gi is T -local (J. P. May & K. Ponto 93)
- Lemma 5.5.7. Localization at T commutes with pullbacks (J. P. May & K. Ponto 93)
- Recall that the upper central series of G is defined inductively by letting Z0 (G) = 1, letting Z1 (G) be the center of G, and letting Zj+1 (G)/Zj (G) be the center of G/Zj (G) (J. P. May & K. Ponto 94)
- It is often necessary to restrict attention to finitely generated modules over the principal ideal domain ZT (J. P. May & K. Ponto 95)
- We give several characterizations of localizations in §6.1, and we use these to study the homotopical behavior of localization with respect to standard constructions on based spaces in (J. P. May & K. Ponto 99)
- Using the homological characterization of localizations, we obtain analogues of the results in §6.2 for wedges, suspensions, cofiber sequences, and smash products. However, in the non-simply connected case, the required preservation of nilpotency is not automatic. Wedges behave badly, for example. The wedge S 1 ∨ S 1 is not nilpotent since a free group on two generators is not nilpotent, and the wedge S 1 ∨S 2 is not nilpotent since π1 does not act nilpotently on π2 . (J. P. May & K. Ponto 107)
- The Künneth theorem also implies that the smash product of localizations is a localization (J. P. May & K. Ponto 107)
- Recall that an H-space, or Hopf space, X is a space together with a product X × X −→ X such that the basepoint ∗ ∈ X is a two-sided unit up to homotopy (J. P. May & K. Ponto 109)
- topological monoids and loop spaces provide the canonical examples (J. P. May & K. Ponto 109)
- Proposition 6.6.2. If Y is an H-space with product µ, then YT is an H-space with product µT such that the localization φ : Y −→ YT is a map of H-spaces (J. P. May & K. Ponto 110)
- The converse does not hold. There are many interesting spaces that are not H-spaces but have localizations that are H-spaces. In fact, in 1960, well before localizations were constructed in general, Adams [2] observed that STn Tn is an Hspace for all odd n and all sets T of odd primes. (J. P. May & K. Ponto 110)
- Since πn (STn ) = [STn , STn ] = ZT (J. P. May & K. Ponto 111)
- In contrast, S n itself is an H-space only if n = 0, 1, 3, or 7, by Adams’ solution to the Hopf invariant one problem [1]. Here the Lie group S 3 is not homotopy commutative and the H-space S 7 given by the unit Cayley numbers is not homotopy associative (J. P. May & K. Ponto 111)
- The study breaks into two main variants. In one of them, one allows general finite T -local H-spaces, not necessarily homotopy associative or commutative, and asks what possible underlying homotopy types they might have. In the other, one studies finite T -local loop spaces, namely spaces X that are of the homotopy type of finite T -CW complexes and are also homotopy equivalent to ΩBX for some T -local space BX. Such X arise as localizations of finite loop spaces. One asks, typically, how closely such spaces resemble compact Lie groups and what limitations the H-space structure forces on the homology and homotopy groups. The structure theorems for Hopf algebras that we give later provide a key starting point for answering such questions (J. P. May & K. Ponto 111)
- A co-H-space Z is a space with a coproduct δ : Z −→ Z ∨ Z such that ∇ ◦ δ is homotopic to the identity, where ∇ : Z ∨ Z −→ Z is the folding map. Suspensions ΣX with their pinch maps provide the canonical examples (J. P. May & K. Ponto 111)
- Proposition 6.6.4. For a simply connected co-H-space Z, the localization ZT can be constructed as the telescope of the sequence of rith ith -copower maps Z −→ Z. (J. P. May & K. Ponto 111)
- Localization at the empty set of primes is called rationalization (J. P. May & K. Ponto 111)
- Many results in algebraic topology that preceded the theory of localization are conveniently proven using the newer theory. We illustrate this with a proof of a basic theorem of Serre on the finiteness of the homotopy groups of spheres. Serre proved the result using (Serre) classes of abelian groups [121]. The proof using rationalization is simpler and more illuminating. Theorem 6.7.1 (Serre). For n ≥ 1, the homotopy groups πq (S n ) are finite with the exceptions of πn (S n ) = Z for all n and π2n−1 (S n ) = Z ⊕ Fn for n even, where Fn is finite. (J. P. May & K. Ponto 112)
- Proposition 6.7.2. The cohomology algebra H ∗ (K(Q, n); Q) is the exterior algebra on ιn if n is odd and the polynomial algebra on ιn if n is even. Proof. For n = 1 and n = 2, this is clear from Theorem 5.2.8 and the fact that S 1 = K(Z, 1) and CP ∞ = K(Z, 2). We proceed by induction on n, using the Serre spectral sequence of the path space fibration K(Q, n) −→ P K(Q, n + 1) −→ K(Q, n + 1). (J. P. May & K. Ponto 112)
- and the spectral sequence is concentrated on the 0th and nth rows. (J. P. May & K. Ponto 112)
- Thus the rationalization of the homotopy group πq (S n ), q > 0, is 0 if q 6= 2n − 1 and is Q if q = 2n − 1. Since π2n−1 (S n ) is a finitely generated abelian group with rationalization Q, it must be the direct sum of Z and a finite group (J. P. May & K. Ponto 112)
- This implies that all homotopy groups πq (S n ), q > n, are in the kernel of rationalization. That is, they are torsion groups. Since they are finitely generated by Theorem 4.5.7, they are finite. (J. P. May & K. Ponto 112)
- Corollary 6.7.3. Consider the rationalization k0 : S n 0n −→ K(Q, n) of the canonical map k : S n −→ K(Z, n). If n is odd, k0 is an equivalence. If n is even, the fiber of k0 is K(Q, 2n − 1). (J. P. May & K. Ponto 113)
- we give an observation that shows, in effect, that phantom maps are usually invisible to the eyes of rational homotopy theory. (J. P. May & K. Ponto 113)
- In Chapter 5, we described how to construct localizations of nilpotent spaces. In the next chapter, we go in the opposite direction and describe how to start with local spaces and construct a “global space” and how to reconstruct a given global space from its localizations. Results such as these are referred to as fracture theorems (J. P. May & K. Ponto 115)
- We are mainly interested in the case when S is empty and localization at S is rationalization (J. P. May & K. Ponto 115)
- We are then starting with a partition of the set of primes in T , and we are most often interested in the case when T is the set of all primes (J. P. May & K. Ponto 115)
- I might be the positive integers and Ti might be the set consisting of just the ith prime number pi . A common situation is when I = {1, 2}, T1 = {p}, and T2 is the set of all other primes. For example, spaces often look very different when localized at 2 and when localized away from 2 (J. P. May & K. Ponto 115)
- It applies to any category that has categorical products. Such categories are said to be cartesian monoidal. Examples include the categories of abelian groups, groups, spaces, and sets. Less obviously, the homotopy category HoT is another example, even though pullbacks do not generally exist in HoT (J. P. May & K. Ponto 118)
- Lemma 7.1.9. In any cartesian monoidal category, a commutative diagram of the following form is a pullback. A (id,gf ) / f A×C f ×id B (id,g) / B × C. (J. P. May & K. Ponto 119)
- Remark 7.2.2. Theorem 7.2.1(ii) was proven under a finite generation hypothesis in Hilton, Mislin, and Roitberg [60], and Hilton and Mislin later noticed that the hypothesis can be removed [59]. That fact is not as well-known as it should be. We learned both it and most of the elegant proof presented here from Bousfield. (J. P. May & K. Ponto 121)
- In §7.2, we started with a T -local group and showed that it was isomorphic to the pullback of some of its localizations (J. P. May & K. Ponto 121)
- (J. P. May & K. Ponto 122)
- The name comes from the fact that when ω exists, it turns out that the map µi : P −→ Gi in (7.3.2) is a localization at Ti , hence the map ν is a localization at S and ω is the localization of (µi ) at S. (J. P. May & K. Ponto 122)
- Definition 7.5.1. Let G be a nilpotent group. The extended genus of G is the collection of isomorphism types of nilpotent groups G ′ such that the localizations Gp and G′ ′p ′p are isomorphic for all primes p. If G is finitely generated, then the genus of G is the set of isomorphism classes of finitely generated nilpotent groups in the extended genus of G. (J. P. May & K. Ponto 125)
- Let Aut(G) denote the group of automorphisms of G. We show that elements of the extended genus are usually in bijective correspondence with double cosets (J. P. May & K. Ponto 125)
- Example 7.5.5. Let A = Z. An automorphism of Q is just a choice of a unit in Q, and similarly for Z(pi ) . (J. P. May & K. Ponto 127)
- Remark 7.5.8. The word “genus” is due to Mislin [102], following an analogy due to Sullivan [129], and has nothing to do with the use of the word elsewhere in mathematics. Rather, the analogy is with genetics or, perhaps better, taxonomy (J. P. May & K. Ponto 127)
- 8.1. Statements of the main fracture theorems (J. P. May & K. Ponto 131)
- Definition 8.1.5. A formal localization associated to the maps ψi : Xi −→ Y is a homotopy class of maps ω : Y −→ ( at i∈I Xi )S which satisfies the following two properties. (i) The composite of ω and Q nd π̃i is homotopic to the identity map for each i ∈ I. i )S ) is the product of an element φS ∗ (x) and an Each element z ∈ π1 (( i∈I X element ω∗ (y), where x ∈ π1 ( h Q i ) i∈I Xi ) and y ∈ π1 (Y ) . (J. P. May & K. Ponto 132)
- This depends on the general observation that homotopy pullbacks of homotopy pullbacks are homotopy pullbacks (J. P. May & K. Ponto 133)
- It says that to check whether or not two maps are homotopic, it suffices to check whether or not they become homotopic after localization at each prime p. (J. P. May & K. Ponto 134)
- Recall too that Proposition 2.2.2 and Corollary 2.2.3 tell us how to compute the homotopy groups of homotopy pullbacks (J. P. May & K. Ponto 136)
- Remark 8.3.7. We repeat Remark 7.2.2, since it applies verbatim here. Theorem 8.1.3 was proven under a finite generation hypothesis in Hilton, Mislin, and Roitberg [60], and Hilton and Mislin later noticed that the hypothesis can be removed [59]. That fact is not as well-known as it should be. We learned both it and most of the elegant proof presented here from Bousfield. (J. P. May & K. Ponto 138)
- Much early work in the theory of localization focused on the concept of genus, which was introduced by Mislin [102] in the context of H-spaces. (J. P. May & K. Ponto 139)
- Having the homotopy type of a finite CW complex is not a generic property, by a counterexample of Mislin [103]. Being of the homotopy type of a space with finitely generated integral homology is generic (J. P. May & K. Ponto 140)
- Theorem 8.5.7. The property of being a finite H-space is generic (J. P. May & K. Ponto 140)
- beautiful worked out example was given by Rector [114]. Theorem 8.5.8. The genus of HP ∞ is uncountably infinite (J. P. May & K. Ponto 140)
- Theorem 8.5.8. The genus of HP ∞ is uncountably infinite. For comparison, McGibbon [96] computed the genus of the finite nilpotent projective spaces RP 2n+1 , CP n and HP n for 1 ≤ n < ∞. For these spaces X, he uses pullbacks over X to give the set G(X) a group structure and proves the following result. Theorem 8.5.9. Let n be a positive integer. (i) G(RP 2n+1 ) = 1, (ii) G(CP n ) = 1, (iii) G(HP n ) ∼ = Z/2 ⊕ . . . ⊕ Z/2, where the number of factors equals the number of primes p such that 2 ≤ p ≤ 2n − 1. (J. P. May & K. Ponto 140)
- As in the case of nilpotent groups, asking how unique formal localizations are gives a starting point of the analysis of the extended genus, and one can then seek the actual genus inside that (J. P. May & K. Ponto 141)
- Many known calculations rely on an understanding of double cosets of homotopy automorphism groups, (J. P. May & K. Ponto 141)
- Rather tautologically, a major theme of algebraic topology is the algebraization of homotopy theory. In some cases, the algebraization is complete, and this is true for rational homotopy theory, as proven by Quillen [110] and Sullivan [130 (J. P. May & K. Ponto 147)
- we show how to give it more topological content via the Samelson product on homotopy groups. This gives a Lie algebra structure on π∗ (X) for a connected H-group X (as defined in Definition 9.2.1) such that the Lie algebra π∗ (X) ⊗ Q is determined by the Hopf algebra H∗ (X; Q (J. P. May & K. Ponto 147)
- This leads to an all too brief discussion of Whitehead products, which are the starting point for serious work in unstable homotopy theory. (J. P. May & K. Ponto 147)
- The problem is the lack of a Künneth theorem for infinite products. (J. P. May & K. Ponto 147)
- Theorem 9.1.3. If A is a commutative, associative, and connected quasi Hopf algebra over Q, then A is isomorphic as an algebra to the tensor product of an exterior algebra on odd degree generators and a polynomial algebra on even degree generators. (J. P. May & K. Ponto 148)
- We calculated the rational cohomology of K(Q; n) in Proposition 6.7.2. It is the polynomial algebra P [ιn ] if n is even and the exterior algebra E[ιn ] if n is odd. (J. P. May & K. Ponto 148)
- Theorem 9.1.4. Rational cohomology defines a contravariant equivalence from the homotopy category of rational connected H-spaces of finite type to the category of commutative, associative, and connected quasi Hopf Algebras of finite type. (J. P. May & K. Ponto 148)
- recall from [89, p. 127] that Eilenberg– Mac Lane spaces can be constructed as commutative topological groups. (J. P. May & K. Ponto 149)
- Definition 9.2.1. An H-monoid is a homotopy associative H-space. An Hgroup is an H-monoid with a map χ providing inverses up to homotopy, so that µ ◦ (id ×χ) ◦ ∆ ≃ ∗ ≃ µ ◦ (χ × id) ◦ ∆, where ∗ denotes the trivial map at the unit element e and µ is the product (J. P. May & K. Ponto 149)
- If we abuse notation by writing χ(x) = x −1 and writing µ(x, y) = xy, then the condition becomes “xx −1 = e = x −1 x” up to homotopy. An H-monoid is grouplike if π0 (X) is a group under the product induced by the product on X (J. P. May & K. Ponto 149)
- More elegantly, an H-space X is an H-monoid if the functor [−, X] is monoid– valued and is an H-group if the functor [−, X] is group–valued (J. P. May & K. Ponto 149)
- This raises the question of whether or not there is a homotopical construction of a Lie bracket on π∗ (X) for an H-group X that is compatible under the Hurewicz homomorphism with the commutator in H∗ (X; Z). The answer is that indeed there is. The relevant product on homotopy groups is called the Samelson product (J. P. May & K. Ponto 150)
- Theorem 9.2.5. Let X be a connected H-group of finite type. Then H∗ (X; Q) is isomorphic as a Hopf algebra to U (π∗ (X) ⊗ Q), where π∗ (X) is regarded as a Lie algebra under the Samelson product. (J. P. May & K. Ponto 150)
- For based spaces J and K, define the generalized Samelson product h−, −i : [J, X] ⊗ [K, X] −→ [J ∧ K, X] (J. P. May & K. Ponto 150)
- by hf, gi = [φ′ ◦(f ∧g)]. Specializing to J = S p and K = S q , this gives the Samelson product h−, −i : πp (X) ⊗ πq (X) −→ πp+q (X). (J. P. May & K. Ponto 151)
- Proposition 9.2.8. For a connected H-group X, π∗ (X) is a Lie algebra under the Samelson product. (J. P. May & K. Ponto 151)
- we briefly describe the Whitehead product, which is fundamental in the deeper parts of unstable homotopy theory and i (J. P. May & K. Ponto 152)
- It is most easily defined as a special case of the Samelson product (J. P. May & K. Ponto 152)
- We rewrite this as [−, −] : [ΣJ, X] ⊗ [ΣK, X] −→ [Σ(J ∧ K), X] and call it the generalized Whitehead product. Taking Taking J = S p−1 and K = S q−1 , this specializes to the Whitehead product [−, −] : πp (X) ⊗ πq (X) −→ πp+q−1 (X). (J. P. May & K. Ponto 152)
- From this point of view, the Whitehead product [ip , iq ] is thought of as a map S p+q−1 −→ S p ∨ S q , and it is the attaching map for the construction of S p × S q from S p ∨ S q . (J. P. May & K. Ponto 153)
- The Whitehead products appear in the EHP-sequence, which is the most important tool for the study of unstable homotopy groups. (J. P. May & K. Ponto 153)
- Thinking of spaces, homotopy types, and spaces in the same genus as analogous to animals, animals in the same species, and animals in the same genus, algebraic topologists are expert at genetic modification to produce different species in the same genus. (J. P. May & K. Ponto 155)
- We usually modify spaces using finite sets I, especially partitions of the primes into two disjoint sets T1 and T2 . In that case, the local to global fraction results go under the name of Zabrodsky mixing (J. P. May & K. Ponto 155)
- They constructed an H-space X that is in the same genus as the Lie group Sp(2) but is not equivalent to it. Both X and Sp(2) are equivalent to S 3 × S 7 away from the primes 2 and 3. As is explained in [60, pp. 122-127], the three H-spaces in sight, X, Sp(2) and S 3 × S 7 , are total spaces of bundles over S 7 with fiber S 3 , and every simply connected finite H-space with rational cohomology E[x3 , x7 ] is equivalent to the total space of such a bundle. (J. P. May & K. Ponto 155)
- Theorem 9.4.6. Let X be a connected homotopy commutative finite H-space. Then X is homotopy equivalent to a torus T = (S 1 ) n for some n. (J. P. May & K. Ponto 155)
- Corollary 9.4.7. A simply connected homotopy commutative finite H-space is contractible. (J. P. May & K. Ponto 155)
- It is usual to focus on a single prime p, and there is no loss of information in doing so since completion at T is the product over p ∈ T of the completions at p, and similarly for all relevant algebraic invariants (J. P. May & K. Ponto 159)
- In contrast to localization, completions of abelian groups can sensibly be defined in different ways, and the most relevant definitions are not standard fare in basic graduate algebra courses (J. P. May & K. Ponto 159)
- Lemma 10.1.2. When restricted to finitely generated abelian groups, the p-adic completion functor is exact (J. P. May & K. Ponto 159)
- .1.1. p-adic completion. It is usual to define the completion of an abelian group A at a given prime p to be the p-adic completion ̂ Âp = lim(A/p r A), (J. P. May & K. Ponto 159)
- This definition will not fully serve our purposes since p-adic completion is neither left nor right exact in general, and exactness properties are essential to connect up with the topology (J. P. May & K. Ponto 159)
- When A = Z, we write Zp instead of Z Ẑ Ẑp for the ring of p-adic integers, and we abbreviate Z/nZ to Z/n. (J. P. May & K. Ponto 159)
- Observe that the p-adic completion functor takes values (J. P. May & K. Ponto 159)
- in the category of Zp -modules (J. P. May & K. Ponto 160)
- Even if we restrict to finitely generated abelian groups, we notice one key point of difference between localization and completion. While a homomorphism of abelian groups between p-local groups is necessarily a map of Z(p) -modules, a homomorphism of abelian groups between p-adically complete abelian groups need not be a map of Zp -modules. (J. P. May & K. Ponto 160)
- When A is finitely generated, p-adic completion is given by the map ψ : A −→ A⊗Zp specified by ψ(a) = a ⊗ 1, this again being a consequence of the Artin-Rees lemma (J. P. May & K. Ponto 160)
- Since Zp is torsion free, it is a flat Z-module, which gives us another way of seeing Lemma 10.1.2 (J. P. May & K. Ponto 160)
- To overcome the lack of exactness of p-adic completion in general, we consider the left derived functors of the p-adic completion functor. For the knowledgable reader, we recall that left derived functors are usually defined only for right exact functors, in which case the 0th left derived functor agrees with the given functor (J. P. May & K. Ponto 160)
- The left derived functors of p-adic completion are given on an abelian group A by first taking a free resolution 0 −→ F ′ −→ F −→ A −→ 0 of A, then applying p-adic completion, and finally taking the homology of the resulting length two chain complex ̂p F̂ ′ p′ −→ ̂ F̂p (J. P. May & K. Ponto 160)
- Since kernels and cokernels of maps of Zp -modules are Zp -modules, since a free abelian group is its own free resolution, and since p-adic completion is exact when restricted to finitely generated abelian groups (J. P. May & K. Ponto 160)
- Lemma 10.1.4. The functors L0 and L1 take values in Zp -modules. If A is either a finitely generated abelian group or a free abelian group, then L0 A = ̂ Âp , L1 A = 0, and φ : A −→ L0 A coincides with p-adic completion (J. P. May & K. Ponto 160)
- Definition 10.1.5. Fix a prime p. We say that the completion of A at p is defined if L1 A = 0, and we then define the completion of A at p to be the homomorphism φ : A −→ L0 A. We say that A is p-complete if φ : A −→ L0 A is an isomorphism. As we shall see in Proposition 10.1.18, if A is p-complete, then L1 A = 0 (J. P. May & K. Ponto 161)
- Example 10.1.6. We have seen that finitely generated and free abelian groups are completable and their completions at p coincide with their p-adic completions (J. P. May & K. Ponto 161)
- Example 10.1.7. Zp ⊗ Zp and Z/p∞ (see below) are Zp -modules that are not p-complete (J. P. May & K. Ponto 161)
- The essential exactness property of our derived functors, which is proven in the same way as the long exact sequences for Tor and Ext, reads as follows (J. P. May & K. Ponto 161)
- Define Z/p ∞ to be the colimit of the groups Z/pr with respect to the homomorphisms p : Z/pr −→ Z/pr+1 given by multiplication by p. (J. P. May & K. Ponto 161)
- Exercise 10.1.9. Verify that Z/p ∞ ∼ = ∼ = Z[p −1 ]/Z. (J. P. May & K. Ponto 161)
- Of course, Ep A = 0 if A is a divisible and hence injective abelian group. Write Hom(Z/pr , A) = Ar for brevity. We may identify Ar with the subgroup of elements of A that are annihilated by pr (J. P. May & K. Ponto 161)
- Notation 10.1.10. For a prime p and an abelian group A, define Ep A to be Ext(Z/p ∞ , A) and define Hp A to be Hom(Z/p ∞ , A). (J. P. May & K. Ponto 161)
- Proposition 10.1.11. There is a natural isomorphism Hp A ∼ = ∼ = lim Ar , (J. P. May & K. Ponto 161)
- Example 10.1.13. Any torsion abelian group A with all torsion prime to p satisfies Hp A = 0 and Ep A = 0. (J. P. May & K. Ponto 162)
- Definition 10.1.20. Fix a nonempty set of primes T and recall that Z[T −1 ] is obtained by inverting the primes in T , whereas ZT is obtained by inverting the primes not in T . (J. P. May & K. Ponto 163)
- In turn, Q/Z is isomorphic to the direct sum over all primes p of the groups Z/p ∞ (J. P. May & K. Ponto 164)
- Indeed, we have seen that K(Z[T −1 ], 1) is a localization of S 1 = K(Z, 1) away from T (J. P. May & K. Ponto 166)
- K(B, n) is T -complete if B is T -complete. (J. P. May & K. Ponto 167)
- The group ̂ F̂p /F is uniquely p-divisible. One can see this, for example, by noting that the canonical map F −→ F ̂ F̂p is a monomorphism of torsion free abelian groups that induces an isomorphism upon reduction mod p (J. P. May & K. Ponto 168)
- In view of Example 10.1.14, we have an interesting explicit example where homotopy groups shift dimension. Example 10.3.3. For a prime p, K(Z/p ∞ , n) ∧ p is an Eilenberg–Mac Lane space K(Zp , n + 1). (J. P. May & K. Ponto 169)
- Analogous dimension shifting examples play a central role in comparing the algebraic K-theory of an algebraically closed field, which is concentrated in odd degrees, to topological K-theory, which is concentrated in even degrees (J. P. May & K. Ponto 169)
- Theorem 10.3.4. Every nilpotent space X admits a completion φ : X −→ X ̂ (J. P. May & K. Ponto 169)
- Thus Ext(B, A) = 0 implies that every such extension splits in the form C ∼ = ∼ = A ⊕ B (J. P. May & K. Ponto 173)
- Profinite completion at p is an exact functor on finitely generated nilpotent groups (J. P. May & K. Ponto 174)
- Proposition 11.1.4. The completion at T of a nilpotent space X is the composite of its localization at T and the completion at T of the localization XT . (J. P. May & K. Ponto 177)
- In fact, the integral homology of completions is so poorly behaved that it is almost never used in practice (J. P. May & K. Ponto 177)
- One might naively hope that, at least if X is f -nilpotent, ̃ H̃∗ ( ̂ X̂T ; Z) might be isomorphic to ̃ H̃∗ (X; Z) ⊗ Z Ẑ ẐT , in analogy with what is true for localization. However, as observed in [20, VI.5.7], that is already false when X = S n . (J. P. May & K. Ponto 177)
- In particular, Hqn (STn ; Z) is an uncountable Q-vector space for q ≥ 2 (J. P. May & K. Ponto 177)
- This allows us to work with the Noetherian ring Zp rather than the ring Ẑ ẐT , which is not Noetherian if the set T is infinite. (J. P. May & K. Ponto 177)
- By Theorem 3.5.4 we may assume that X and Y are Postnikov towers and that f is a cocellular map. (J. P. May & K. Ponto 178)
- The problem is that, while the kernel and cokernel of a map of Zp modules between p-complete abelian groups are Zp -modules, they still need not be p-complete (J. P. May & K. Ponto 178)
- . Notably, it is not true that ΣX is p-complete when X is p-complete (J. P. May & K. Ponto 180)
- However, using the characterization of completion in terms of mod p homology, we can obtain correct statements simply by completing the constructions that fail to be complete. (J. P. May & K. Ponto 180)
- Clearly, in view of these results, it is unreasonable to expect to have a cellular construction of completions analogous to the cellular construction of localizations given in §6.5. (J. P. May & K. Ponto 181)
- There is a large body of interesting work on p-complete H-spaces. Here again, some of the interest is in seeing how much like compact Lie groups they are (J. P. May & K. Ponto 181)
- One also wants them to be equivalent to loop spaces (J. P. May & K. Ponto 181)
- Of course, this holds for any topological group G, since G is equivalent to the loops on its classifying space BG. (J. P. May & K. Ponto 181)
- This notion was introduced and studied by Dwyer and Wilkerson [41], who showed how remarkably similar to compact Lie groups these X are. The completion of a compact Lie group is an example, but there are many others. Like compact Lie groups, p-compact groups have versions of maximal tori, normalizers of maximal tori, and Weyl groups. A complete classification, analogous to the classification of compact Lie groups, has recently been obtained (J. P. May & K. Ponto 181)
- Definition 11.5.2. A p-compact group is a triple (X, BX, ε), where BX is a p-complete space, ε : X −→ ΩBX is a homotopy equivalence, and the mod p cohomology of X is finite dimensional. It is often assumed that BX is simply connected, so that X is connected. (J. P. May & K. Ponto 181)
- is the q th Bockstein operation (J. P. May & K. Ponto 182)
- The compatibility condition involves a notion of formal completion, and we also give a brief discussion of the what we call the adèlic genus of a nilpotent group (J. P. May & K. Ponto 183)
- . Conceptually, the point is that the group theory knows only about Eilenberg-Mac Lane spaces K(G, 1), but the topology knows how to use two-stage Postnikov towers to construct completions of Eilenberg-Mac Lane spaces K(G, 1) for nilpotent groups G that are not completable algebraically (J. P. May & K. Ponto 183)
- We begin by showing that completion preserves certain pullbacks. This is in contrast to the case of localization, where all pullbacks are preserved. Of course, the difference is a consequence of the failure of exactness for completion (J. P. May & K. Ponto 183)
- Lemma 12.2.1. If an abelian group A is completable at T , then the kernel of the completion φ̂ φ̂ : A → ET A is Hom(Z[T −1 ], A). If G is an f ZT -nilpotent group, then the completion φ̂ φ̂ : G → ET G is a monomorphism. (J. P. May & K. Ponto 185)
- . Since rationalization is exact (J. P. May & K. Ponto 186)
- It is natural to think of completion at p as the composite of localization at p and completion at p (J. P. May & K. Ponto 188)
- Our algebraic local to global result reads as follows. Its hypotheses seem to be minimal, but it is instructive to compare it with Theorem 13.3.1, where the topology allows us to generalize to groups that are not completable at T . (J. P. May & K. Ponto 189)
- We first develop conditions on the input that ensure that our local to global fracture theorem delivers f Ẑ ẐT -nilpotent groups as output. This is subtle since finite generation conditions on the input are not always sufficient. There are finitely generated T -complete groups that cannot be realized as the completions of finitely generated T -local groups (J. P. May & K. Ponto 190)
- Definition 12.4.6. The adèlic genus of a finitely generated nilpotent group G is the set of isomorphism classes of finitely generated nilpotent groups G ′ such that G0 is isomorphic to G′ ′0 ′0 and ̂ Ĝp is isomorphic to ̂ Ĝ′ ′p ′p for all primes p. The complete genus of G is defined by dropping the requirement that G0 be isomorphic to G′ 0 0 . The name “adèlic” is suggested by Sullivan’s analogy [131, 129] with the theory of adèles in number theory. (J. P. May & K. Ponto 193)
- The conclusion would be that the adèlic genus of G is in bijective correspondence with the double cosets Aut H Aut (J0 )/ Aut J (J. P. May & K. Ponto 194)
- The following general observation is elementary but important. It helps explain why one should expect to be able to reconstruct T -local spaces from their rationalizations and their completions at T . Lemma 13.2.1. Let f : X −→ Y be any map. Then f∗ : H∗ (X; ZT ) −→ H∗ (Y ; ZT ) is an isomorphism if and only if f∗ : H∗ (X; Q) −→ H∗ (Y ; Q) and, for all p ∈ T , f∗ : H∗ (X; Fp ) −→ H∗ (Y ; Fp ) are isomorphisms (J. P. May & K. Ponto 197)
- Indeed, homology commutes with sums and colimits of coefficient groups since this already holds on the chain level. (J. P. May & K. Ponto 197)
- We saw in §12.4 that the lack of a functorial formal completion of nilpotent groups impeded the naive analysis of the complete genus of f -nilpotent groups (J. P. May & K. Ponto 203)
- The idea is to form the tensor product of a space X and a ring R to obtain a space “X ⊗R” such that π∗ (X ⊗R) is naturally isomorphic to π∗ (X)⊗R. (J. P. May & K. Ponto 203)
- Since we want tensoring over R to be an exact functor on abelian groups, we also insist that the underlying abelian group of R be torsion free. (J. P. May & K. Ponto 203)
- For any set of primes S ⊂ T , the localization of X at S can be thought of as X ⊗ ZS and the completion of X at S can be thought of as X ⊗ Ẑ ẐS . We didn’t need the finite type hypothesis to conclude that π∗ (XS ) ∼ = ∼ = π∗ (X) ⊗ ZS , but we did need it to conclude that π∗ ( ̂ X̂S ) ∼ = ∼ = π∗ (X) ⊗ Ẑ ẐS . In both cases, with our X ⊗ R notation, we started with K(A, n) ⊗ R = K(A ⊗ R, n) and inducted up the Postnikov tower of X to obtain the construction. We explain how the same construction goes in our more general situation (J. P. May & K. Ponto 203)
- When T is the empty set, the realification of rational spaces case has been studied using the algebraization of rational homotopy theory (J. P. May & K. Ponto 204)
- The difficulty is that, in this generality, the cohomology of the EilenbergMac Lane spaces K(B ⊗ R, n) can be quite badly behaved. Thus we may not have enough cohomological control to start and continue the induction (J. P. May & K. Ponto 204)
- Of course, it suffices to define the functor on countable CW complexes since we are working in homotopy categories (J. P. May & K. Ponto 205)
- Remember that localizations and completions are defined by universal properties in the homotopy category and so are not uniquely specified (J. P. May & K. Ponto 207)
- Definition 13.6.5. The adèlic genus of an f -nilpotent space X is the set of homotopy classes of f -nilpotent spaces Y such that X0 is equivalent to Y0 and X ̂ X̂p is isomorphic to Y ̂ Ŷp for all primes p. The complete genus of X is defined by dropping the requirement that X0 be isomorphic to Y0 . Write ̂ Ĝ0 (X) for the adèlic genus of X and ̂ Ĝ(X) for the complete genus. Recall that G(X) denotes the (local) genus of X. If two spaces are in the same local genus, they are in the same adèlic genus, and if two spaces are in the same adèlic genus they are in the same complete genus. Therefore we have inclusions G(X) ⊂ G ̂ Ĝ0 (X) ⊂ G ̂ Ĝ(X). (J. P. May & K. Ponto 208)
- Theorem 13.6.6. For a simple space X of finite type, there is a canonical bijection between ̂ Ĝ0 (X) and the set of double cosets hAut(X0 )(F X0 )/hAut(F X). (J. P. May & K. Ponto 209)
- Theorem 13.6.7. If X is a simply connected finite CW complex, then ̂ Ĝ(X) and therefore ̂ Ĝ0 (X) and G(X) are finite sets. It is natural to ask if these three notions of genus are genuinely different (J. P. May & K. Ponto 210)
- Recall that the function space F (X, Y ) is nilpotent when X is finite and Y is of finite type (J. P. May & K. Ponto 210)
- Taking Y = X, this suggests that the automorphism groups above should be nilpotent, or nearly so, when X is finite and that their analysis should be closely related to the algebraic analysis of the genus of a finitely generated nilpotent group (J. P. May & K. Ponto 210)
- Theorem 13.6.8. Let X be an f -nilpotent space. If X0 admits an H-space structure, then ̂ Ĝ0 (X) = ̂ Ĝ(X). If, further, X is either π∗ -finite or H∗ -finite, then G(X) = ̂ Ĝ(X). However, there are simply connected examples such that ̂ Ĝ0 (X) 6= ̂ Ĝ(X), and these can be chosen to be π∗ -finite or H∗ -finite. Similarly, there are simply connected examples such that G(X) 6= ̂ Ĝ0 (X), and these too can be chosen to be π∗ -finite or H∗ -finite (J. P. May & K. Ponto 210)
- Say that X is π∗ -finite or H∗ -finite if it has only finitely many non-zero homotopy groups or only finitely many non-zero homology groups, all of them assumed to be finitely generated (J. P. May & K. Ponto 210)
- Model category theory is due to Quillen [112]. Nice introductions are given in [40, 51], and there are two expository books on the subject [63, 64] (J. P. May & K. Ponto 213)
- In fact, by now the very term “homotopy theory” admits of two interpretations. There is the homotopy theory of topological spaces, which is the core of algebraic topology, and there is also homotopy theory as a general methodology applicable to many other subjects (J. P. May & K. Ponto 213)
- Let M be a category. We insist that categories have sets of morphisms between pairs of objects; category theorists would say that M is locally small. Similarly, we understand limits and colimits to be small, (J. P. May & K. Ponto 213)
- meaning that they are defined with (J. P. May & K. Ponto 213)
- respect to functors out of small categories D. We assume once and for all that M is bicomplete. This means that M is complete (has all limits) and cocomplete (has all colimits). In particular, it has an initial object ∅ and a terminal object ∗ (the coproduct and product of the empty set of objects). (J. P. May & K. Ponto 214)
- A model structure on M consists of three interrelated classes of maps (W , C , F ), called the weak equivalences, the cofibrations, and the fibrations (J. P. May & K. Ponto 214)
- homotopy category HoM that is obtained by inverting the weak equivalences. This is a localization process that is analogous to the localization of rings at multiplicatively closed subsets, and it is characterized by an analogous universal property. Formally, this means that there must be a functor γ : M −→ HoM such that γ(w) is an isomorphism if w ∈ W and γ is initial with respect to this property. (J. P. May & K. Ponto 214)
- One might attempt to construct HoM by means of words in the morphisms of M and formal inverses of the morphisms of W , but the result of such a construction is not locally small in general (J. P. May & K. Ponto 214)
- Definition 14.1.1. A class K of maps in M is closed under retracts if, when given a commutative diagram (J. P. May & K. Ponto 214)
- The following observation is often applied to classical homotopy categories, where it shows that a retract of a homotopy equivalence is a homotopy equivalence (J. P. May & K. Ponto 214)
- there can be many model structures on M with the same weak equivalences. (J. P. May & K. Ponto 214)
- Lemma 14.1.2. In any category, if f is a retract of g and g is an isomorphism, then f is an isomorphism (J. P. May & K. Ponto 215)
- Transfinite colimits play a substantial role in the foundational literature of model category theory, and they are crucial to the construction of Bousfield localizations (J. P. May & K. Ponto 215)
- Definition 14.1.7. Let L be a class of maps in M . We say that L is left saturated if the following (redundant) closure properties hold (J. P. May & K. Ponto 216)
- Definition 14.1.10. An ordered pair (L , R) of classes of morphisms of M factors M if every morphism f : X −→ Y factors as a composite X i(f ) / Z(f ) p(f ) /Y with i(f ) ∈ L and p(f ) ∈ R. (J. P. May & K. Ponto 216)
- Mapping cylinders and mapping path fibrations [89, pp 43, 48] give the original source for the following idea, but it also arises from analogous categorical contexts. Definition 14.1.11. A weak factorization system (WFS) of M is an ordered pair (L , R) of classes of morphisms of M that factors M and satisfies both L = R and R = L (J. P. May & K. Ponto 217)
- The required equalities say that the maps in L are precisely the maps that have the LLP with respect to the maps in R and the maps in R are precisely the maps that have the RLP with respect to the maps in L . (J. P. May & K. Ponto 217)
- Category theorists also study strong factorization systems, for which the relevant lifts λ are required to be unique. 3 The difference is analogous to the difference between the class of fibrations and the class of covering maps as the choice of R. (J. P. May & K. Ponto 217)
- Definition 14.2.1. A model structure on M consists of classes (W , C , F ) of morphisms of M , the weak equivalences, cofibrations, and fibrations, such that (i) W has the two out of three property. (ii) (C , F ∩ W ) is a (functorial) weak factorization system. (iii) (C ∩ W , F ) is a (functorial) weak factorization system. (J. P. May & K. Ponto 218)
- The maps in F ∩ W are called acyclic (or trivial) fibrations; those in C ∩ W are called acyclic (or trivial) cofibrations. The definition requires every map to factor both as the composite of a cofibration followed by an acyclic fibration and as an acyclic cofibration followed by a fibration (J. P. May & K. Ponto 218)
- It also requires there to be a lift in any commutative square A i E p X > ~> ~ ~ ~ /B in which i is a cofibration, p is a fibration, and either i or p is acyclic (J. P. May & K. Ponto 218)
- By (14.2.2) and (14.2.3, to specify a model structure on a category with a chosen class of weak equivalences that satisfies the two out of three property, we need only specify either the cofibrations or the fibrations, not both (J. P. May & K. Ponto 219)
- (14.2.4) C (F ∩ W ) and F (C ∩ W ). It is usual to define model categories by requiring (14.2.4) and requiring F , C , and W to be closed under retracts. The following observation (due to Joyal) shows that our axioms imply that W is closed under retracts and are therefore equivalent to the usual ones (J. P. May & K. Ponto 219)
- Lemma 14.2.5. The class W as well as the classes C , C ∩ W , F , and F ∩ W in a model structure are subcategories that contain all isomorphisms and are closed under retracts. Therefore W is a subcategory of weak equivalences in the sense of Definition 14.1.4. (J. P. May & K. Ponto 219)
- Definition 14.2.7. An object X of a model category M is cofibrant if the unique map ∅ −→ X is a cofibration. An acyclic fibration q : QX −→ X in which QX is cofibrant is called a cofibrant approximation or cofibrant replacement 5 of X. We can obtain q by factoring ∅ −→ X (J. P. May & K. Ponto 220)
- Given two cofibrant replacements q : QX −→ X and q ′ : Q′ X −→ X, the lifting property gives a weak equivalence ξ : QX −→ Q′ X such that q ′ ◦ ξ = q (J. P. May & K. Ponto 221)
- In many model categories, either all objects are fibrant or all objects are cofibrant (but rarely both). For example, all objects are fibrant in the model structures that we shall define on U , and all objects are cofibrant in the usual model structure on simplicial sets (J. P. May & K. Ponto 221)
- For example, the following result becomes especially helpful when all objects are cofibrant or all objects are fibrant. Lemma 14.2.9 (Ken Brown’s lemma). Let F : M −→ N be a functor, where M is a model category and N is a category with a subcategory of weak equivalences. If F takes acyclic cofibrations between cofibrant objects to weak equivalences, then F takes all weak equivalences between cofibrant objects to weak equivalences (J. P. May & K. Ponto 221)
- in most examples there is a familiar and classical notion of a homotopy between maps. It is defined in terms of canonical cylinder and path objects, such as X × I and Map(I, X) in the case of spaces. Quillen [112] developed a notion of homotopy in general model categories and showed how to derive many familiar results using the model theoretic notion. (J. P. May & K. Ponto 222)
- objects (J. P. May & K. Ponto 222)
- Definition 14.3.1. A cylinder object for X ∈ M is an object Cyl X together with maps i0 : X −→ Cyl X, i1 : X −→ Cyl X, and p : Cyl (X) −→ X such that p ◦ i0 = id = p ◦ i1 and p is a weak equivalence; by the two out of three property, i0 and i1 are also weak equivalences. A cylinder object is good if the map i = i0 + i1 : X ∐ X −→ Cyl (X) is a cofibration. A good cylinder object Cyl (X) is very good if p is an acyclic fibration. Factorization of the folding map X ∐ X −→ X shows that every X has at least one very good cylinder object. A left homotopy between maps f, g : X −→ Y is a map h : Cyl (X) −→ Y such that h ◦ i0 = f and (J. P. May & K. Ponto 222)
- ◦ i1 = g, where Cyl (X) is any cylinder object for X; h is good or very good if Cyl (X) is good or very good. Define π ℓ (X, Y ) to be the set of equivalence classes of maps X −→ Y under the equivalence relation generated by left homotopy (J. P. May & K. Ponto 223)
- Lemma 14.3.3. If h is a left homotopy from f to g and either f or g is a weak equivalence, then so is the other. (J. P. May & K. Ponto 223)
- We emphasize that the definition of left homotopy allows the use of any cylinder object and that the notion of left homotopy and its good and very good variants are not equivalence relations in general (J. P. May & K. Ponto 223)
- Even in some categories with canonical cylinders, such as the category of simplicial sets, homotopy is not an equivalence relation in general (J. P. May & K. Ponto 223)
- For example, in the standard model structure on topological spaces of §17.2, X × I is a cylinder object, but it is not good unless X is cofibrant, and similarly for categories of chain complexes. (J. P. May & K. Ponto 223)
- Lemma 14.3.7 (HEP). Let i : A −→ X be a cofibration and Y be a fibrant object. Then i satisfies the right homotopy extension property with respect to Y . That is, for any good path object Cocyl Y and any maps f and h that make the following square commute, there is a lift h̃ h̃ that makes the triangles commute. A h / i Cocyl Y p0 X f h̃ h̃ ; v; v v v v Y (J. P. May & K. Ponto 224)
- Lemma 14.3.8 (CHP). Let p : E −→ B be a fibration and X be a cofibrant object. Then p satisfies the left covering homotopy property with respect to X. That is, for any good cylinder object Cyl X and any maps f and h that make the following square commute, there is a lift h̃ h̃ that makes the triangles commute. X f i0 E p Cyl X h h̃ h̃ < x< x x x x B. (J. P. May & K. Ponto 224)
- Perhaps the real force of Quillen’s approach to homotopies is the comparison between left and right homotopies. Proposition 14.3.11. Consider maps f, g : X −→ Y . (i) If X is cofibrant and f is left homotopic to g, then f is right homotopic to g. (ii) If Y is fibrant and f is right homotopic to g, then f is left homotopic to g. (J. P. May & K. Ponto 225)
- Definition 14.3.12. When X is cofibrant and Y is fibrant, we say that f is homotopic to g, written f ≃ g, if f is left or, equivalently, right homotopic to g. We then write π(X, Y ) for the set of homotopy classes of maps X −→ Y . (J. P. May & K. Ponto 226)
- Corollary 14.3.13. Let X be cofibrant and Y be fibrant. Fix a good cylinder object Cyl X and a good path object Cocyl Y . If f ≃ g, then f is left homotopic to g via a homotopy defined on Cyl X and f is right homotopic to g via a homotopy mapping to Cocyl Y . (J. P. May & K. Ponto 226)
- Theorem 14.3.14. The following versions of the dual Whitehead theorems hold. (i) If X is cofibrant and p : Z −→ Y is an acyclic fibration, then the function p∗ : π ℓ (X, Z) −→ π ℓ (X, Y ) is a bijection. (ii) If Y is fibrant and i : W −→ X is an acyclic cofibration, then the function i ∗ : π r (X, Y ) −→ π r (W, Y ) is a bijection (J. P. May & K. Ponto 226)
- Remark 14.3.16. We have used HELP and coHELP in several places, notably §3.3. The first author has long viewed them to be a central organizational convenience in classical homotopy theory (J. P. May & K. Ponto 227)
- Consider a map f : X −→ Y . Choose cofibrant replacements q : QX −→ X and q : QY −→ Y and fibrant replacements r : X −→ RX and r : Y −→ RY . (J. P. May & K. Ponto 227)
- Definition 14.4.1. Consider the full categories Mc , Mf , and Mcf of cofibrant, fibrant, and bifibrant objects of M . Define their homotopy categories hMc , hMf , and hMcf to be the categories with the same objects and with morphisms the equivalence classes of maps with respect to right homotopy, left homotopy, and homotopy, respectively. In the first two cases, we understand equivalence classes under the equivalence relation generated by right or left homotopy. (J. P. May & K. Ponto 227)
- RY . Then we can obtain lifts Qf : QX −→ QY and Rf : RX −→ RY in the diagrams (J. P. May & K. Ponto 227)
- Thus we have a kind of point set level naturality of q and r even when we do not have functors Q and R (J. P. May & K. Ponto 228)
- and we agree to write RQ for the functor hR ◦ Q : M −→ hMcf induced by chosen objectwise cofibrant and fibrant replacements, (J. P. May & K. Ponto 228)
- Definition 14.4.5. Define the homotopy category HoM to have objects the objects of M and morphism sets HoM (X, Y ) = hMcf (RQX, RQY ) = π(RQX, RQY ), with the evident composition. Define γ : M −→ HoM (X, Y ) to be the identity on objects and to send a map f to RQf . Observe that HoM is equivalent to hMcf via the functor that sends X to RQX and f to RQf . Proposition 14.4.6. The class of maps f such that γ(f ) is an isomorphism is precisely W , and every map in HoM is a composite of morphisms in γ(M ) and inverses of morphisms in γ(W ). (J. P. May & K. Ponto 228)
- Theorem 14.4.7. The functor γ : M −→ HoM is a localization of M at W . (J. P. May & K. Ponto 229)
- Theorem 14.4.8 (Whitehead). The following versions of the dual Whitehead theorems hold. (i) A map p : Z −→ Y between fibrant objects is a weak equivalence if and only if p∗ : π(X, Z) −→ π(X, Y ) is a bijection for all cofibrant objects X. (ii) A map i : W −→ X betweeen cofibrant objects is a weak equivalence if and only if i ∗ : π(X, Y ) −→ π(W, Y ) is a bijection for all fibrant objects Y . (J. P. May & K. Ponto 229)
- In any category C , a map f : A −→ B is an isomorphism if and only if either f∗ : C (C, A) −→ C (C, B) or f ∗ : C (B, C) −→ C (A, C) is an isomorphism for one C in each isomorphism class of objects in C . In fact, we need only test on objects C and D that are isomorphic to A and B (J. P. May & K. Ponto 229)
- It is important to understand when functors defined on M are homotopy invariant, in the sense that they take homotopic maps to the same map. There are three results along this line, the most obvious of which is the least useful. (J. P. May & K. Ponto 229)
- Lemma 14.4.9. Any functor F : M −→ H that takes weak equivalences to isomorphisms identifies left or right homotopic maps. (J. P. May & K. Ponto 230)
- However, the hypothesis on F here is too strong and rarely holds in practice. (J. P. May & K. Ponto 230)
- Lemma 14.4.10. Any functor F : Mc −→ H that takes acyclic cofibrations to isomorphisms identifies right homotopic maps. (J. P. May & K. Ponto 230)
- There is a standard construction of WFS’s and model categories, which is based on Quillen’s “small object argument”. The latter is a general method for starting with a set, I say, of maps of M and constructing from I a functorial WFS ( (I ), I ). (J. P. May & K. Ponto 231)
- The small object argument is often repeated in the model category literature, but it admits a useful variant that we feel has not been sufficiently emphasized in print. 1 We call the variant the compact object argument (J. P. May & K. Ponto 231)
- In many basic examples, such as topological spaces, chain complexes, and simplicial sets, only sequential colimits are required. When this is the case, we obtain a more concrete type of cofibrantly generated model category called a compactly generated model category. In such cases we are free to ignore transfinite cell complexes. Compactly generated model categories are attractive to us since the relevant cell theory is much closer to classical cell theory in algebraic topology (e.g. [89]) and in homological algebra (e.g. [74]) than the transfinite version (J. P. May & K. Ponto 231)
- We feel that the general case of cofibrantly generated model categories is overemphasized in the model category literature, and we urge the reader not to get bogged down in the details of the requisite smallness condition (J. P. May & K. Ponto 231)
- We describe model structures in over and under “slice categories” in §15.3. This gives a frequently used illustration of how one creates new model structures from given ones (J. P. May & K. Ponto 231)
- The essential starting point is to define I-cell complexes. When I is the set {S n −→ Dn+1 } of standard cell inclusions, all CW complexes will be examples of I-cell complexes as we define them (J. P. May & K. Ponto 232)
- Remark 15.1.12. There is a general property of a category, called local presentability [12, §5.2], that ensures that any set I permits the small object argument. In such categories, this leads to a more uniform and aesthetically satisfactory treatment of the small object argument. It is satisfied by most algebraically defined categories. It is not satisfied by the category of compactly generated topological spaces. (J. P. May & K. Ponto 236)
- However, if instead of using all compact Hausdorff spaces in the definition of compactly generated spaces [89, p. 37], one only uses standard simplices, one obtains the locally presentable category of “combinatorial spaces”. It appears that one can redo all of algebraic topology with combinatorial spaces replacing compactly generated spaces (J. P. May & K. Ponto 236)
- Definition 15.2.1. A model category M is cofibrantly generated if there are (small) sets I and J of maps such that C = C (I) and C ∩W = C (J ) and therefore F = J and F ∩W = I . The sets I and J are called the generating cofibrations and generating acyclic cofibrations. We say that M is compactly generated if, further, I and J are compact, in which case only ordinary sequential cell complexes are needed to define C (I) and C (J ) (J. P. May & K. Ponto 236)
- Remark 15.2.2. A combinatorial model category is a locally presentable category that is also a cofibrantly generated model category [9]. (J. P. May & K. Ponto 236)
- For example, Theorem 15.2.3 is often used to transport a model structure across an adjunction, as we formalize in Theorem 16.2.5 below, and then only the acyclicity condition need be verified (J. P. May & K. Ponto 237)
- The acyclicity condition clearly holds if J ⊂ W and W is a left saturated class of maps in the sense of Definition 14.1.7. This rarely applies since pushouts of weak equivalences are generally not weak equivalences. However, pushouts of coproducts of maps in J often are weak equivalences, and verifying that is usually the key step in verifying the acyclicity condition. (J. P. May & K. Ponto 237)
- Definition 15.3.1. Let M be a category and B be an object of M . The slice category B/M of objects under B has objects the maps i = iX : B −→ X in M and morphisms the maps f : X −→ Y such that f ◦ iX = iY (J. P. May & K. Ponto 238)
- Definition 15.4.1. A model category M is left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence. This means that if i is a cofibration and f is a weak equivalence in a pushout diagram A f i B j X g /Y then g is a weak equivalence. (J. P. May & K. Ponto 240)
- The following concept will play a significant role in our discussion of Bousfield localization in Chapter 19, and it is important throughout model category theory. (J. P. May & K. Ponto 240)
- Over and under model structures often play a helpful technical role in proofs, as is well illustrated by their use in the proof of the following result. One point is that appropriate maps can be viewed as cofibrant or fibrant objects in model categories, allowing us to apply results about such objects to maps. (J. P. May & K. Ponto 240)
- Proposition 15.4.2. Let M be a model category. Then any pushout of a weak equivalence between cofibrant objects along a cofibration is a weak equivalence, hence M is left proper if every object of M is cofibrant (J. P. May & K. Ponto 240)
- Definition 15.4.3. A model category M satisfies the left gluing lemma if, for any commutative diagram A f C i o g k /B h A ′ o C ′ j o ℓ B′ in which i and j are cofibrations and f , g, and h are weak equivalences, the induced map of pushouts A ∪C B −→ A ′ ∪C ′ B ′ is a weak equivalence. (J. P. May & K. Ponto 241)
- Proposition 15.4.4. A model category M is left or right proper if and only if it satisfies the left or right gluing lemma (J. P. May & K. Ponto 241)
- In our examples, we sometimes prove the gluing lemma directly, because it is no more difficult. However, in deeper examples it is often easier to check that the model structure is left proper and to use the previous result to deduce that the left gluing lemma holds (J. P. May & K. Ponto 242)
- Let i : A −→ X be a map in M and Z be an object of M . Consider the induced map (15.5.1) i ∗ : [X, Z] −→ [A, Z] (J. P. May & K. Ponto 242)
- of hom sets in HoM . We describe how we can sometimes deduce that i ∗ is a bijection directly from lifting properties and, conversely, how we can sometimes deduce lifting properties when i ∗ is a bijection. While i will be a cofibration and Z will be fibrant, the force of these results comes from the fact that the relevant lifting properties concern pairs of maps, neither of which need be a weak equivalence (J. P. May & K. Ponto 243)
- it well illustrates how important it is to know whether or not a model structure is left or right proper (J. P. May & K. Ponto 243)
- Definition 15.5.2. Define the homotopy pushout (or double mapping cylinder) M (f, g) of a pair of maps f : A −→ X and g : A −→ Y to be the pushout of f ∐ g : A ∐ A −→ X ∐ Y along the cofibration i0 + i1 : A ∐ A −→ Cyl A of a good cylinder object Cyl A (J. P. May & K. Ponto 243)
- Lemma 15.5.6. Any map f : X −→ Y has a cofibrant approximation. (J. P. May & K. Ponto 243)
- Left and right Quillen adjoints are the most common source of left and right derived functors (J. P. May & K. Ponto 247)
- we describe the 2-categorical way of understanding the passage to derived homotopy categories and we explain that double categories, rather than categories or 2-categories, give the appropriate conceptual framework for understanding maps between model categories (J. P. May & K. Ponto 247)
- One small point that deserves more emphasis than it receives in the literature is that there is a familiar classical notion of homotopy in the enriched categories that appear in nature, and the model categorical notion of homotopy and the classical notion of homotopy can be used interchangably in such contexts. (J. P. May & K. Ponto 247)
- Having defined model categories, we want next to define functors and natural transformations between them in such a way that they give “derived” functors and natural transformations on passage to their derived homotopy categories (J. P. May & K. Ponto 247)
- We say that a functor F : M −→ N between categories with weak equivalences is homotopical if it takes weak equivalences to weak equivalences, and we say that a functor F : M −→ HoN or more generally a functor F : M −→ H for any other category H is homotopical if it takes weak equivalences to isomorphisms. (J. P. May & K. Ponto 247)
- . The universal property of localization then gives a functor ̃ F̃ : HoM −→ HoN in the first case or ̃ F̃ : HoM −→ H in the general case (J. P. May & K. Ponto 247)
- The functor ̃ F̃ is a derived functor of F . However, this will not suffice for the applications. As we have already said, most functors F that we encounter are not homotopical, and then it is too much to expect that diagrams such as those above commute; rather, we often obtain diagrams like these that commute up to a natural transformation that is characterized by a universal property (J. P. May & K. Ponto 248)
- We note parenthetically that there are also functors that in some sense deserve the name of a derived functor and yet are neither left nor right derived in the sense we are about to define. The theory of such functors is not well understood, but they appear in applications (e.g. [93]) and have been given a formal description in [124]. (J. P. May & K. Ponto 248)
- Definition 16.1.1. A left derived functor of a functor F : M −→ H is a functor LF : HoM −→ H together with a natural transformation µ : LF ◦ γ −→ F such that for any functor K : HoM −→ H and natural transformation ξ : K ◦γ −→ F , there is a unique natural transformation σ : K −→ LF such that the composite µ ◦ (σ · γ) : K ◦ γ −→ LF ◦ γ −→ F coincides with ξ. That is, (LF, µ) is terminal among pairs (K, ξ). (J. P. May & K. Ponto 248)
- We use · to denote the composite of a natural transformation and a functor; categorically, that is often called whiskering (J. P. May & K. Ponto 248)
- Since left derived functors are characterized by a universal property, they are unique up to canonical isomorphism if they exist. Confusingly, they are examples of what are known categorically as right Kan extensions (J. P. May & K. Ponto 248)
- Proposition 16.1.3. If F : M −→ H takes acyclic cofibrations between cofibrant objects to isomorphisms, then the left derived functor (LF, µ) exists. Moreover, for any cofibrant object X of M , µ : LF X −→ F X is an isomorphism (J. P. May & K. Ponto 249)
- Definition 16.2.1. Let F : M −→ N and U : N −→ M be left and right adjoint. The pair (F, U ) is a Quillen adjunction if the following equivalent conditions are satisfied. (J. P. May & K. Ponto 251)
- The Quillen adjunction (F, U ) is a Quillen equivalence if for any map f : F X −→ Y with adjoint g : X −→ U Y , where X is cofibrant and Y is fibrant, f is a weak equivalence in N if and only if g is a weak equivalence in M . (J. P. May & K. Ponto 251)
- Many of the applications concern adjunctions (F, U ) where F specifies free structured objects (such as monoids, algebras, etc) and U forgets the structure on these objects. (J. P. May & K. Ponto 251)
- Proposition 16.2.2. If (F, U ) is a Quillen adjunction, then the total derived functors LF and RU exist and form an adjoint pair. If (F, U ) is a Quillen equivalence, then (LF, RU ) is an adjoint equivalence between HoM and HoN . (J. P. May & K. Ponto 251)
- The criterion of Proposition 16.2.3(v) is especially useful since a standard way to build a model category structure on N is to use an adjunction (F, U ) to create it from a model structure on M , setting WN = U −1 (WM ) and FN = U −1 (FM ). The following result, which is [63, 11.3.2], is frequently used for this purpose. (J. P. May & K. Ponto 252)
- More deeply, it masks one of the greatest difficulties of model category theory and of the theory of derived functors in general. Good properties of left adjoints are preserved under composition of left adjoints. However, it is in practice very often necessary to compose left adjoints with right adjoints. Results about such composites are hard to come by. They are often truly deep mathematics. (J. P. May & K. Ponto 253)
- However, the categorical considerations of greatest relevance to the applications of model category theory concern enrichment from hom sets to hom objects (J. P. May & K. Ponto 253)
- A monoidal structure on a category V is a product, ⊗ say, and a unit object I such that the product is associative and unital up to coherent natural isomorphisms (J. P. May & K. Ponto 253)
- Informally, coherence means that diagrams that intuitively should commute do in fact commute. (The symmetry coherence admits a weakening that gives braided monoidal categories, but those will not concern us.) A symmetric monoidal category V is closed if it has internal hom objects V (X, Y ) in V together with adjunction isomorphisms V (X ⊗ Y, Z) ∼ = ∼ = V (X, V (Y, Z)). These isomorphisms of hom sets imply isomorphisms of internal hom objects in V V (X ⊗ Y, Z) ∼ = ∼ = V (X, V (Y, Z)). (J. P. May & K. Ponto 254)
- From now on, we let V be a bicomplete closed symmetric monoidal category. Such categories appear so often in nature that category theorists have invented a name for them: such a category is often called a “cosmos”. (J. P. May & K. Ponto 254)
- When ⊗ is the cartesian product, we say that V is cartesian closed, but the same category V can admit other symmetric monoidal structures (J. P. May & K. Ponto 254)
- (i) The category Set of sets is closed cartesian monoidal. (ii) The category U of (compactly generated) spaces is cartesian closed. (J. P. May & K. Ponto 254)
- The category U∗ of based spaces is closed symmetric monoidal under the smash product. The smash product would not be associative if we used just spaces, rather than compactly generated spaces (J. P. May & K. Ponto 254)
- The category sSet of simplicial sets is cartesian closed. A simplicial set is a contravariant functor ∆ −→ Set, where ∆ is the category of sets n = {0, 1, · · · , n} and monotonic maps. There are n-simplices ∆[n] in sSet, and n 7→ ∆[n] gives a covariant functor ∆ −→ sSet (J. P. May & K. Ponto 254)
- For a commutative ring R, the category MR of R-modules is closed symmetric monoidal under the functors ⊗R and HomR ; in particular, the category Ab of abelian groups is closed symmetric monoidal (J. P. May & K. Ponto 254)
- For a commutative ring R, the category ChR of Z-graded chain complexes of R-modules (with differential lowering degree) is closed symmetric monoidal under the graded tensor product and hom functors (X ⊗R Y )n = Σp+q=n Xp ⊗R Yq ; d(x ⊗ y) = d(x) ⊗ y + (−1) p x ⊗ d(y). HomR (X, Y )n = Πi HomR (Xi , Yi+n ); d(f )i = d ◦ fi − (−1) n fi−1 ◦ d. Here the symmetry γ : X ⊗ Y −→ Y ⊗ X is defined with a sign, γ(x ⊗ y) = (−1) pq y ⊗ x for x ∈ Xp and y ∈ Yq . (J. P. May & K. Ponto 254)
- The category Cat of small categories is cartesian closed (J. P. May & K. Ponto 254)
- Definition 16.3.3. Let V be a symmetric monoidal category. A V -category M , or a category M enriched over V , consists of (i) a class of objects, with typical objects denoted X, Y , Z; (ii) for each pair of objects (X, Y ), a hom object M (X, Y ) in V ; (iii) for each object X, a unit map idX : I −→ M (X, X) in V ; (iv) for each triple of objects (X, Y, Z), a composition morphism in V M (Y, Z) ⊗ M (X, Y ) −→ M (X, Z). (J. P. May & K. Ponto 255)
- Categories as usually defined are categories enriched in Set. (ii) Categories enriched in Ab are called Ab-categories. They are called additive categories if they have zero objects and biproducts [77, p. 196]. They are called abelian categories if, further, all maps have kernels and cokernels, every monomorphism is a kernel, and every epimorphism is a cokernel (J. P. May & K. Ponto 256)
- (iii) Categories enriched in U are called topological categories. (iv) Categories enriched in sSet are called simplicial categories. (v) Categories enriched in ChR for some R are called DG-categories. (vi) Categories enriched in Cat are called (strict) 2-categories and, inductively, categories enriched in the cartesian monoidal category of (n − 1)-categories are called (strict) n-categories. (J. P. May & K. Ponto 256)
- For any ring R, not necessarily commutative, the category of left R-modules is abelian (J. P. May & K. Ponto 256)
- Most of the model category literature focuses on simplicial categories (J. P. May & K. Ponto 256)
- However, there are important enriched categorical notions that take account of the extra structure given by the enrichment and are not just reinterpretations of ordinary categorical notions. In particular, there are weighted (or indexed) colimits and limits. The most important of these (in non-standard notation) are tensors X ⊙V (sometimes called copowers) and cotensors (sometimes called powers) Φ(V, X) in M for objects X ∈ M and V ∈ V . These are characterized by natural isomorphisms (16.3.6) M (X ⊙ V, Y ) ∼ = ∼ = V (V, M (X, Y )) ∼ = ∼ = M (X, Φ(V, Y )) of hom sets. (J. P. May & K. Ponto 257)
- The functor V (I, −) is a right adjoint and therefore preserves limits, such as pullbacks; (J. P. May & K. Ponto 259)
- We have defined categories enriched in a symmetric monoidal category V , and we have observed that one such V is the cartesian monoidal category Cat, the category of categories and functors (J. P. May & K. Ponto 262)
- We have also observed that a category enriched in Cat is called a 2-category. An example is Cat itself. (J. P. May & K. Ponto 262)
- We think of functors as morphisms between categories and natural transformations as morphisms between functors. We also have two 2-categories ModCatℓ and ModCatr of model categories. Their objects (or 0-cells) are model categories, their morphisms (or 1-cells) are Quillen left adjoints and Quillen right adjoints, respectively, and their morphisms between morphisms (or 2-cells) are natural transformations in both cases. (J. P. May & K. Ponto 262)
- A pseudo-functor F : C −→ D between 2-categories (J. P. May & K. Ponto 262)
- assigns 0-cells, 1-cells, and 2-cells in D to 0-cells, 1-cells, and 2-cells in C . For each fixed pair X, Y of 0-cells, F specifies a functor C (X, Y ) −→ D(FX, FY ). However, for 0-cells X, F(IdX ) need not equal IdFX and, for composable 1-cells F and G, F(G ◦ F ) need not equal FG ◦ FF . Rather, there are 2-cell isomorphisms connecting these. These isomorphisms are subject to coherence axioms asserting that certain associativity and left and right unity diagrams commute (J. P. May & K. Ponto 263)
- Proposition 16.5.1. Passage from model categories to their derived homotopy categories HoM , derived functors, and derived natural transformations specify pseudo-functors L : ModCatℓ −→ Cat and R : ModCatr −→ Cat. (J. P. May & K. Ponto 263)
- The proper framework is given by viewing model categories as forming not a pair of 2-categories, but rather as a single double category (J. P. May & K. Ponto 263)
- Just as we can define a category enriched in any symmetric monoidal category V , we can define an internal category in any complete category V . It has object and morphism objects Ob and Mor in V together with maps S, T : Mor −→ Ob, I : Ob −→ Mor, and C : Mor ×Ob Mor −→ Mor in V , called source, target, identity, and composition (J. P. May & K. Ponto 263)
- An ordinary category is an internal category in the category of sets. Internal categories in the category U of topological spaces appear frequently. A double category is just an internal category in Cat. A 2-category can be viewed as a double category whose object category is discrete, meaning that it has only identity morphisms (J. P. May & K. Ponto 263)
- It has 0-cells, namely the objects of the category Ob, it has both “vertical” and “horizontal” 1-cells, namely the morphisms of Ob and the objects of Mor, and it has 2-cells, namely the morphisms of Mor. (J. P. May & K. Ponto 263)
- Thus let M be a category with a subcategory W of weak equivalences. We form a double category D(M , W ) whose objects are the objects of M , whose horizontal and vertical 1-cells are the morphisms of M and of W , and whose 2-cells are commutative diagrams X f v Y w X ′ g / Y′ in which v, w ∈ W . Thinking in terms of the arrow category of M , we can view this square as a morphism v −→ w between vertical arrows or as a morphism f −→ g between horizontal arrows. (J. P. May & K. Ponto 264)
- The reason for introducing this language is that it gives a way of expressing L and R as part of a single kind of functor, namely a double pseudofunctor F : D −→ E between double categories. Such an F assigns 0-cells, vertical 1-cells, horizontal 1-cells, and 2-cells in D to the corresponding kinds of cells in C . (J. P. May & K. Ponto 264)
- Corollary 16.5.4. If F is both a Quillen left adjoint and a Quillen right adjoint, then LF ∼ = ∼ RF . = The real force of Theorem 16.5.3 concerns the comparisons it induces between composites of left and right derived adjoint base change functors (J. P. May & K. Ponto 264)
- Theorem 16.5.3. Model categories are the 0-cells of a double category Mod whose vertical and horizontal 1-cells are the Quillen left and right adjoints and whose 2-cells are the natural transformations. There is a double pseudofunctor F : Mod −→ Q(Cat) such that Fv = L and Fh = R. (J. P. May & K. Ponto 264)
- This result encodes many relationships between left and right Quillen adjoints in a form that is familiar in other categorical contexts. The Quillen adjunctions (F, U ) are examples of the correct categorical notion of an adjunction in a double category, called a conjunction in (J. P. May & K. Ponto 264)
- The central point we want to make is that there are three intertwined model structures on the relevant categories, and that the least familiar, which is called the mixed model structure and was introduced relatively recently by Michael Cole [31], is in some respects the most convenient (J. P. May & K. Ponto 265)
- In fact, we shall argue that algebraic topologists, from the very beginning of the subject, have by preference worked implicitly in the mixed model structure on topological spaces (J. P. May & K. Ponto 265)
- Theorem 15.3.6, which shows how to construct model structures on the category of based spaces from model structures on the category of unbased spaces. (J. P. May & K. Ponto 265)
- The most obvious homotopical notion of a weak equivalence is an actual homotopy equivalence. We call such maps h-equivalences for short. It is natural to expect there to be a model category structure on the category U of compactly generated spaces in which the weak equivalences are the h-equivalences (J. P. May & K. Ponto 265)
- The cofibrations are the maps i : A −→ X that satisfy the homotopy extension property (HEP) (J. P. May & K. Ponto 265)
- The fibrations are the maps p : E −→ B that satisfy the covering homotopy property (CHP). This means that for all spaces A they satisfy the RLP with respect to the inclusion i0 : A −→ A × I. (J. P. May & K. Ponto 265)
- This means that for all spaces B they satisfy the LLP with respect to the map p0 : B I −→ B given by evaluation at 0. (J. P. May & K. Ponto 265)
- call these Hurewicz cofibrations and Hurewicz fibrations, conveniently abbreviated to h-cofibrations and h-fibrations (J. P. May & K. Ponto 266)
- These cofibrations and fibrations were not considered by Quillen in his paper introducing model categories [112], but Strøm [128] later proved a version of the following result. Technically, he worked in the category of all spaces, not just the compactly generated ones, and in that category the model theoretic cofibrations are the Hurewicz cofibrations that are closed inclusions. However, as we left as an exercise in [89, p. 46], the Hurewicz cofibrations in U are closed inclusions (J. P. May & K. Ponto 266)
- Theorem 17.1.1 (h-model structure). The category U is a monoidal model category whose weak equivalences, cofibrations, and fibrations, denoted (Wh , Ch , Fh ), are the h-equivalences, h-cofibrations, and h-fibrations. All spaces are both h-fibrant and h-cofibrant, hence this model structure is proper (J. P. May & K. Ponto 266)
- Corollary 17.1.2. The category U∗ of based spaces in U is a proper model category whose weak equivalences, cofibrations, and fibrations are the based maps that are h-equivalences, h-cofibrations, and h-fibrations when regarded as maps in U . The pair (T, U ), where T is given by adjoining disjoint basepoints and U is the forgetful functor, is a Quillen adjunction relating U to U∗ . Moreover, U∗ is a monoidal model category with respect to the smash product. (J. P. May & K. Ponto 266)
- Remark 17.1.3. In the first half of the book, we worked thoughout in the category T of nondegenerately based spaces in U . These are precisely the hcofibrant objects in the h-model category of all based spaces. There is an elementary cofibrant approximation functor, called whiskering. (J. P. May & K. Ponto 266)
- The left adjoint (−) × I commutes with colimits (J. P. May & K. Ponto 268)
- We have completed the proof that U is a model category (J. P. May & K. Ponto 268)
- In the first half of the book, we assumed that all spaces had the homotopy types of CW-complexes. Therefore there was no distinction between a homotopy equivalence and a weak homotopy equivalence. (J. P. May & K. Ponto 268)
- Note that the most natural kind of based cell complexes would start with based cells S n −→ Dn+1 for chosen basepoints of spheres and would have based attaching maps. While such cell complexes are useful, they can only model connected based spaces (J. P. May & K. Ponto 271)
- When we specialize to spaces, the statements in the rest of the section give model theoretic refinements of classical results, and the reader may wish to skip to the next section, referring back to this one as needed. For example, the first part of the following result refines the Whitehead theorem that a weak equivalence between CW complexes is a homotopy equivalence (J. P. May & K. Ponto 272)
- Therefore the category of CW homotopy types in T used in the first half of this book is the full subcategory of the category U∗ whose objects are the mcofibrant based spaces (J. P. May & K. Ponto 278)
- It has long been accepted that the q-equivalences give the definitively right weak equivalences for classical homotopy theory. (J. P. May & K. Ponto 278)
- The fibrations most frequently used in practice are the Hurewicz fibrations rather than the Serre fibrations. There are good reasons for this. For example, Hurewicz fibrations, but not Serre fibrations, are determined locally, in the sense of the following result (e.g. [84, 3.8]). It generalizes a theorem of Hurewicz [67]. Theorem 17.4.4. Let p : E −→ B be a surjective map and assume that B has a numerable open cover {U } such that each restriction p−1 U −→ U is an h-fibration. Then p is an h-fibration. (J. P. May & K. Ponto 278)
- The proof uses the characterization of h-fibrations in terms of path lifting functions [89, §7.2], and that characterization itself often gives an easy way of checking that a map is an h-fibration. Paradoxically, this means that it is often easiest to prove that a map is a q-fibration by proving the stronger statement that it is an h-fibration (J. P. May & K. Ponto 278)
- These considerations argue for the h-fibrations, Fh , as our preferred subcategory of fibrations (J. P. May & K. Ponto 278)
- The mixed model structure has all of the good formal properties of the q-model structure. It is almost certainly not cofibrantly generated, but that is irrelevant to the applications (J. P. May & K. Ponto 278)
- Any map with the name cofibration should at least be a classical Hurewicz cofibration, and an m-cofibration is an h-cofibration that is a q-cofibration up to homotopy equivalence (J. P. May & K. Ponto 278)
- Proposition 17.3.4(i) generalizes the relative version of the Whitehead theorem that a weak equivalence between cell complexes is a homotopy equivalence (J. P. May & K. Ponto 278)
- Cell complexes are more general than CW complexes, but any cell complex is homotopy equivalent to a CW complex. Therefore, in the mixed model structure, we can use cell and CW complexes interchangeably (J. P. May & K. Ponto 279)
- One is to first approximate any space X by a weakly equivalent CW complex, (e.g. [89, §10.5]), and then use the Whitehead theorem (e.g. [89, §10.3]) to show that any cell approximation of X is homotopy equivalent to the constructed CW approximation. The other is to inductively deform any given cell complex to a homotopy equivalent CW complex by cellular approximation of its attaching maps. The geometric realization of the total singular complex of X gives a particularly nice functorial CW approximation (e.g. [89, §16.2]), and it is a functorial cofibrant approximation in both the q and the m-model structures. (J. P. May & K. Ponto 279)
- A central reason for preferring the m to the q-model structure is that it is generally quite hard to check whether or not a given space is actually homeomorphic to a cell or CW complex, whereas there are powerful classical theorems that characterize spaces of the homotopy types of CW complexes. (J. P. May & K. Ponto 279)
- For an elementary example of this, there are many contractible spaces that cannot be homeomorphic to cell complexes; cones on badly behaved spaces give examples (J. P. May & K. Ponto 279)
- Although the standard model structure on simplicial sets is very convenient and useful, the proofs that it is indeed a model structure are notoriously involved (J. P. May & K. Ponto 279)
- The senior author has long believed that simpler proofs should be possible (J. P. May & K. Ponto 279)
- In retrospect, each gives a fibrant replacement functor on simplicial sets with unusually nice properties: both of these functors preserve fibrations and finite limits. Our proof of the model axioms requires use of these properties, but it is irrelevant which of the functors we use. (J. P. May & K. Ponto 280)
- Our starting point is the following definition, part (iii) of which is not but ought to be the standard definition of a weak equivalence of simplicial sets (J. P. May & K. Ponto 280)
- Definition 17.5.1. Let f : X −→ Y be a map of simplicial sets. (i) f is a cofibration if the map fq : Xq −→ Yq of q-simplices is an injection for each q; equivalently, f is a categorical monomorphism. In particular, every simplicial set is cofibrant. (ii) f is a fibration if it is a Kan fibration. This means that f satisfies the RLP with respect to all horns Λ k [n] −→ ∆[n]. In particular, the fibrant objects are the Kan complexes. (iii) f is a weak equivalence if f ∗ : π(Y, Z) −→ π(X, Z) is a bijection for all Kan complexes Z. (J. P. May & K. Ponto 280)
- Let (T, S) denote the adjoint pair between S and U given by geometric realization and the total singular complex; T X is usually denoted |X|, (J. P. May & K. Ponto 280)
- Note that in contrast with spaces, homotopy groups do not enter into the definition of weak equivalences. One reason is that they are not easily defined for general simplicial sets. However, they admit a direct combinatorial definition for Kan complexes (J. P. May & K. Ponto 280)
- The equivalence of (ii) and (iii) is formal. Since the Kan complexes are the bifibrant objects in S , this result should be viewed as the simplicial analogue of the Whitehead theorem that a weak equivalence between CW complexes is a homotopy equivalence (J. P. May & K. Ponto 281)
- Lemma 17.5.5. The functor S takes spaces to Kan complexes, takes Serre fibrations to Kan fibrations, and preserves limits (J. P. May & K. Ponto 281)
- Lemma 17.5.6. The functor T preserves finite limits, hence so does the composite functor ST . (J. P. May & K. Ponto 281)
- Theorem 17.5.10. For a Kan complex X, the unit η : X −→ ST X of the adjunction is a combinatorial weak equivalence and therefore a homotopy equivalence. Corollary 17.5.11. For a space Y , the counit ε : T SY −→ Y of the adjunction is a q-equivalence (J. P. May & K. Ponto 282)
- Proof. As with any adjunction, the composite SY η / ST SY Sε SY is the identity map (J. P. May & K. Ponto 282)
- Corollary 17.5.13. For any simplicial set X, the unit η : X −→ ST X is a weak equivalence (J. P. May & K. Ponto 283)
- Proposition 17.5.12 means that the left adjoint T creates the weak equivalences in S . In most other adjoint pair situations, it is the right adjoint that creates the weak equivalences. The difference is central to the relative difficulty in proving the model category axioms in S . (J. P. May & K. Ponto 283)
- As a left adjoint, T preserves all colimits (J. P. May & K. Ponto 284)
- Remark 17.6.7. There is no h-model structure on simplicial sets in the literature, and it does not seem sensible to try to define one. One point is just that the unit interval simplicial set I = ∆[1] is asymmetric and the obvious notion of homotopy is not an equivalence relation. (J. P. May & K. Ponto 288)
- Again, as in topology, we believe that the m-model structure is central and deserves much more attention than it has received in the literature (J. P. May & K. Ponto 289)
- The category ChR is bicomplete. Limits and colimits in ChR are just limits and colimits of the underlying R-modules, constructed degreewise, with the naturally induced differentials. Here we use the term R-module for a graded R-module, without differentials (or with differential identically zero). (J. P. May & K. Ponto 289)
- Recall from §16.3 that a cosmos is a bicomplete closed symmetric monoidal category. The category ChZ is a cosmos under ⊗ and Hom (J. P. May & K. Ponto 289)
- The category ChR is enriched, tensored and cotensored over ChZ . The chain complex of morphisms X −→ Y is HomR (X, Y ), where HomR (X, Y ) is the subcomplex of Hom(X, Y ) consisting of those maps f that are maps of underlying R-modules. (J. P. May & K. Ponto 289)
- Observe that since the zero module 0 is both an initial and terminal object in ChR , the analogy to make is with based rather than unbased spaces. For n ∈ Z, we define S n , the n-sphere chain complex, to be Z concentrated in degree n with zero differential. For any integer n, we define the n-fold suspension Σn X of an R-chain complex X to be X ⊗ S n . Thus (Σn X)n+q ∼ = Xq . The notation is motivated by the observation that if we define πn (X) to be the abelian group of chain homotopy classes of maps S n −→ X (ignoring the R-module structure), then πn (X) = Hn (X). The motto is that homology is a special case of homotopy and that homological algebra is a special case of homotopical algebra. (J. P. May & K. Ponto 290)
- We used the notations ⊙ and Φ for tensors and cotensors earlier, but we will use ⊗ and Hom here, where these again mean ⊗Z and HomZ (J. P. May & K. Ponto 290)
- Analogously, we define D n+1 to be the (n + 1)-disk chain complex. It is Z in degrees n and n + 1 and zero in all other degrees. There is only one differential that can be non-zero, and we choose that differential to be the identity map Z −→ Z. The copy of Z in degree n is identified with S n and is quite literally the boundary of Dn+1 . We agree to write SR n R n+1 = R ⊗ S n and DR R = R ⊗ Dn+1 . (J. P. May & K. Ponto 290)
- We define I to be the cellular chains of the unit interval. It is the chain complex with one basis element [I] in degree 1, two basis elements [0] and [1] in degrees 0, and differential d([I]) = [0] − [1]. We define a homotopy f ≃ g between maps of R-chain complexes X −→ Y to be a map of R-chain complexes h : X ⊗ I −→ Y that restricts to f and g on X ⊗ [0] and Y ⊗ [1]. (J. P. May & K. Ponto 290)
- In ChR , coproducts are direct sums and the pushout of maps f : A −→ X and g : A −→ Y is the “difference cokernel” (X ⊕ Y )/Im(f − g). (J. P. May & K. Ponto 290)
- Definition 18.1.1. Let f : X −→ Y be a map of R-chain complexes. Define the mapping cylinder M f to be the pushout Y ∪f (X ⊗ I) of the diagram Yo X f i0 X ⊗ I. Define the mapping cocylinder N f to be the pullback X ×f Hom(I, Y ) of the diagram X f /Y o Hom(I, Y ). (J. P. May & K. Ponto 290)
- We have two natural categories of weak equivalences in ChR . The h-equivalences are the homotopy equivalences of R-chain complexes, and the q-equivalences are the quasi-isomorphisms, namely those maps of R-chain complexes that induce an isomorphism on passage to the homology of the underlying chain complexes (J. P. May & K. Ponto 290)
- chain homotopic maps induce the same map on homology (J. P. May & K. Ponto 291)
- We let hChR denote the ordinary homotopy category of ChR and call it the classical homotopy category of ChR . It is obtained from ChR by passing to homotopy classes of maps or, equivalently, by inverting the homotopy equivalences (J. P. May & K. Ponto 291)
- We let HoChR denote the category obtained from ChR , or equivalently from hChR , by formally inverting the quasi-isomorphisms (J. P. May & K. Ponto 291)
- Definition 18.2.1. An h-cofibration is a map i : A −→ X in ChR that satisfies the homotopy extension property (HEP). That is, for all B ∈ ChR , i satisfies the LLP with respect to the map p0 : B I −→ B given by evaluation at the zero cycle [0]. An h-fibration is a map p : E −→ B that satisfies the covering homotopy property (J. P. May & K. Ponto 291)
- Consider the category A/ChR of chain complexes under a chain complex R. Let X and Y be R-chain complexes under A, with given maps i : A −→ X and j : A −→ Y . Two maps f, g : X −→ Y under A are said to be homotopic under A, or relative to A, if there is a homotopy h : X ⊗ I −→ Y between them such that h(a ⊗ [I]) = 0 for a ∈ A. That is, h restricts on A ⊗ I to the algebraic version of the constant homotopy at j. A cofiber homotopy equivalence is a homotopy equivalence under A. (J. P. May & K. Ponto 292)
- Corollary 18.2.7. Let p : E −→ B be an h-acyclic h-fibration. Then ker(p) is contractible and p is isomorphic over B to the projection B ⊕ ker(p) −→ B. (J. P. May & K. Ponto 293)
- Lemma 18.2.8. Let 0 /X f /Y g /Z /0 be an exact sequence of R-chain complexes whose underlying exact sequence of Rmodules splits degreewise. Then f is a homotopy equivalence if and only if Z is contractible and g is a homotopy equivalence if and only if X is contractible. (J. P. May & K. Ponto 293)
- Since the functor HomR (A, −) preserves finite direct sums of underlying graded R-modules (J. P. May & K. Ponto 293)
- more sophisticated algebraic situations, there are h, r, and q-model structures, and they are all different. This happens, for example, if R is a commutative ring, A is a DG R-algebra, and we consider model structures on the category of differential graded A-modules 2 . In this situation, A need not be projective as an R-module, and then functors such as ⊗A and HomA rarely preserve exact sequences. (J. P. May & K. Ponto 294)
- Relative homological algebra rectifies this by restricting the underlying notion of an exact sequence of A-modules to sequences that are degreewise split exact as sequences of graded R-modules (J. P. May & K. Ponto 294)
- Definition 18.3.3. A map f : X −→ Y of R-chain complexes is an r-cofibration if it is a degreewise split monomorphism; it is an r-fibration if it is a degreewise split epimorphism (J. P. May & K. Ponto 294)
- Of course, such splittings are given by maps of underlying graded R-modules which need not be maps of chain complexes. The following result (due to Cole [30]) shows that the splittings can be deformed to chain maps if the given R-split maps are homotopy equivalences (J. P. May & K. Ponto 294)
- Proposition 18.3.4. Let 0 /X f /Y g /Z /0 be an exact sequence of R-chain complexes whose underlying exact sequence of graded R-modules splits. If f or g is a homotopy equivalence, then the sequence is isomorphic under X and over Z to the canonical exact sequence of R-chain complexes 0 /X /X ⊕ Z /Z / 0. (J. P. May & K. Ponto 295)
- In topology, cellular approximation of maps allows us to replace cell complexes by equivalent CW complexes. In algebra, the comparison is much simpler and the difference is negligible since an “attaching n map” SR R −→ Fq X for a cell D n+1 R necessarily has image in the elements (Fq X)n of degree n. When we restrict attention to those X such that Xq = 0 for q < 0, as in classical projective resolutions, we may as well also restrict attention to those I-cell complexes such that Fq X = X≤q (J. P. May & K. Ponto 299)
- This gives X a second filtration that corresponds to the skeletal filtration of CW complexes in topology (J. P. May & K. Ponto 299)
- An I-cell complex X has an increasing filtration given by its successive terms, which we shall here denote by Fq X (J. P. May & K. Ponto 299)
- Of course, regarding an ungraded R-module M as a DG R-module concentrated in degree 0, a q-cofibrant approximation of M is exactly a projective resolution of M . (J. P. May & K. Ponto 303)
- Remark 18.5.5. Hovey [66] has studied abelian categories with model structures in which the cofibrations are the monomorphisms with cofibrant cokernel and the fibrations are the epimorphisms with fibrant kernel. (J. P. May & K. Ponto 304)
- The mixed model structure has all of the good formal properties of the q-model structure. It is proper, it is monoidal when R is commutative, and it is an ChZ -model structure in general. The identity functor on ChR is a right Quillen equivalence from the m-model structure to the q-model structure and therefore a left Quillen equivalence from the q-model structure to the m-model structure. The class Cm of m-cofibrations is very well-behaved (J. P. May & K. Ponto 304)
- Homotopy invariant constructions that start with complexes of projective R-modules automatically give m-cofibrant objects, but not necessarily degreewise projective objects. In particular, it is very often useful to study the perfect complexes, namely the objects of ChR that are homotopy equivalent to bounded complexes of finitely generated projective R-modules. These are the dualizable objects of HoChR (J. P. May & K. Ponto 304)
- From the point of view of classical homological algebra, it is interesting to think of projective resolutions homotopically as analogues of approximation by CW complexes, well-defined only up to homotopy equivalence (J. P. May & K. Ponto 305)
- Of course, using either a q-cofibrant or m-cofibrant approximation P of an R-module M , regarded as a chain complex concentrated in degree 0, we have Tor R ∗ (N, M ) = H∗ (N ⊗R P ) and Ext∗ R (M, N ) = H ∗ HomR (P, N ), the latter regraded cohomologically. We can think of these as obtained by first applying the derived functors of N ⊗R (−) and HomR (−, N ) and then taking homology n groups or, equivalently, thinking in terms of spheres SR R , homotopy groups. (J. P. May & K. Ponto 305)
- Indeed, this gives a new perspective on the construction of the homotopy category HoU of spaces from the naive homotopy category hU obtained by identifying homotopic maps and of the derived homotopy category HoChR of chain complexes from the naive homotopy category hChR . (J. P. May & K. Ponto 307)
- Localization model structures codify Bousfield localization, which vastly generalizes the constructions in the first half of this book (J. P. May & K. Ponto 307)
- There we concentrated on arithmetic localizations and completions that are closely related to standard algebraic constructions. We restricted our constructions to nilpotent spaces since that is the natural range of applicability of our elementary methodology and since most applications focus on such spaces. There are several different ways to generalize to non-nilpotent spaces, none of them well understood calculationally. Bousfield localization gives a general conceptual understanding of the most widely used of these, and it gives the proper perspective for generalizations to categories other than the category of spaces and (J. P. May & K. Ponto 307)
- In modern algebraic topology, especially in stable homotopy theory, Bousfield localizations at generalized homology theories play a fundamental role. (J. P. May & K. Ponto 307)
- they are central to the structural study of the stable homotopy category. (J. P. May & K. Ponto 307)
- §19.3 and give a geodesic approach to the construction of localizations of spaces at generalized homology theories. (J. P. May & K. Ponto 307)
- §19.4, where we place the localization of spaces at a homology theory in a wider context of localization at a map, or at a class of maps. (J. P. May & K. Ponto 307)
- We are thinking of the weak homotopy equivalences in U and the quasi-isomorphisms in ChR (J. P. May & K. Ponto 308)
- since we are thinking of actual homotopy equivalences (J. P. May & K. Ponto 308)
- As will become clear, the resolvant objects in U are the spaces of the homotopy types of CW complexes, and this codifies the Whitehead theorem for such spaces (J. P. May & K. Ponto 308)
- Similarly, the (bounded below) resolvant objects in ChR are the chain complexes that are of the homotopy type of chain complexes of projective modules, and this codifies the analogous Whitehead theorem for such chain complexes (J. P. May & K. Ponto 308)
- Writing “up to homotopy” to mean “in HoM”, if f : D −→ X is any map from a resolvant object D to X, there is a map f ˜ f˜, unique up to homotopy, such that the following diagram commutes up to homotopy (J. P. May & K. Ponto 308)
- Definition 19.2.2. For spaces X and a homology theory E∗ on spaces, say that ξ : X −→ Y is an E-equivalence if it induces an isomorphism E∗ (X) −→ E∗ (Y ) and let LE denote the class of E-equivalences. Say that a space Z is E-local if it is LE -local. Say that a map φ : X −→ XE from X into an E-local space XE is a localization at E if φ is an E-equivalence. Example 19.2.3. When E∗ (X) = H∗ (X; ZT ), these definitions agree up to nomenclature with our definition of localization at T in §5.2. Similarly, when E∗ (X) = H∗ (X; FT ), where FT = ×p∈T Fp , these definitions agree up to nomenclature with our definition of completion at T (J. P. May & K. Ponto 310)
- Observe that the names “localization” and “completion” that are carried over from algebra in the earlier parts of the book are really the names of two examples of a generalized notion of localization (J. P. May & K. Ponto 311)
- the essential idea of Bousfield localization is to try to construct a model structure on M of the form (L , C , FL ). (J. P. May & K. Ponto 311)
- This is a general and conceptually pleasing way to construct localizations (J. P. May & K. Ponto 312)
- Theorem 19.2.11. Under suitable hypotheses, every map f : X −→ Y factors as an L -acyclic cofibration i : X −→ E followed by an L -fibration p : E −→ Y . (J. P. May & K. Ponto 312)
- As an historical aside, Bousfield’s original treatment [15] first introduced the methods of transfinite induction into model category theory, and only later were his ideas codified into the notion of a cofibrantly generated model structure (J. P. May & K. Ponto 314)
- Little is known about the behavior on homotopy groups of localizations and completions of non-nilpotent spaces. Of course, for generalized homology theories E∗ , no such concrete descriptions can be expected, even for nilpotent spaces. The determination of the homotopy groups of particular E-localizations of spectra is a major part of stable homotopy theory. (J. P. May & K. Ponto 317)
- already for S 1 ∨ S n , the Bousfield-Kan completion at p does not induce an isomorphism on mod p homology. We reiterate that, with any definition, relatively little is known about the behavior on homotopy groups in general (J. P. May & K. Ponto 317)
- As usual, we let F (X, Y ) denote the function space of based maps X −→ Y and let X+ denote the union of an unbased space X and a disjoint basepoint (J. P. May & K. Ponto 319)
- Theorem 19.5.2. Let f : A −→ B be a map between CW complexes and let Z be any based space. Then f ∗ : F (B, Z) −→ F (A, Z) is a weak equivalence if and only if the induced functions [S n + , F (B, Z)] −→ [S n + , F (A, Z)], (J. P. May & K. Ponto 319)
- or equivalently [B ∧ S n + , Z] −→ [A ∧ S n + , Z], are bijections for all n ≥ 0 (J. P. May & K. Ponto 320)
- More recently, Jeff Smith (unpublished) has proven that the f -localization model structure exists for any map f in any left proper combinatorial simplicial model category. (J. P. May & K. Ponto 320)
- There is an extensive literature on f -localizations for a map f . The first existence proof is due to Bousfield [16], and the books [52, 63] study the foundations in detail. These sources work with simplicial sets. Dror Fajoun’s monograph [38] gives several variant existence proofs, and he explains how either simplicial sets or topological spaces can be used. His monograph analyzes many interesting examples in detail. Bousfield [18] gives a nice overview of this area, with many references. (J. P. May & K. Ponto 320)
- Remark 19.5.6. One can ask whether or not localizations at classes, rather than sets, of maps always exist. Remarkably, Casacuberta, Scevenels, and Smith [25] prove that this holds if Vopěnka’s principle (a certain large cardinal axiom) is valid, but that it cannot be proven using only the usual ZFC axioms of set theory (J. P. May & K. Ponto 320)
- We end our discussion of model categories and localization with a philosophical remark that contains a puzzling and interesting open problem. (J. P. May & K. Ponto 320)
- In particular, in the categories of spaces or chain complexes, we have the qmodel structure and the m-model structure with the same weak equivalences and therefore with equivalent homotopy categories. In these cases, all objects are fibrant and so the definition of L -local objects is the same in the two cases. However, the question of the existence of the L -model structure is still model dependent. Clearly, asking whether or not (L , C , FL ) is a model structure is a different question for (J. P. May & K. Ponto 320)
- Remark 19.5.7. The notion of localizing a category M , or a homotopy category HoM , at a subcategory L of weak equivalences makes sense as a general matter of homotopical algebra, independent of model category theory. However, the general theory here depends on the chosen model structure since we have required L -local objects to be fibrant in a given model structure on M . Thus our definition of an L -local object really defines the notion of being L -local relative to a given model structure. For a fixed ambient category of weak equivalences W , there may be several model structures (W , C , F ) on M with different good properties (J. P. May & K. Ponto 320)
- The known existence proofs for L -local (or K -local) model structures on spaces start from the q-model structure. We actually do not know whether or not (L , Cm , Fℓ ) is a model structure even when we do know that (L , Cq , Fℓ ) is a model structure. We regard this as a quite unsatisfactory state of affairs, but we leave the existence of model structures (L , Cm , Fℓ ) as an open problem (J. P. May & K. Ponto 321)
- C = Cq and for C = Cm . Of course, FL depends on which choice we make, even though the fibrant objects are the same with both choices (J. P. May & K. Ponto 321)
- Thus the familiar concepts of algebra and module dualize to concepts of coalgebra and comodule, and the structures of algebra and coalgebra combine to give the notion of a bialgebra. Incorporating antipodes (sometimes called conjugations), we obtain the notion of a Hopf algebra. In the cocommutative case, bialgebras and Hopf algebras can be viewed as monoids and groups in the symmetric monoidal category of cocommutative coalgebras (J. P. May & K. Ponto 325)
- Of course, ⊗ is associative and unital (with unit R) up to natural isomorphism and has the natural commutativity isomorphism γ :A⊗B →B⊗A deg a deg b specified by γ(a ⊗ b) = (−1) b ⊗ a. (J. P. May & K. Ponto 325)
- In categorical language, the category MR of graded R-modules is symmetric monoidal, and it is closed in the sense that there is a natural isomorphism Hom(A ⊗ B, C) ∼ = ∼ = Hom(A, Hom(B, C)); it sends f to g, where g(a)(b) = f (a ⊗ b) (J. P. May & K. Ponto 325)
- We say that A is projective if each Ai is projective (over R), and we say that A is of finite type if each Ai is finitely generated (over R). We say that A is bounded if it is non-zero in only finitely many degrees. Thus A is finitely generated if and only if it is bounded and of finite type. We say that A is bounded below (or above) if Ai = 0 for i sufficiently small (or large). Then ν is an isomorphism if A is bounded and either A or C is projective of finite type, ρ is an isomorphism if A is projective of finite type, and the last map α is an isomorphism if A and B are bounded below and A or B is projective of finite type (J. P. May & K. Ponto 326)
- The tensor product of bigraded modules is given by (A ⊗ B)p,q = od i+j=p,k+l=q Ai,k ⊗ Bj,l . (J. P. May & K. Ponto 326)
- A filtration {Fp A} of a graded module A is an expanding sequence of submodules Fp A. A filtration is said to be complete if A ∼ = ∼ = colimFp A and A ∼ = ∼ = limA/Fp A (J. P. May & K. Ponto 326)
- In most cases that we shall encounter, we shall have either Fp A = A for p ≥ 0 and ∩p Fp A = 0 or Fp A = 0 for p < 0 and A = ∪p Fp A (J. P. May & K. Ponto 326)
- We say that a filtration of A is flat if each A/Fp A is a flat R-module; we say that a filtration is split if each sequence 0 → Fp A → A → A/Fp A → 0 (J. P. May & K. Ponto 326)
- is split exact over R. Of course, these both hold automatically when R is a field. (J. P. May & K. Ponto 326)
- The associated bigraded module E 0 A of a filtered module A is specified by E 0 p,q A = (Fp A/Fp−1 A)p+q . Of course, E 0 is a functor from filtered modules to bigraded modules. (J. P. May & K. Ponto 327)
- Chases of congeries of exact sequences give the following comparison assertion. Proposition 20.1.2. Let A and B be filtered R-modules such that A and B are either both split or both flat. Then the natural map E 0 A ⊗ E 0 B → E 0 (A ⊗ B) is an isomorphism of bigraded R-modules (J. P. May & K. Ponto 328)
- Definition 20.2.1. An R-algebra A = (A, φ, η) is a graded R-module A together with a product φ : A ⊗ A → A and unit η : R → A such that the following diagrams commute (J. P. May & K. Ponto 328)
- An augmentation of A is a morphism of algebras ε : A → R. Given ε, ker ε is denoted IA and called the augmentation ideal of A; since εη = id, A ∼ = ∼ = R ⊕ IA. (J. P. May & K. Ponto 328)
- An algebra A is commutative if and only if φ : A ⊗ A → A is a map of algebras. (J. P. May & K. Ponto 329)
- Definition 20.2.3. Let A be a flat R-module. A bialgebra (A, φ, ψ, η, ε) is an algebra (A, φ, η) with augmentation ε and a coalgebra (A, ψ, ε) with unit η such that the following diagram is commutative (J. P. May & K. Ponto 329)
- That is, φ is a morphism of coalgebras or, equivalently, ψ is a morphism of algebras. If the associativity of φ and coassociativity of ψ are deleted from the definition, then A is said to be a quasi bialgebra 1 (J. P. May & K. Ponto 329)
- The flatness of A is usually not assumed but holds in practice; in its absence, the notion of bialgebra is perhaps too esoteric to be worthy of contemplation (J. P. May & K. Ponto 329)
- Let A be an augmented algebra. Define the R-module QA of indecomposable elements of A by the exact sequence IA ⊗ IA φ / IA / QA / 0. Note that QA is well-defined even if A is not associative (J. P. May & K. Ponto 330)
- Let C be a unital coalgebra. Define the R-module P C of primitive elements of C by the exact sequence 0 / PC / JC ψ / JC ⊗ JC. Let IC = ker ε. We say that x ∈ IC is primitive if its image in JC lies in P C. Note that P C is well-defined even if C is not coassociative (J. P. May & K. Ponto 330)
- Definition 20.2.8. Let A be a quasi bialgebra. Define ν : P A → QA to be the composite PA / JA ∼ = IA / QA (or, equivalently, the restriction of IA → QA to P A if P A is regarded as contained in A). A is said to be primitive, or primitively generated, if ν is an epimorphism; A is said to be coprimitive if ν is a monomorphism (J. P. May & K. Ponto 330)
- In the first case, that is a familiar fact from classical algebra since the intersection of the powers of a (two-sided) ideal in a ring can be non-zero [6, p. 110]. (J. P. May & K. Ponto 331)
- For a monoid G, the monoid ring R[G] is a bialgebra with product, coproduct, unit and counit induced by the product, diagonal, identity element, and trivial function G −→ {pt}. If G is a group, its inverse function induces an antipode on R[G], in the sense of the following definition (J. P. May & K. Ponto 331)
- Definition 20.3.1. An antipode χ on a bialgebra A is a map χ : A → A of R-modules such that the following diagrams commute. A ⊗O A id ⊗χ / A⊗A φ A ⊗O A χ⊗id A⊗A φ A ψ ε /R η /A A ψ O ε /R η /A A Hopf algebra is a bialgebra with a given antipode. (J. P. May & K. Ponto 331)
- . Historically, the concept of Hopf algebra originated in algebraic topology, where the term “Hopf algebra” was used for what we are calling a bialgebra (J. P. May & K. Ponto 332)
- the bialgebras that usually appear in algebraic topology automatically have antipodes (J. P. May & K. Ponto 332)
- We have followed the algebraic literature in using the name antipode and distinguishing between bialgebras and Hopf algebras because of the more recent interest in Hopf algebras of a kind that do not seem to appear in algebraic topology, such as quantum groups (J. P. May & K. Ponto 332)
- It is a standard and easy observation that the tensor product is the categorical coproduct in the category of commutative algebras (J. P. May & K. Ponto 332)
- Recall that, in any category with products, we have the notion of a monoid, namely an object with an associative and unital product, and of a group, namely a monoid with an antipode. (J. P. May & K. Ponto 332)
- Proposition 20.3.4. The tensor product is the categorical product in the category C of commutative coalgebras. A cocommutative bialgebra is a monoid in C , and a cocommutative Hopf algebra is a group in C . (J. P. May & K. Ponto 332)
- Construction 20.3.5. Let C be a coalgebra and A be an algebra. Then Hom(C, A) is an algebra, called a convolution algebra. Its unit element is the composite C ε ε − →R η − → A and its product is the composite ∗ : Hom(C, A) ⊗ Hom(C, A) α α − → Hom(C ⊗ C, A ⊗ A) Hom(ψ,φ) −−−−−−→ Hom(C, A). (J. P. May & K. Ponto 332)
- If C is unital with unit η and A is augmented with augmentation ε, then the set G(C, A) of maps of R-modules f : C −→ A such that f η = η and εf = ε is a submonoid of Hom(C, A) under the convolution product ∗. (J. P. May & K. Ponto 332)
- Remark 20.3.6. Visibly, when A is a bialgebra, an antipode is a (two-sided) inverse to the identity map A −→ A in the monoid G(A, A) (J. P. May & K. Ponto 332)
- . With kernels and cokernels defined degreewise, the category of left A-modules is abelian. There is an analogous abelian category of right A-modules. For a right A-module (M, λ) and left A-module (N, ξ), the tensor product M ⊗A N , which of course is just an R-module, can be described as the cokernel of λ ⊗ id − id ⊗ξ : M ⊗ A ⊗ N → M ⊗ N ; (J. P. May & K. Ponto 333)
- Definition 20.4.1. Let (A, φ, η) be an algebra. A left A-module (N, ξ) is an R-module N and action ξ : A ⊗ N → N such that the following diagrams commute. (J. P. May & K. Ponto 333)
- Since ⊗ is right but not left exact, the category of left C-comodules does not admit kernels in general; it is abelian if C is a flat R-module (J. P. May & K. Ponto 333)
- For a right C-comodule (M, µ) and a left C-comodule (N, ν), define the cotensor product M C N to be the kernel of (J. P. May & K. Ponto 333)
- Definition 20.4.2. Given an augmentation ε : A → R of A, regard R as a (left and right) A-module via ε and define QA N = R ⊗A N = N/IA · N ; QA N is called the module of A-indecomposable elements of N and is abbreviated QN (J. P. May & K. Ponto 333)
- For a general algebra A, the tensor product (over R) of A-modules is an A ⊗ A-module, (J. P. May & K. Ponto 334)
- Note that any bialgebra C which contains A as a sub bialgebra is certainly a left A-coalgebra. (J. P. May & K. Ponto 334)
- PC N is called the module of C-primitive elements of N and is abbreviated P N (J. P. May & K. Ponto 334)
- An R-module A such that Ai = 0 for i < 0 (as we have tacitly assumed throughout) and A0 = R is said to be connected. Note that a connected algebra admits a unique augmentation and a connected coalgebra admits a unique unit. (J. P. May & K. Ponto 337)
- We shall see in (J. P. May & K. Ponto 337)
- We shall see in §21.3 that a connected bialgebra always admits a unique antipode (J. P. May & K. Ponto 337)
- For example, the homology of a connected homotopy associative H-space X is a connected Hopf algbra (J. P. May & K. Ponto 337)
- The homology of non-connected but grouplike (π0 (X) is a group) homotopy associative H-spaces leads to the more general notion of a component Hopf algebra. (J. P. May & K. Ponto 337)
- When concentrated in degree zero, these are just the classical group algebras R[G]. These too have unique antipodes (J. P. May & K. Ponto 337)
- To illustrate the power of these beautiful but elementary algebraic results, we show how they can be used to prove Thom’s calculation of unoriented cobordism and Bott’s periodicity theorem for BU in §21.5 and §21.6. (J. P. May & K. Ponto 337)
- We here prove various special properties that hold in the connected case but do not hold in general. However, they generally do apply to bigraded objects that are connected to the eyes of one of the gradings, and such structures can arise from filtrations of objects that are not connected. (J. P. May & K. Ponto 337)
- Lemma 21.1.1. Let A be a connected coprimitive quasi Hopf algebra. Then A is associative and commutative. If the characteristic of R is a prime p, then the pth power operation ξ (defined only on even degree elements of A if p > 2) is identically zero on IA. (J. P. May & K. Ponto 337)
- A Prüfer ring is an integral domain all of whose ideals are flat. A Noetherian Prüfer ring is a Dedekind ring. (J. P. May & K. Ponto 338)
- Lemma 21.1.2. A connected Hopf algebra A over a Prufer ring R is the colimit of its sub Hopf algebras of finite type. (J. P. May & K. Ponto 338)
- The following result is a version of “Nakayama’s lemma”. It and its dual are used constantly in algebraic topology. Lemma 21.1.5. If A is a connected algebra and N is a left A-module, then N = 0 if and only if QN = 0. Proof. Clearly QN = 0 if and only if IA ⊗ N → N is an epimorphism, and this implies that N is zero by induction on degrees (J. P. May & K. Ponto 338)
- We here prove the basic results of Milnor and Moore on tensor product decompositions of connected Hopf algebras. These play a key role in many calculations, for example in the calculation of the cobordism rings of manifolds. (J. P. May & K. Ponto 339)
- Then there is an isomorphism f : B → A ⊗ QB which is a map of both left A-modules and right QB-comodules (J. P. May & K. Ponto 339)
- Since a direct summand of a flat module is flat, the assumption on π implies that QB is flat if B is flat. Of course, when R is a field, as is the case in most applications, the only assumption is that ι : A → B be a monomorphism. (J. P. May & K. Ponto 339)
- (i) An augmented algebra A is said to be grouplike if the set ε −1 (1) of degree 0 elements is a group under the product of A (J. P. May & K. Ponto 341)
- We define grouplike algebras and component coalgebras (J. P. May & K. Ponto 341)
- Say that C is a component coalgebra if C0 is R-free, each Cg is a sub coalgebra of C, and C is the direct sum of the Cg (J. P. May & K. Ponto 341)
- If C is unital then it has a privileged component, namely C1 (J. P. May & K. Ponto 341)
- If X is a based space, then H∗ (X; R), if R-flat, is a unital component coalgebra. Similarly, H∗ (ΩX; R), if R-flat, is a grouplike component Hopf algebra; it is connected if and only if X is simply connected (J. P. May & K. Ponto 341)
- The homology Hopf algebras H∗ (BU ; Z) and H∗ (BO; F2 ) enjoy a very special property: they are self-dual, so that they are isomorphic to the cohomology Hopf algebras H ∗ (BU ; Z) and H ∗ (BO; F2 ) (J. P. May & K. Ponto 342)
- We assume that the reader knows that the cohomology Hopf algebras are given by (21.4.1) H ∗ (BU ; Z) = P {ci | i ≥ 1} with ψ(cn ) = giv i+j=n ci ⊗ cj and (21.4.2) H ∗ (BO; F2 ) = P {wi | i ≥ 1} with ψ(wn ) = i+j=n wi ⊗ wj . (J. P. May & K. Ponto 342)
- The calculations of H ∗ (BU (n); Z) and H ∗ (BO(n); F2 ) are summarized in [89, pp 187, 195], and passage to colimits over n gives the stated conclusions. Thus determination of the homology algebras is a purely algebraic problem in dualization (J. P. May & K. Ponto 342)
- Recall that the dual coalgebra of a polynomial algebra P [x] over R is written Γ[x]; when P [x] is regarded as a Hopf algebra with x primitive, Γ[x] is called a divided polynomial Hopf algebra. (J. P. May & K. Ponto 342)
- Clearly H ∗ (BU (1); Z) = P [c1 ] and H ∗ (BO(1); F2 ) = P [w1 ] are quotient al). (J. P. May & K. Ponto 343)
- Definition 21.4.6. We define some universal Hopf algebras. (i) A universal enveloping Hopf algebra of a coalgebra C is a Hopf algebra LC together with a morphism i : C −→ LC of coalgebras which is universal with respect to maps of coalgebras f : C −→ B, where B is a Hopf algebra. That is, any such f factors uniquely as f ˜ f˜ ◦ i for a morphism f ˜ f˜: LC −→ B of Hopf algebras. (ii) A universal covering Hopf algebra of an algebra A is a Hopf algebra M A together with a morphism p : M A −→ A of algebras which is universal with respect to maps of algebras f : B −→ A, where B is a Hopf algebra. That is, (J. P. May & K. Ponto 343)
- any such f factors uniquely as p ◦ f ˜ f˜ for a morphism f ˜ f˜: B −→ M A of Hopf algebras. Lemma 21.4.7. Universal Hopf algebras exist and are unique. That is, (i) any coalgebra C admits a universal enveloping Hopf algebra i : C −→ LC; (ii) any algebra A admits a universal covering Hopf algebra p : M A −→ A. (J. P. May & K. Ponto 344)
- For an R-module X, let X n denote the n-fold tensor product of X with itself. With the usual sign (−1)deg x deg y inserted when x is permuted past y, the symmetric group Σn acts on X n . (J. P. May & K. Ponto 344)
- If X is an algebra or coalgebra, then so is X n , and Σn acts as a group of automorphisms (J. P. May & K. Ponto 344)
- In [89, Ch. 25], we explained Thom’s classical computation of the real cobordism of smooth manifolds (J. P. May & K. Ponto 346)
- Knowing the splitting theorems of §21.2 and the self-duality theorem of §21.4, the senior author simply transcribed the first and quoted the second to give the main points of the calculation (J. P. May & K. Ponto 346)
- A punch line, explained at the end of the section, is that the conceptual argument applies to much more sophisticated cobordism theories, where the actual calculations are far more difficult. (J. P. May & K. Ponto 346)
- The Thom prespectrum T O and its associated Thom spectrum M O are described in [89, pp. 216, 229] (J. P. May & K. Ponto 347)
- The ring structure on T O gives its homology an algebra structure, and the Thom isomorphism Φ : H∗ (T O) −→ H∗ (BO) is an isomorphism of algebras (J. P. May & K. Ponto 347)
- The Thom space T O(1) of the universal line bundle is equivalent to RP ∞ (J. P. May & K. Ponto 347)
- this implies that M O is a product of suspensions of Eilenberg-Mac Lane spectrum HF2 and that π∗ (M O) ∼ = ∼ = N∗ as an algebra (J. P. May & K. Ponto 347)
- This gives the now standard way of obtaining Thom’s calculation [133] of π∗ (M O (J. P. May & K. Ponto 347)
- the theory of PL-manifolds was designed to get around the lack of obvious transversality in the theory of topological manifolds, (J. P. May & K. Ponto 347)
- the P L-cobordism groups are isomorphic to the homotopy groups of a Thom prespectrum T P L. (J. P. May & K. Ponto 347)
- The calculation of H∗ (BP L; Fp ) at all primes p is described in [78, 29], (J. P. May & K. Ponto 348)
- The self duality of H ∗ (BU ) described in (21.4.1) and Theorem 21.4.3 also plays a central role in a quick proof of (complex) Bott periodicity (J. P. May & K. Ponto 348)
- Theorem 21.6.1. There is a map β : BU −→ ΩSU of H-spaces which induces an isomorphism on homology (J. P. May & K. Ponto 348)
- It follows from the dual Whitehead theorem that β must be an equivalence (J. P. May & K. Ponto 348)
- of the 1970’s story of infinite loop space theory and E ∞ ring spectra; see [86] for a 1970’s overview and [92] for a modernized perspective (J. P. May & K. Ponto 348)
- Its image will then be open by invariance of domain and closed by the compactness of ΣS 2n (J. P. May & K. Ponto 350)
- hence will be all of S 2n+1 since S 2n+1 is connected. (J. P. May & K. Ponto 350)
- Using the Serre spectral sequence of the path space fibration over SU (n + 1), we conclude that (J. P. May & K. Ponto 350)
- The conclusion follows by the comparison theorem, Theorem 24.6.1 (J. P. May & K. Ponto 351)
- All of the structure theorems for Hopf algebras in common use in algebraic topology are best derived by filtration techniques from the Poincaré-Birkhoff-Witt theorem for graded Lie algebras and restricted Lie algebras (J. P. May & K. Ponto 353)
- We next show that primitive (= primitively generated) Hopf algebras in characteristic zero are the universal enveloping algebras of their Lie algebras of primitive elements. We then use this fact to study the algebra structure of commutative Hopf algebras in characteristic zero (J. P. May & K. Ponto 353)
- Definition 22.1.1. A (graded) Lie algebra over R is a (graded) R-module L together with a morphism of R-modules L ⊗ L → L, denoted [−, −] and called the bracket operation, such that there exists an associative R-algebra A and a monomorphism of R-modules j : L → A such that j([x, y]) = [jx, jy] for x, y ∈ L, where the bracket operation in A is the (graded) commutator, [a, b] = ab − (−1) deg a deg b ba. (J. P. May & K. Ponto 353)
- A morphism of Lie algebras is a morphism of R-modules which commutes with the bracket operation (J. P. May & K. Ponto 353)
- Lemma 22.1.2. Let L be a Lie algebra and let x ∈ Lp , y ∈ Lq , and z ∈ Lr . Then the following identities hold. (J. P. May & K. Ponto 353)
- We shall see that, at least if R is a field (J. P. May & K. Ponto 353)
- any R-module with a bracket operation satisfying these identities can be embedded in a bracket-preserving way in an associative algebra and is therefore a Lie algebra. (J. P. May & K. Ponto 354)
- This is not true for a general R. (J. P. May & K. Ponto 354)
- Of course, for any R, any associative alegbra is a Lie algebra under the commutator operation (J. P. May & K. Ponto 354)
- Definition 22.1.3. The universal enveloping algebra of a Lie algebra L is an associative algebra U (L) together with a morphism of Lie algebras i : L → U (L) such that, for any morphism of Lie algebras f : L → A, where A is an associative algebra, there exists a unique morphism of algebras f ˜ f˜ : U (L) → A such that f ˜ f˜i = f . Clearly U (L) is unique up to canonical isomorphism, if it exists. (J. P. May & K. Ponto 354)
- Clearly U (L) is unique up to canonical isomorphism, if it exists. Proposition 22.1.4. Any Lie algebra L has a universal enveloping algebra U (L), and i : L → U (L) is a monomorphism whose image generates U (L) as an algebra. Moreover, U (L) is a primitive Hopf algebra (J. P. May & K. Ponto 354)
- the tensor algebra, or free associative algebra, generated Proof. Let T (L) b by L. Explicitly, T (L) = P th n≥0 Tn (L), where T0 (L) = R and Tn (L) = L ⊗ . . . ⊗ L, (J. P. May & K. Ponto 354)
- The Poincaré-Birkhoff-Witt theorem gives a complete description of the associated graded Hopf algebra of U (L) with respect to a suitable filtration (under appropriate hypotheses) and therefore gives a complete description of the additive structure of U (L) (J. P. May & K. Ponto 355)
- Theorem 22.2.3 (Poincaré-Birkhoff-Witt). Let L be an R-free Lie algebra. Assume that char R = 2, or that 2 is invertible in R, or that L = L+ so that L is concentrated in even degrees. Then f : A(L) → E ⊕ U (L) is an isomorphism of Hopf algebras. (J. P. May & K. Ponto 355)
- A quick calculation shows that the R-module P A of primitive elements of a Hopf algebra A is a sub Lie algebra (J. P. May & K. Ponto 358)
- Throughout this section and the next, R is assumed to be a field of characteristic zero. However, all of the results remain valid if R is any ring of characteristic zero in which 2 is invertible and all R-modules in sight are R-free (J. P. May & K. Ponto 358)
- Corollary 22.3.2. If A is a commutative primitive Hopf algebra, then A is isomorphic as a Hopf algebra to the free commutative algebra generated by P A (J. P. May & K. Ponto 359)
- Among other things, our next corollary shows that a connected Hopf algebra is primitive if and only if it is cocommutative (J. P. May & K. Ponto 359)
- Again, let R be a field of characteristic zero. We prove the classical structure theorems for commutative Hopf algebras in characteristic zero (J. P. May & K. Ponto 360)
- If we write E(X) for the exterior algebra generated by an R-module X concentrated in odd degrees and P (X) for the polynomial algebra generated by an R-module X concentrated in even degrees, then, for a general R-module X, A(X) = E(X − ) ⊗ P (X + ), where X − and X + denote the submodules of X concentrated in odd and even degrees, respectively (J. P. May & K. Ponto 360)
- As before, A(X) denotes the free commutative algebra generated by an R-module X (J. P. May & K. Ponto 360)
- Theorem 22.4.1 (Leray). Let A be a connected, commutative, and associative quasi Hopf algebra. Let σ : QA → IA be a morphism of R-modules such that πσ = id, where π : IA → QA is the quotient map. Then the morphism of algebras f : A(QA) → A induced by σ is an isomorphism (J. P. May & K. Ponto 360)
- The following immediate consequence of the previous theorem was the theorem of Hopf which initiated the study of Hopf algebras. Corollary 22.4.2 (Hopf). Let A be a connected, commutative, and associative quasi Hopf algebra such that Qn A = 0 if n is even. Then A ∼ = ∼ = E(QA) as an algebra. In particular, the conclusion holds if An = 0 for all sufficiently large n. (J. P. May & K. Ponto 360)
- Theorem 22.4.4. Let A be a connected, commutative, and cocommutative Hopf algebra. Then A ∼ = ∼ = E((P A) − ) ⊗ P ((P A)+ ) as a Hopf algebra. (J. P. May & K. Ponto 361)
- We next show that primitive Hopf algebras in characteristic p are the universal enveloping algebras of their restricted Lie algebras of primitive elements. We then use this fact to study the algebra structure of commutative Hopf algebras in characteristic p (J. P. May & K. Ponto 363)
- Definition 23.1.1. A restricted Lie algebra over R is a Lie algebra L together with a function ξ : L+ → L+ with ξ(Ln ) ⊂ Lpn , such that there exists an associative algebra A and a monomorphism of Lie algebras j : L → A such that jξ(x) = ξj(x), where ξ : A+ → A+ is the pth power operation. A morphism of restricted Lie algebras is a morphism of Lie algebras which commutes with the “restrictions” ξ. (J. P. May & K. Ponto 363)
- In this section, R is assumed to be a perfect field of characteristic p. We need R to be perfect since relations of the form xp q = ry p , where r has no pth root, would lead to counterexamples to the main results (J. P. May & K. Ponto 367)
- Define the height of y by ht(y) = t if y p t = 0 but y p t −1 6= 0, or ht(y) = ∞ if there exists no such t (J. P. May & K. Ponto 368)
- Theorem 23.4.7. If A is a primitive commutative Hopf algebra and A0 is finitely generated as an algebra, then A is isomorphic as a Hopf algebra to a tensor product of monogenic Hopf algebras. Theorem 23.4.8 (Borel). If A is a connected, commutative, and associative quasi Hopf algebra, then A is isomorphic as an algebra to a tensor product of monogenic Hopf algebras (J. P. May & K. Ponto 371)
- It is a well kept secret that the further one goes into the subject, the less one uses such complexes for actual calculation. Rather, one starts with a few spaces whose homology and cohomology groups can be computed by hand, using explicit chain complexes. One then bootstraps up such calculations to the vast array of currently known computations using a variety of spectral sequences (J. P. May & K. Ponto 373)
- McCleary’s book [95] is a good encyclopedic reference for the various spectral sequences in current use. Other introductions can be found in many texts in algebraic topology and homological algebra [76, 120, 138] (J. P. May & K. Ponto 373)
- Definition 24.1.1. A homologically graded spectral sequence E = {E r } consists of a sequence of Z-bigraded R modules E r = {Ep,q r }r≥1 together with differentials d r : E p r p,q → E r p−r,q+r−1 ∼ H∗ (E r ). A morphism f : E → E ′ of spectral sequences is a such that E r+1 = family of morphisms of complexes f r : E r → E ′r such that f r+1 is the morphism H∗ (f r ) induced by f r . (J. P. May & K. Ponto 373)
- These identifications give a sequence of submodules 0 = B 0 ⊂ B 1 ⊂ . . . ⊂ Z 2 ⊂ Z 1 ⊂ Z 0 = E 1 . Define Z ∞ = ∩∞ r= r=1 ∞ r Z , B = ∪∞ r= r=1 r ∞ B , and Ep,q p,q = Z ∞ p,q /B ∞ p,q , writing E ∞ ∞ = {Ep,q p,q }. (J. P. May & K. Ponto 374)
- r Note that elements of Ep p,0 cannot be boundaries for r ≥ 2 since the differential d r : E p r p+r,−r+1 −→ E r p,0 has domain the 0 group (J. P. May & K. Ponto 374)
- r Similarly, for r ≥ 1, E0 0,q consists only of cycles and so there are epimorphisms (J. P. May & K. Ponto 374)
- The maps eB and eF are called edge homomorphisms (J. P. May & K. Ponto 374)
- This map is called the transgression. It is an additive relation [76, II.6] from a 2 submodule of Ep p,0 2 to a quotient module of E0 0,p−1 . (J. P. May & K. Ponto 374)
- This leads to the most elementary example, called the Bockstein spectral sequence. (J. P. May & K. Ponto 375)
- Definition 24.2.1. Let D and E be modules. An exact couple C = hD, E; i, j, ki is a diagram (J. P. May & K. Ponto 375)
- Example 24.2.3. Let C be a torsion free chain complex over Z. From the short exact sequence of groups 0 /Z p /Z / Z/pZ /0 we obtain a short exact sequence of chain complexes 0 /C /C / C ⊗ Z/pZ /0. The induced long exact homology sequence is an exact couple (J. P. May & K. Ponto 375)
- The resulting spectral sequence is called the mod p Bockstein spectral sequence . (J. P. May & K. Ponto 376)
- When r = 1, this is the universal coefficient exact sequence for calculating Hn (C; Fp ), and we may view it as a higher universal coefficient exact sequence in general (J. P. May & K. Ponto 376)
- We conclude that complete knowledge of the Bockstein spectral sequences of C for all primes p allows a complete description of H∗ (C) as a graded abelian group (J. P. May & K. Ponto 376)
- The previous example shows that if X is a space whose homology is of finite type and if one can compute H∗ (X; Q) and H∗ (X; Fp ) together with the mod p Bockstein spectral sequences for all primes p, then one can read off H∗ (X; Z). For this reason, among others, algebraic topologists rarely concern themselves with integral homology but rather focus on homology with field coefficients (J. P. May & K. Ponto 376)
- This is one explanation for the focus of this book on rationalization and completion at primes. (J. P. May & K. Ponto 376)
- Let A be a Z-graded complex of modules. An (increasing) filtration of A is a sequence of subcomplexes (J. P. May & K. Ponto 377)
- Theorem 24.3.2. If A = ∪p Fp A and for each n there exists s(n) such that Fs(n) An = 0, then E ∞ p,q 0 A = Ep p,q H∗ (A). (J. P. May & K. Ponto 377)
- The proof is tedious, but elementary. We give it in the last section of the chapter for illustrative purposes. The conclusion of the theorem, E ∞ p,q A ∼ = ∼ 0 = Ep,q H∗ (A), is often written E 2 p,q A ⇒ Hp+q (A), and E r is said to converge to H∗ (A). (J. P. May & K. Ponto 377)
- Dually, cohomology spectral sequences arise naturally from decreasing filtrations of complexes (J. P. May & K. Ponto 377)
- In practice, we often start with a homological filtered complex and dualize it to obtain a cohomological one, setting A ∗ = Hom(A, R) and filtering it by F p A ∗ = Hom(A/Fp−1 A, R) At least when R is a field, the resulting cohomology spectral sequence is dual to the homology spectral sequence. (J. P. May & K. Ponto 378)
- Recall that a differential graded algebra (DGA) A over R is a graded algebra with a product that is a map of chain complexes, so that the Leibnitz formula d(xy) = d(x)y + (−1) degx xd(y) is satisfied. When suitably filtered, A often gives rise to a spectral sequence of DGA’s, meaning that each term E r is a DGA. It is no exaggeration to say that the calculational utility of spectral sequences largely stems from such multiplicative structure. (J. P. May & K. Ponto 378)
- Leibnitz formula for each r ≥ 1. Example 24.4.3. The cup product in the singular cochains C ∗ (X) gives rise to the product φ : H ∗ (X; Fp ) ⊗ H ∗ (X; Fp ) → H ∗ (X; Fp ). Regarding H ∗ (X; Fp ) as the E1 term of the Bockstein spectral sequence of the cochain complex C ∗ (X), we find that φ satisfies µ. Therefore each E r in the mod p cohomology Bockstein spectral sequence of X is a DGA, and E r+1 = H ∗ (E r ) as an algebra (J. P. May & K. Ponto 379)
- In applications, the important thing is to understand what the properties say. Their proofs generally play no role. In fact, this is true of most spectral sequences in algebraic topology (J. P. May & K. Ponto 380)
- It is usual to construct the Serre spectral using the singular (or, in Serre’s original work, cubical) chains of all spaces in sight. We give a more direct homotopical construction that has the advantage that it generalizes effortlessly to a construction of the Serre spectral sequence in generalized homology and cohomology theories. (J. P. May & K. Ponto 380)
- Using [89, p. 48], we may as well replace p by a Hurewicz fibration. This is convenient since it allows us to exploit relationship between cofibrations and fibrations that does not hold for Serre fibrations (J. P. May & K. Ponto 380)
- Using [89, p. 75], we may choose a based weak equivalence f from a CW complex with a single vertex to B (J. P. May & K. Ponto 380)
- Having a CW base space gives a geometric filtration with which to work, and having a single vertex fixes a canonical basepoint and thus a canonical fiber. (J. P. May & K. Ponto 380)
- Give B its skeletal filtration, Fp B = B p , and define Fp E = p−1 (Fp B) (J. P. May & K. Ponto 380)
- Theorem 24.5.1 (Homology Serre spectral sequence). There is a first quadrant homological spectral sequence {E r , dr }, with E 1 p,q ∼ = ∼ = Cp (B; Hq (F ; π)) and E 2 p,q ∼ = ∼ = Hp (B; Hq (F ; π)) that converges to H∗ (E; π). It is natural with respect to maps (J. P. May & K. Ponto 381)
- and it is here that the local coefficient systems Hq (F ) enter into the picture. These groups depend on the action of π1 (B, b) on F . (J. P. May & K. Ponto 381)
- most of the usual applications, π1 (B, b) acts trivially on F and Hq (F ) is just the ordinary homology group Hq (F ), (J. P. May & K. Ponto 381)
- A short exact sequence 1 −→ G ′ −→ G −→ G ′′ −→ 1 of (discrete) groups gives a fibration sequence K(G ′ , 1) −→ K(G, 1) −→ K(G ′′ , 1), and there result Serre spectral sequences in homology and cohomology (J. P. May & K. Ponto 383)
- This spectral sequence can also be constructed purely algebraically, and it is then sometimes called the Lyndon spectral sequence. It is an example where local coefficients are essential (J. P. May & K. Ponto 383)
- Proposition 24.5.3 (Lyndon-Hochschild-Serre spectral sequence). Let G ′ be a normal subgroup of a group G with quotient group G ′′ and let π be a G-module. Then there is a spectral sequence with E ∼ = ∼ = H p (G ′′ ; H q (G ′ ; π)) that converges to H ∗ (G; A). (J. P. May & K. Ponto 383)
- The algebraic action of G ′′ on G (J. P. May & K. Ponto 383)
- Theorem 24.6.1 (Comparison Theorem, [76, XI.11.1]). Let f : E −→ ′ E be a homomorphism of first quadrant spectral sequences of modules over a commutative (J. P. May & K. Ponto 383)
- ring. Assume that E2 and ′ E2 admit universal coefficient exact sequences as displayed in the following diagram, and that, on the E2 level, f is given by a map of short exact sequences as displayed. (J. P. May & K. Ponto 384)
- r Write fp p,q r : Ep p,q r −→ ′ E p p,q . Then any two of the following imply the third. 2 (i) fp p,0 2 : Ep p,0 2 −→ ′ E p p,0 is an isomorphism for all p ≥ 0. 2 (ii) f0 0,q 2 : E0 0,q 2 −→ ′ E 0 0,q is an isomorphism for all q ≥ 0. (iii) f ∞ p,q : E ∞ p,q ′ −→ E ∞ p,q is an isomorphism for all p and q. (J. P. May & K. Ponto 384)
- Details can be found in [76, XI.11]. (J. P. May & K. Ponto 384)
- The comparison theorem is particularly useful for the Serre spectral sequence when the base and fiber are connected and the fundamental group of the base acts trivially on the homology of the fiber (J. P. May & K. Ponto 384)
- If we had given a chain level construction of the Serre spectral sequence, its convergence would be a special case (J. P. May & K. Ponto 384)
- r The formula can be turned around to give a construction of {E A} that avoids the use of exact couples (J. P. May & K. Ponto 385)
- Proof of convergence of the Serre spectral sequence (J. P. May & K. Ponto 386)

Last Annotation: 04/05/2019

- Mobius bundle example (Daniel Dugger 73)
- Super cool result!! (Daniel Dugger 73)
- Try this with a spectral sequence (Daniel Dugger 74)

- I first learned Serre’s definition of intersection multiplicity from Mel Hochster, back when I was an undergraduate. I was immediately intrigued by this surprising connection between homological algebra and geometry (Daniel Dugger 3)
- If you play around with some simple examples, an idea for defining intersection multiplicities comes up naturally. It is i(Z, W ; P ) = dim C h C [x1 , . . . , xn ]/(f1 , . . . , fk , g1 , . . . , gl ) i P (1.1) i(Z, W ; P ) = dimC C[x1 , . . . , xn ]/(f1 , . . . , fk , g1 , . . . , gl ) . Here the subscript P indicates localization of the given ring at the maximal ideal (x1 − p1 , . . . , xn − pn ) where P = (p1 , . . . , pn ). (Daniel Dugger 5)
- Example 1.2. Let f = y − x2 and g = y. This is our example of the parabola and the tangent line at its vertex. The point P = (0, 0) is the only intersection point, and our definition tells us to look at the ring C [x, y]/(y − x 2 , y) ∼ = ∼ = C[x]/(x 2 ). As a vector space over C this is two-dimensional, with basis 1 and x. So our definition gives i(Z, W ; P ) = 2 as desired. [Note that technically we should localize at the ideal (x, y), which corresponds to localization at (x) in C [x]/(x 2 ); however, this ring is already local and so the localization has no effect] (Daniel Dugger 5)
- Note the appearance of (x − 1) with multiplicity two in the above factorization. The fact that we had a tangent line at x = 1 guaranteed that the multiplicity would be strictly larger than one (Daniel Dugger 6)
- Serre discovered the correct formula for the interesection multiplicity [S]. His formula is as follows. If we set R = C [x1 , . . . , xn ] then i(Z, W ; P ) = ∞ X j=0 (−1) j dim h Tor R j R/(f1 , . . . , fk ), R/(g1 , . . . , g l ) i P . (Daniel Dugger 7)
- An algebraist who looks at (1.5) will immediately notice some possible generalizations. The R/(f ) and R/(g) terms can be replaced by any finitely-generated module M and N , as long as the Torj (M, N ) modules are finite-dimensional over C . For this it turns out to be enough that M ⊗R N be finite-dimensional over C . Also, we can replace C [x1 , . . . , xn ] with any ring having the property that all finitely-generated modules have finite projective dimension—necessary so that the alternating sum of (1.5) is finite. Such rings are called regular. (Daniel Dugger 7)
- Let R be a regular, local ring (all rings are assumed to be commutative and Noetherian unless otherwise noted). Let M and N be finitely-generated modules over R such that M ⊗R N has finite length. This implies that all the Torj (M, N ) modules also have finite length. Define e(M, N ) = ∞ X j=0 (−1) j X (−1)j ` Torj (M, N ) (1.6) e(M, N ) = and call this the intersection multiplicty of the modules M and N . (Daniel Dugger 8)
- Based on geometric intuition, Serre made the following conjectures about the above situation: (1) dim M + dim N ≤ dim R always (2) e(M, N ) ≥ 0 always (3) If dim M + dim N < dim R then e(M, N ) = 0. (4) If dim M + dim N = dim R then e(M, N ) > 0. Serre proved all of these in the case that R contains a field, the so-called “geometric case” (some non-geometric examples for R include power series rings over the padic integers Zp ). Serre also proved (1) in general. Conjecture (3) was proven in the mid 80s by Roberts and Gillet-Soule (independently), using some sophisticated topological ideas that were imported into algebra. Conjecture (2) was proven by Gabber in the mid 90s, using some high-tech algebraic geometry. Conjecture (4) is still open. (Daniel Dugger 8)
- There are certain generalized cohomology theories—called complex-oriented — which have a close connection to geometry and intersection theory. Any such cohomology can be used to detect intersection multiplicities. (Daniel Dugger 8)
- Topological K-theory is a complex-oriented cohomology theory. Elements of the groups K ∗ (X) are specified by vector bundles on X, or more generally by bounded chain complexes of vector bundles on X. Fundamental classes for complex submanifolds of X are given by resolutions. (Daniel Dugger 8)
- When X is an algebraic variety there is another version of K-theory called algebraic K-theory, which we might denote K ∗ alg (X). The analogs of vector bundles are locally free coherent sheaves, or just finitely-generated projective modules when X is affine. Thus, in the affine case elements of K ∗ alg (X) can be specified by bounded chain complexes of finitely-generated projective modules. This is the main connection between homological algebra and K-theory (Daniel Dugger 8)
- This (Daniel Dugger 8)
- (4) Serre’s definition of intersection multiplicities essentially comes from the intersection product in K-homology, which is the cup product in K-cohomology translated to homology via Poincaré Duality. (Daniel Dugger 9)
- Theorem 2.1 (Hilbert Syzygy Theorem). Let k be a field and let R be k[x1 , . . . , xn ] (or any localization of this ring). Then every finitely-generated R-module has a free resolution of length at most n. (Daniel Dugger 10)
- We mention it here because it implies that Torj (M, N ) = 0 for j > n. Therefore the sum in Serre’s formula is actually finite. More generally, a ring is called regular if every finitely-generated module has a finite, projective resolution. It is a theorem that localizations of regular rings are again regular. Hilbert’s Syzygy Theorem simply says that polynomial rings over a field are regular. (Daniel Dugger 10)
- The importance of this observation is that it tells us that the Tor’s in Serre’s formula may all be taken over the ring RP . So we might as well work over this ring from beginning to end (Daniel Dugger 10)
- Lemma 2.2. Suppose that 0 → M 0 → M → M 00 → 0 is a short exact sequence of R-modules. Then e(M, N ) = e(M 0 , N )+e(M 00 , N ), assuming all three multiplicities are defined (that is, under the assumption that dim C (M ⊗ N ) < ∞ and similarly with M replaced by M 0 and M 00 ). (Daniel Dugger 11)
- Lemma 2.2 is referred to as the additivity of intersection multiplicities. Of course the additivity holds equally well in the second variable, by the same argument. While exploring ideas in this general area, Grothendieck hit upon the idea of inventing a group that captures all the additive invariants of modules. Any invariant such as e(−, N ) would then factor through this group. Here is the definition: Definition 2.3. Let R be any ring. Let F(R) be the free abelian group with one generator [M ] for every isomorphism class of finitely-generated R-module M . Let G(R) be the quotient of F(R) by the subgroup generated by all elements [M ] − [M 0 ] − [M 00 ] for every short exact sequence 0 → M 0 → M → M 00 → 0 of finitely-generated Rmodules. The group G(R) is called the Grothendieck group of finitely-generated R-modules. (Daniel Dugger 11)
- 1) Suppose R = F , a field. Clearly G(F ) is generated by [F ], since every finitelygenerated F -module has the form F n (Daniel Dugger 12)
- More generally, suppose that R is a domain. The rank of an R-module M is defined to be the dimension of M ⊗R QF (R) over QF (R), where QF (R) is the quotient field. (Daniel Dugger 12)
- Let G be a finite group, and let R = C [G] be the group algebra. So R-modules are just representations of G on complex vector spaces (Daniel Dugger 12)
- For a not-so-simple example, let R be the ring of integers in a number field. It turns out that G(R) ∼ = ∼ = Z ⊕ Cl(R), where Cl(R) is the ideal class group of R. This class group contains some sophisticated number-theoretic information about R. It is known to always be torsion, and it is usually nontrivial. (Daniel Dugger 12)
- All finitely-generated R-modules have a finite composition series, and so we can take the Jordan-Hölder length; this is the same as
`\(A\) = dim Z/p A/pA + dim Z/p pA\. With some trouble one can check that this is indeed an additive invariant \(or refer to the Jordan-Hölder theorem\), and of course`

( Z /p) = 1 (Daniel Dugger 13) - Definition 2.7. Let R be any ring. Let FK (R) be the free abelian group with one generator [P ] for every isomorphism class of finitely-generated, projective Rmodule M . Let K(R) be the quotient of FK (R) by the subgroup generated by all elements [P ] − [P 0 ] − [P 00 ] for every short exact sequence 0 → P 0 → P → P 00 → 0 of finitely-generated projectives. The group K(R) is called the Grothendieck group of finitely-generated projective modules. Every short exact sequence of projectives is actually split, so we could also have defined K(R) by imposing the relations [P ⊕ Q] = [P ] + [Q] for every two finitelygenerated projectives P and Q. This makes it a little easier to understand when two modules represent the same class in K(R (Daniel Dugger 13)
- Proposition 2.8. Let P and Q be finitely-generated projective R-modules. Then [P ] = [Q] in K(R) if and only if there exists a finitely-generated projective module W such that P ⊕ W ∼ = ∼ = Q ⊕ W . In fact, the same remains true if we require W to be free instead of projective. (Daniel Dugger 14)
- For the last statement in the proposition, just observe that since W is projective it is a direct summand of a free module. That is, there exists a module W 0 such that W ⊕W 0 is finitely-genereated and free (Daniel Dugger 14)
- Since projective modules are flat, the product [P ]·[Q] = [P ⊗R Q] is additive and so extends to a product K(R)⊗K(R) → K(R). (Daniel Dugger 14)
- Remark 2.9. Given the motivation of having the tensor product give a ring structure, one might wonder why we used projective modules to define K(R) rather than flat modules. We could have done so, but for finitely-generated modules over commutative, Noetherian rings, being flat and projective are equivalent notions (Daniel Dugger 14)
- Theorem 2.10. If R is regular, then α : K(R) → G(R) is an isomorphism. (Daniel Dugger 14)
- Suppose Q• → M → 0 is another finite, projective resolution of M . Use the Comparison Theorem of homological algebra to produce a map of chain complexes (Daniel Dugger 15)
- Let T• be the mapping cone of f : P• → Q• . Recall this means that Tj = Qj ⊕ Pj−1 , with the differential defined by dT (a, b) = dQ (a) + (−1) |b| f (b), dP (b) . (Daniel Dugger 15)
- There is a short exact sequence of chain complexes 0 → Q ,→ T → ΣP → 0 where ΣP denotes a copy of P in which everything has been shifted up a dimension (so that (ΣP )n = Pn−1 ). (Daniel Dugger 15)
- For any ring R, we have the group K(R) which also comes to us with an easily-defined ring structure ⊗. We also have the group G(R)—but this does not have any evident ring structure. When R is regular, there is an isomorphism K(R) → G(R) which allows one to transplant the ring structure from K(R) onto G(R): and this leads us directly to our alternating-sum-of-Tors (Daniel Dugger 16)
- from K(R) onto G(R): and this leads us directly to our alternating-sum-of-Tors. This situation is very reminiscent of something you have seen in a basic algebraic topology course. When X is a (compact, oriented) manifold, there were early attempts to put a ring structure on H∗ (X) coming from the intersection product. This is technically very difficult. In modern times one avoids these technicalities by instead introducing the cohomology groups H ∗ (X), and here it is easy to define a ring structure: the cup product. When X is a compact, oriented manifold one has the Poincaré Duality isomorphism H ∗ (X) → H∗ (X) given by capping with the fundamental class, and this lets one transplant the cup product onto H∗ (X). This is the modern approach to intersection theory. (Daniel Dugger 16)
- The parallels here are intriguing: K(R) is somehow like H ∗ (X), and G(R) is somehow like H∗ (X). The regularity condition is like being a manifold. (Daniel Dugger 16)
- [The reader might wonder what happened to the assumptions of compactness and orientability. Neither of these is really needed for Poincaré Duality, as long as one does things correctly. For the version of Poincaré Duality for noncompact manifolds one needs to replace ordinary homology with Borel-Moore homology—this is similar to singular homology, but chains are permitted to have infinitely many terms if they stretch out to infinity. For non-orientable manifolds one needs to use twisted coefficients (Daniel Dugger 16)
- out (Daniel Dugger 16)
- Hilbert’s Nullstellensatz says that points of C n are in bijective correspondence with maximal ideals in R: the bijection sends q = (q1 , . . . , qn ) to mq = (x1 −q1 , . . . , xn −qn ). With a little work one can generalize this bijection. If S ⊆ C n is any subset, define I(S) = {f ∈ R | f (x) = 0 for all x ∈ S}. This is an ideal in R, in fact a radical ideal (meaning that if f n ∈ I(S) then f ∈ I(S)). In the other direction, if I ⊆ R is any ideal then define V (I) = {x ∈ C n | f (x) = 0 for all f ∈ I}. Notice that V (mq ) = {q} and I({q}) = mq . (Daniel Dugger 17)
- An algebraic set in C n is any subset of the form V (I) for some ideal I ⊆ R. The algebraic sets form the closed sets for a topology on C n , called the Zariski topology. One form of the Nullstellensatz says that V and I give a bijection between algebraic sets and radical ideals in R. Under this bijection the prime ideals correspond to irreducible algebraic sets—ones that cannot be written as X ∪ Y where both X and Y are proper closed subsets. Algebraic sets are also called algebraic subvarieties. The above discussion is summarized in the following table: Geometry Algebra C n or An nC nC C [x1 , . . . , xn ] = R Points (q1 , . . . , qn ) Maximal ideals (x1 − q1 , . . . , xn − qn ) Algebraic sets Radical ideals Irreducible algebraic sets Prime ideals (Daniel Dugger 17)
- The ring R is best thought of as the set of maps of varieties A n → A 1 , with pointwise addition and multiplication. If we restrict to some irreducible subvariety X = V (P ) ⊆ A n instead, then the ring of functions X → A 1 is R/P . This ring of functions is commonly called the coordinate ring of X. Much of the dictionary between A n and R discussed above adapts verbatim to give a dictionary between X and its coordinate ring: Geometry Algebra X = V (P ) C [x1 , . . . , xn ]/P = R/P Points in X Maximal ideals in R/P Algebraic subsets V (I) ⊆ X Radical ideals in R/P Irreducible algebraic sets V (Q) ⊆ X Prime ideals in R/P . Note that ideals in R/P correspond bijectively to ideals in R containing P , and likewise for prime (respectively, radical) ideals. (Daniel Dugger 17)
- We need one last observation. Passing from A n to A n+1 corresponds algebraically to passing from R to R[t]. If X = V (P ) ⊆ A n is an irreducible algebraic set, then X × A 1 ⊆ A n+1 is V (P [t]) where P [t] ⊆ R[t]. That is, the coordinate ring of X is R/P and the coordinate ring of X × A 1 is R[t]/P [t] = (R/P )[t]. We supplement our earlier tables with the following line: Geometry Algebra X X × A 1 S S[t] We have defined G(−) and K(−) as functors taking rings as their inputs, but we could also think of them as taking varieties (or schemes) as their inputs (Daniel Dugger 17)
- Theorem 2.15. If R is Noetherian, the Grothendieck group G(R) is generated by the set of elements [R/P ] where P ⊆R is prime. Before proving this result let us comment on the significance. When X is a topological space, the groups H∗ (X) have a geometric presentation in terms of “cycles” and “homologies”. The cycles are, of course, generators for the group. The definition of G(R) doesn’t look anything like this, but Theorem 2.15 says that the group is indeed generated by classes that have the feeling of “algebraic cycles” on the variety Spec R (Daniel Dugger 18)
- It is worth pointing out that in H∗ (X) the cycles are strictly separated by dimension—the dimensions i cycles are confined to the single group Hi (X)—whereas in G(R) the cycles of different dimensions are all inhabiting the same group. This is one of the main differences between K-theory and singular homology/cohomology. (Daniel Dugger 18)
- Lemma 2.16. Let R be a Noetherian ring. For any finitely-generated R-module M , there exists a prime ideal P ⊆R and an embedding R/P ,→ M . Equivalently, there is some z∈M whose annihilator is prime. (Daniel Dugger 18)
- If M is an R-module, write M [t] for the R[t]-module M ⊗R R[t]. The functor M→ 7 M [t] is exact, because R[t] is flat over R (in fact, it is even free). So we have an induced map α : G(R) → G(R[t]) given by [M ] 7→ [M [t]]. Theorem 2.18 (Homotopy invariance). If R is Noetherian, α : G(R) → G(R[t]) is an isomorphism. We comment on the name “homotopy invariance” for the above result. If X = Spec R then Spec R[t] = X × A 1 , so the result says that G(−) gives the same values on X and X × A 1 . This is reminiscent of a functor on topological spaces giving the same values on X and X × I. (Daniel Dugger 19)
- Recall that a module is projective if and only if it is a direct summand of a free module. So free modules are projective, and for almost all applications in homological algebra one can get by with using only free modules. Consequently, it is common not to know many examples of non-free projectives. We begin this section by remedying this. (Daniel Dugger 20)
- Let R = Z /6. Since Z /2 ⊕ Z /3 ∼ = ∼ = Z/6, both Z/2 and Z/3 are projective R-modules—and √ they are clearly√ free. not (Daniel Dugger 21)
- This example generalizes: if D is a Dedekind domain (such as the ring of integers in an algebraic number field) then every ideal I ⊆ D is projective. Non-principal ideals are never free. (Daniel Dugger 21)
- Let R = R [x, y, z]/(x 2 + y 2 + z 2 − 1). If C(S 2 ) denotes the ring of continuous functions S 2 → R , note that we may regard R as sitting inside of C(S 2 ): it is the subring of polynomial functions on the 2-sphere. The connections with the topology of the 2-sphere will be important below. Let π : R 3 → R be the map π(f, g,h) = xf + yg + zh. That is, π is leftmultiplication by the matrix x y z . Let T be the kernel of π: 0 → T ,→ R 3 π π −→ R → 0. (Daniel Dugger 21)
- The map π is split via χ : R → R3 sending 1 7→ (x, y, z). We conclude that T ⊕R ∼ = ∼ = R 3 , so T is projective. (Daniel Dugger 21)
- Note that T is, in some sense, an algebraic analog of the tangent bundle of S 2 . These parallels between projective modules and vector bundles are very important, (Daniel Dugger 22)
- This example is based on the Möbius bundle over S 1 . Let S = R [x, y]/(x 2 + y 2 − 1) and let R ⊆ S be the span of the even degree monomials. One should regard S as the ring of polynomial functions on the circle, and R is the ring of polynomial functions f (x, y) satisfying f (x, y) = f (−x, −y). So R is trying to be the ring of polynomial functions on R P 1 (which happens to be homeomorphic to S 1 ). (Daniel Dugger 22)
- A projective module P is called stably free if there exists a free module F such that P ⊕ F is free (Daniel Dugger 22)
- . It turns out that K(R) can be used to tell us whether such modules exist or not (Daniel Dugger 22)
- Define the reduced Grothendieck group of R to be K e K(R) e = K(R)/h[R]i. (Daniel Dugger 22)
- In the 1950s, Serre conjectured that every finitely-generated projective over F [x1 , . . . , xn ] is actually free. As we will see later (Remark 11.5 below), the motivation for this conjecture is inspired by topology and the connection between vector bundles and projective modules. Quillen and Suslin independently proved Serre’s conjecture in the 1970s. (Daniel Dugger 23)
- Proposition 3.1. Let R be a commutative ring. The following are equivalent: (1) K(R) ∼ = ∼ =Z (2) K e K(R) = 0 e (3) Every finitely-generated, projective R-module is stably-free. (Daniel Dugger 23)
- Remark 4.3. Theorem 4.1 gives another parallel between G(−) and singular homology. If X = Spec R then A = Spec R/f is a closed subscheme, and Spec f −1 R = X − A is the open complement (Daniel Dugger 24)
- So the sequence in Theorem 4.1 can be written as G(A) → G(X) → G(X − A) → 0. This is somewhat reminiscent of the long exact sequence in singular homology · · · → H∗ (A) → H∗ (X) → H∗ (X, A) → · · · (Daniel Dugger 24)
- The second thing is to recall something you probably learned in a basic algebra class, namely the Jordan-Hölder Theorem. This says that given any two filtrations of M we can refine each one so that the two refinements have the same quotients up to reindexing. (Daniel Dugger 28)
- Definition 5.2. A map of chain complexes C• → D• is a quasi-isomorphism if the induced maps Hi (C• ) → Hi (D• ) are isomorphisms for all i ∈ Z. Two chain complexes C• and D• are quasi-isomorphic, written C• ’ D• , if there is a zig-zag of quasi-isomorphisms C• ∼ −→ J 1 ∼ ←− J 2 ∼ −→ · · · ∼ −→ J n ∼ ←− D• (Daniel Dugger 29)
- Lemma 5.3. If P and Q are bounded below complexes of projectives, then every quasi-isomorphism is a chain homotopy equivalence. (Daniel Dugger 29)
- This lemma lets us replace the words “chain homotopy equivalence” with “quasiisomorphism” in any statement about bounded, projective complexes (Daniel Dugger 30)
- ). The advantage of doing this is simply that quasi-isomorphisms are somewhat easier to identify than chain homotopy equivalences. (Daniel Dugger 30)
- why complicate things by making the defintion using complexes of arbitrary length? The answer comes from algebraic geometry. Let X be a scheme and let U be an open subset of X. Then the ‘correct’ way to define a relative K-theory group K(X, U ) is to use bounded chain complexes of locally free sheaves on X that are exact on U . When X = Spec R and U = Spec S −1 R then it happens that one can get the same groups using only complexes of length one—as we saw above. But even for X = Spec R not every open subset is of this form. A general subset will have the form U = (Spec S −1 R) ∪ (Spec S −1 R) ∪ · · · ∪ (Spec S −1 R), but to get the same relative K-group here one must use complexes of length at most d (Daniel Dugger 34)
- When R is a regular ring all localizations S −1 R are also regular (Daniel Dugger 34)
- Let R be a Noetherian ring, and let Z ⊆ Spec R be a Zariski closed set. Recall that an R-module M is said to be supported on Z if MP = 0 for all primes P ∈ / Z. One usually defines Supp M , the support of M , to be {P ∈ Spec R | MP 6= 0}. (Daniel Dugger 34)
- Let R be a discrete valuation ring (a regular local ring whose maximal ideal is principal), (Daniel Dugger 46)
- Let D be a Dedekind domain—a regular ring of dimension one (Daniel Dugger 47)
- In such a ring all nonzero primes are maximal ideals (Daniel Dugger 47)
- The quotient of P 6=0 Z by the classes div(x) is called the divisor class group of D; it is isomorphic to the ideal class group from algebraic number theory. Our short exact sequence shows that K e K e 0 (D) is also isomorphic to this group. (Daniel Dugger 48)
- Our next goal in these notes is to explore the idea of doing linear algebra locally over a fixed base space X. To be slightly more precise, our objects of interest will be maps of spaces E → X where the fibers carry the structure of vector spaces; a map from E → X to E 0 → X is a continuous map F : E → E 0 , commuting with the maps down to X, such that F is a linear transformation on each fiber. It turns out that much of linear algebra carries over easily to this enhanced setting. But there are more isomorphism types of objects here, because the topology of X allows for some twisting in the vector space structure of the fibers. The surprise is that studying these ‘twisted vector spaces’ over a base space X quickly leads to interesting homotopy invariants of X! Topological K-theory is a cohomology theory for topological spaces that arises out of this study of fibrewise linear algebra (Daniel Dugger 50)
- Definition 8.1. A family of (real) vector spaces is a map p : E → X together with operations + : E ×X E → E and · : R × E → E making the two diagrams E ×X E HH H$ H H H H H H H E H + E ~ ~ ~ ~ ~ ~ ~ R × EF · F# F F F F F F F /E ~ ~ ~ ~ ~ ~ ~ ~ X X commute, and such that the operations make each fiber p −1 (x) into a real vector space over X. One could write down the above definition completely category-theoretically, in terms of maps and commutative diagrams. Essentially one is defining a “vector space object” in the category of spaces over X. (Daniel Dugger 50)
- notion is too wild to be of much use: there are too many ‘crazy’ families of vector spaces like this one. One fixes this by adding a condition that forces the fibers to vary continuously, in a certain sense. This is done as follows: Definition 8.3. A vector bundle is a family of vector spaces p : E → X such that for each x ∈ X there is a neighborhood x ∈ U ⊆ X, an n ∈ Z≥0 , and an isomorphism of families of vector spaces p −1 (U ) F # F F F F F F F ∼ = ∼ = / U × R n w{ w w w w w w w w U Usually one simply says that a vector bundle is a family of vector spaces that is locally trivial. The isomorphism in the above diagram is called a “local trivialization”. (Daniel Dugger 51)
- The family of vector spaces from Example 8.2(c) perhaps makes it clear that this notion is too wild to be of much use: there are too many ‘crazy’ families of vector (Daniel Dugger 51)
- Let X = R . Let e1 , e2 be the standard basis for R 2 . Let E ⊆ X × R 2 be the union of {(x, re1 ) | x ∈ Q, r ∈ R} and {(x, re2 ) | x ∈ X, r ∈ R}. Recall from (a) that X × R 2 → X is a family of vector spaces, and note that E becomes a sub-family of vector spaces under the same operations. (Daniel Dugger 51)
- Remark 8.4. Note that the n appearing in Definition 8.3 depends on the point x. It is called the rank of the vector bundle at x, and denoted rankx (E). It is easy to prove that the rank is constant on the connected components of X. (Daniel Dugger 51)
- Notation 8.5. If p : E → X is a family of vector spaces and A ,→ X is a subspace, then p −1 (A) → A is also a family of vector spaces. We will usually write this restriction as E|A . Note that if E is a vector bundle then so is E|A , by a simple argument. The construction E|A is a special case of a pullback bundle, which we will discuss in Section 8.9. Of the families of vector spaces (Daniel Dugger 51)
- we considered in Example 8.2, only the trivial family from (a) is a vector bundle. Before discussing more interesting examples, it will be useful to have a mechanism for deciding when a family of vector spaces is trivial. If p : E → X is a family of vector spaces, a section of p is a map s : X → E such that ps = id. The set of sections is denoted Γ(E), and this becomes a vector space using pointwise addition and multiplication in the fibers of E. A collection of sections s1 , . . . , sr is linearly independent if the vectors s1 (x), s2 (x), . . . , sr (x) are linearly independent in Ex for every x ∈ X. (Daniel Dugger 51)
- Proposition 8.6. Let E → X be a family of vector spaces of constant rank n. Then the family is trivial if and only if there is a linearly independent collection of sections s1 , s2 , . . . , sn . (Daniel Dugger 52)
- (a) Let φ : R n → R n be a vector space isomorphism. Let E 0 = [0, 1] × R n and let E be the quotient of E 0 by the relation (0, v) ∼ (1, φ(v)). Identifying S 1 with the quotient of [0, 1] by 0 ∼ 1, we obtain a map E → S 1 that is clearly a family of vector spaces. We claim this is a vector bundle. If x ∈ (0, 1) then it is evident that E is locally trivial at x, so the only point of concern is x = 0 = 1 ∈ S 1 . Let e1 , . . . , en be the standard basis for R n , and let si : [0, 41 41 ] → E 0 be the constant section whose value is ei . Likewise, let s0 i i : ( 4 , 1] → E 0 be the constant section whose value is φ(ei ). Projecting into E we obtain si (0) = s 0i (1), and so the sections si and s0 i patch together to give a section Si : U → E, where U = [0, 41 1 ) ∪ ( 34 , 1]. The sections S1 , . . . , Sn are independent and therefore give a local trivialization of E over U . When n = 1 and φ(x) = −x the resulting bundle is the Möbius bundle M , depicted below: (Daniel Dugger 52)
- Let X = R P n , and let L ⊆ X × R n+1 be the set L = {(l, v) | l ∈ R P n , x ∈ l}. Then L is a subfamily of the trivial family, and we claim that it is a line bundle over X. To see this, for any l ∈ X we must produce a local trivialization. By symmetry it suffices to do this when l = he1 i. Let U ⊆ RP n be the set of lines whose orthogonal projection to he1 i is nonzero. Such a line contains a unique vector of the form e1 + u where e1 · u = 0. Define s : U → L by sending l to (l, e1 + u) where e1 + u is the unique point on l described above. This section is clearly nonzero everywhere, so it gives a trivialization of L|U . Thus, we have proven that L is locally trivial and hence a vector bundle. The bundle L is called the tautological line bundle over R P n . Do not confuse this with the canonical line bundle over R P n that we will define shortly (they are duals of each other). Note that when n = 1 the bundle L is isomorphic to the Möbius bundle on S 1 . (Daniel Dugger 52)
- Projection to the first coordinate π : η → Grk (V ) makes η into a rank k vector bundle, called the tautological bundle over Grk (V ). (Daniel Dugger 53)
- Pullback bundles can be slightly non-intuitive. Let M → S 1 be the Möbius bundle, and let f : S 1 → S 1 be the map z 7→ z 2 . We claim that f ∗ M ∼ = 1. This is easiest to see if one uses the following model for M : M = n e iθ , re i θ θ 2 θ ∈ [0, 2π], r ∈ R o (Daniel Dugger 53)
- We may form a new bundle E ⊕ E 0 , whose underlying topological space is just the pullback E ×X E 0 . So a point in E ⊕ F is a pair (e, e0 ) where p(e) = p0 (e0 ). The rules for vector addition and scalar multiplication are the evident ones. Note that the fiber of E ⊕ E 0 over a point x is simply Ex ⊕ Ex0 . (Daniel Dugger 54)
- More generally, any canonical construction one can apply to vector spaces may be extended to apply to vector bundles. So one can talk about the bundles E ⊗ E 0 , the dual bundle E ∗ , the hom-bundle Hom(E, E 0 ), the exterior product bundle / i E, and so on (Daniel Dugger 54)
- Recall that if fA : A → Y and fB : B → Y are continuous maps that agree on A ∩ B then we may patch these together to get a continuous map f : X → Y provided that either (i) A and B are both closed, or (ii) A and B are both open. This is a basic fact about topological spaces. (Daniel Dugger 55)
- closed, or (ii) A and B are both open. This is a basic fact about topological spaces. The analogous facts for vector bundles are very similar in the case of an open cover, but more subtle for closed covers. Proposition 8.15. Let E → X be a family of vector spaces. (a) If {Uα } is an open cover of X and each E|Uα is a vector bundle, then E is a vector bundle. (b) Suppose {A, B} is a cover of X by closed subspaces, and that for every x ∈ A∩B and every open neighborhood x ∈ U ⊆ X there exists a neighborhood x ∈ V ⊆ U such that V ∩ A ∩ B ,→ V ∩ B has a retraction. Then if E|A and E|B are both vector bundles, so is E. (Daniel Dugger 55)
- (ii) The two isomorphisms φγ,α and φγ,β ◦ φβ,α agree on their common domain of definition, which is Eα |Uα ∩Uβ ∩Uγ . (Daniel Dugger 56)
- Condition (ii) above is usually called the cocycle condition. (Daniel Dugger 56)
- To see why, consider the case where all of the Eα ’s are trivial bundles of rank n. Then the data in the φα,β maps is really just the data of a map gα,β : Uα ∩ Uβ → GLn (R). These (Daniel Dugger 56)
- gα,β maps are called transition functions. (Daniel Dugger 57)
- Condition (ii) is the requirement that the transition functions assemble to give a C Č Čech 1-cocycle with values in the group GLn (R) (Daniel Dugger 57)
- We will see in a moment (Corollary 8.23) that for real vector bundles over paracompact Hausdorff spaces one always has E ∼ = ∼ = E ∗ , although the isomorphism is not canonical. This is not true for complex or quaternionic bundles, however (Daniel Dugger 57)
- Let L → C P n be the tautological complex line bundle over C P n . Its (complex) dual L ∗ is called the canonical line bundle over C P n . (Daniel Dugger 57)
- Whereas from a topological standpoint neither L nor L ∗ holds a preferential position over the other, in algebraic geometry there is an important difference between the two. The difference comes from the fact that L ∗ has certain “naturally defined” sections, whereas L does not (Daniel Dugger 57)
- For a point z = [z0 : · · · : zn ] ∈ CP n , Lz is the complex line in C n+1 spanned by (z0 , . . . , zn ). Given only z ∈ CP n there is no evident way of writing down a point on Lz , without making some kind of arbitrary choice; said differently, the bundle L does not have any easily-described sections. In contrast, it is much easier to write down a functional on Lz . For example, let φi be the unique functional on Lz that sends the point (z0 , . . . , zn ) to zi . Notice that this description depends only on z ∈ C P n , not the point (z0 , . . . , zn ) ∈ Cn+1 that represents it; that is, the functional sending (λz0 , . . . , λzn ) to λzi is the same as φi . In this way we obtain an entire C n+1 ’s worth of sections for L ∗ , by taking linear combinations of the φi ’s. (Daniel Dugger 57)
- To be clear, it is important to realize that L has plenty of sections—it is just that one cannot describe them by simple formulas. The slogan to remember is that the bundle L ∗ has algebraic sections, whereas L does not. In algebraic geometry the bundle L ∗ is usually denoted O(1), whereas L is denoted O(−1). More generally, O(n) denotes (L∗ )⊗n when n ≥ 0 (so that O(0) is the trivial line bundle), and denotes L ⊗(−n) when n < 0 (Daniel Dugger 57)
- Definition 8.21. Let E → X be a real vector bundle. An inner product on E is a map of vector bundles E ⊗ E → 1 that induces a positive-definite, symmetric, bilinear form on each fiber Ex . A vector bundle with an inner product is usually called an orthogonal vector bundle. (Daniel Dugger 58)
- Every complex vector space may be equipped with a nondegenerate, symmetric bilinear form (Daniel Dugger 58)
- Inner products on R n are in bijective correspondence with symmetric, positive-definite matrices A ∈ Mn×n (R), (Daniel Dugger 59)
- by sending an inner product h−, −i to the matrix aij = hei , ej i (Daniel Dugger 59)
- Now consider the fibration sequence On ,→ GLn (R) → GLn (R)/On . The projection map sends a matrix P to P In P T = P P T . The inclusion On ,→ GLn (R) is a homotopy equivalence by Gram-Schmidt, and so GLn (R)/On is weakly contractible. Standard techniques show that this homogeneous space may be given the structure of a CW-complex (Daniel Dugger 59)
- Suppose that E → X is a rank n real vector bundle with an inner product. Choose a trivializing open cover {Uα }, and for each α fix an inner-productpreserving trivialization fα : E|Uα → Uα × Rn where the codomain has the standard inner product (this is possible by Proposition 8.25). The transition functions gα,β : Uα ∩ Uβ → GLn (R) therefore factor through On , as they must preserve the inner product. This process is usually referred to as reduction of the structure group. (Daniel Dugger 59)
- We claim that the projection map p1 : Z → W is a fiber bundle with fiber Rk(n−k) , but defer the proof for just a moment. The fact that the fiber is contractible then shows that p1 is weak homotopy equivalence (Daniel Dugger 61)
- Proposition 9.2. Let X be a paracompact space. Then any surjection of bundles E F has a splitting. (Daniel Dugger 61)
- Proof. Briefly, we choose local splittings and then use a partition of unity to patch them together (Daniel Dugger 61)
- Proposition 9.3. Let X be any space, and let f : E → F be a map of vector bundles over X. If f has constant rank then ker f , coker f , and im f are vector bundles. (Daniel Dugger 62)
- Corollary 9.4. Let X be a paracompact space. Then any injection of bundles E ,→ F has a splitting. (Daniel Dugger 62)
- Proposition 9.5. Suppose that X is compact and Hausdorff. Then every bundle is a subbundle of some trivial bundle. (Daniel Dugger 62)
- Lemma 9.7. Let E α α −→ F β −→ G be an exact sequence of vector bundles. Then im α (which equals ker β) is a vector bundle. (Daniel Dugger 63)
- In this section we explore our first connection between topology and algebra. We will see that vector bundles are closely related to projective modules. (Daniel Dugger 64)
- The assignment E 7→ Γ(E) gives a functor from vector bundles to C(X)-modules. (Daniel Dugger 64)
- When X is a space let C(X) denote the ring of continuous functions from X to R , where the addition and multiplication are pointwise (Daniel Dugger 64)
- It is easy to check that Γ is a left-exact functor (Daniel Dugger 64)
- If E → X is a vector bundle then of course the modules of the form Γ(E) are not just arbitrary C(X)-modules; there is something special about them. It is easiest to say what this is under some assumptions on X: Proposition 10.1. If X is compact and Hausdorff, and E is a vector bundle over X, then Γ(E) if a finitely-generated, projective module over C(X). (Daniel Dugger 64)
- That is, Γ(E) is a direct summand of a free module; hence it is projective. (Daniel Dugger 64)
- For the rest of this section we will assume that our base spaces are compact and Hausdorff. Let hhVect(X)ii denote the category of vector bundles over X, and let hhMod −C(X)ii denote the category of modules over the ring C(X). (Daniel Dugger 64)
- hhProj −C(X)ii denote the full subcategory of finitely-generated, projective modules. Then Γ is a functor hhVect(X)ii → hhProj −C(X)ii. It is proven in [Sw] that this is actually an equivalence: Theorem 10.2 (Swan’s Theorem). Let X be a compact, Hausdorff space. Then Γ : hhVect(X)ii → hhProj −C(X)ii is an equivalence of categories. (Daniel Dugger 65)
- To prove this result we need to verify two things: • Every finitely-generated projective over C(X) is isomorphic to Γ(E) for some vector bundle E. • For every two vector bundles E and F , the induced map Γ : HomVect(X) (E, F ) → HomC(X) (ΓE, ΓF ) is a bijection. (Daniel Dugger 65)
- That is to say, we need to prove that Γ is surjective on isomorphism classes, and is fully faithful (Daniel Dugger 65)
- Note that we have the evaluation map evx : Γ(E) → Ex . This map clearly sends the submodule mx Γ(E) to zero. Lemma 10.4. Assume that X is paracompact Hausdorff. Then for any vector bundle E → X and any x ∈ X, the map evx : Γ(E)/mx Γ(E) ∼ = ∼ = = −→ Ex is an isomorphism. (Daniel Dugger 66)
- Our final goal is to prove that Γ is fully faithful. To do this, it is useful to relate the fibers Ex of our bundle to an algebraic construction based on the module Γ(E). For each x ∈ X consider the evaluation map evx : C(X) → R, and let mx be the kernel. The ideal mx ⊆ C(X) is maximal, since the quotient is a field. (Daniel Dugger 66)
- Proposition 10.5. Assume that X is paracompact Hausdorff. Then for any vector bundles E and F over X, the map Γ : HomVect(X) (E, F ) → HomC(X) (ΓE, ΓF ) is a bijection. (Daniel Dugger 66)
- For a fixed n, let Vectn (X) denote the set of isomorphism classes of vector bundles on X. It turns out that when X is a finite complex this set is always countable, and often finite. It actually gives a homotopy invariant of the space X. (Daniel Dugger 67)
- Example 11.3. To give an idea how we will apply these results, let us think about vector bundles on S 1 . Divide S 1 into an upper hemisphere D+ and a lower hemisphere D− , intersecting in two points. Each of D+ and D− are contractible, so any vector bundle will be trivializable when restricted to these subspaces. Given two elements α, β ∈ GLn (R), let En (α, β) be the vector bundle on S 1 obtained by taking n D+ and n D− and gluing them together via α and β at the two points on the equator. The considerations of the previous paragraph tell us that every vector bundle on S 1 is of this form (Daniel Dugger 68)
- In (2) we have used the fact that π0 (GLn (R)) = Z/2, with the isomorphism being given by the sign of the determinant. (Daniel Dugger 68)
- To summarize, from (1) and (2) it follows that isomorphism types for rank n bundles over S 1 are in bijective correspondence with the path components of GLn (R). (Daniel Dugger 69)
- The methods of the above example apply in much greater generality, and with little change allow one to get control over vector bundles on any suspension. (Daniel Dugger 69)
- Remark 11.5. We have seen that all bundles on contractible spaces are trivial, and that there is a close connection between vector bundles and projective modules. Recall that when k is a field then k[x1 , . . . , xn ] is the algebraic analog of affine space A n , and that projectives over this ring correspond to algebraic vector bundles. The analogy with topology is what led Serre to conjecture that all finitely-generated projectives over k[x1 , . . . , xn ] are free (Daniel Dugger 70)
- We have proven that if E is a vector bundle on X × I then i∗0 (E) ∼ = ∼ i = ∗ 1 (E). It is natural to wonder if this result has a converse, but stating such a thing is somewhat tricky (Daniel Dugger 70)
- So we find ourselves in somewhat of a muddle. Perhaps there is an interesting question here, but we don’t quite know how to ask it. One approach is to restrict to a class of bundles where “equality” is something we can better control (Daniel Dugger 70)
- We may view a vector bundle as a family of vector spaces indexed by the base space. In general, we may view a map X → Y as a family of blah if each fiber is a blah. (Daniel Dugger 71)
- We naively hope that families of some mathematical object over X are in bijection with maps from X to some space, called the moduli space corresponding to that mathematical object. (Daniel Dugger 71)
- some space, called the moduli space corresponding to that mathematical object. With this naive idea, we would hope that families over ∗ are in bijective correspondence with points of our moduli space. However, this does not work since the moduli space of R n ’s is ∗. (Daniel Dugger 71)
- V ⊆ W then we get an induced inclusion of Grassmannians Grk (V ) ,→ Grk (W ). Consider the standard chain of inclusions of Euclidean spaces R k , and define the infinite Grassmannian Grn (R ∞ ) to be the colimit of the induced sequence of finite Grassmannians: Grn (R ∞ ) = colim k→∞ [Grn (R k )]. (Daniel Dugger 71)
- Define γn → Grn (R∞ ) by γn = {(V, x) | V ⊂ R∞ , dim(V ) = n, x ∈ V }. This is the tautological vector bundle on the infinite grassmanian. (Daniel Dugger 71)
- To any map f : X → Grn (R∞ ) we associate the pullback bundle f ∗ γn / γn X f / Grn (R ∞ ). The assignment f 7→ f ∗ γn gives a map Hom(X, Grn (R∞ )) → Vectn (X). Observe that if f, g : X → Grn (R∞ ) are homotopic maps, then f ∗ γn ∼ = g ∗ γn by Corollary 11.2(a). In this way we have constructed a map φ : [X, Grn (R ∞ )] → Vectn (X). We will show that this is an isomorphism when X is compact and Hausdorff (Daniel Dugger 71)
- Theorem 11.8. The map φ : [X, Grn (R ∞ )] → Vectn (X) is always injective, and is bijective when X is compact and Hausdorff. (Daniel Dugger 71)
- Proposition 11.10. Let X be a finite-dimensional CW-complex. For real vector bundles, Vectn (X) → Vectn+1 (X) is a bijection for n ≥ dim X + 1 and a surjection for n = dim X. For complex bundles, VectC n n C (X) → Vectn n+1 (X) is a bijection for n ≥ 21 dim X and a surjection for n ≥ 12 (dim X − 1). (Daniel Dugger 72)
- Fix a space X. If E → X is a vector bundle of rank n, then of course E ⊕ 1 is a vector bundle of rank n + 1. We get a sequence of maps Vect0 (X) ⊕1 −→ Vect1 (X) ⊕1 −→ Vect2 (X) ⊕1 −→ · · · Are these maps injective? Surjective? Are there more and more isomorphism classes of vector bundles as one goes up in rank, or is it the case that all “large” rank vector bundles actually come from smaller ones via addition of a trivial bundle? (Daniel Dugger 72)
- rank vector bundles actually come from smaller ones via addition of a trivial bundle? A homotopical analysis of classifying spaces allow us give some partial answers here. (Daniel Dugger 72)
- this section we explore the set of isomorphism classes Vectn (S k ) for various values of k and n. There are two important points. First, for a fixed k these sets stablize for n 0. Secondly, Bott was able to compute these stable values completely and found an 8-fold periodicity (with respect to k) in the case of real vector bundles, and a 2-fold periodicity in the case of complex bundles. Bott’s periodicity theorems are of paramount importance in modern algebraic topology (Daniel Dugger 72)
- In (Daniel Dugger 72)
- 12.1. The clutching construction. Let X be a pointed space, and let C+ and C− be the positive and negative cones in ΣX. Fix n ≥ 0. For a map f : X → GLn (R), let En (f ) be the vector bundle obtained by gluing n|C+ and n|C− via the map f (we use Corollary 8.17(b) here). Precisely, if x ∈ X and v belongs to the fiber of n C+ over x then we glue v to f (x) · v in the fiber of n C− over x. This procedure for constructing vector bundles on ΣX is called clutching, and every bundle on ΣX arises in this way (Daniel Dugger 73)
- Let us apply the above result when X is a sphere S k−1 . We obtain a bijection Vectn (S k ) ’ πk−1 GLn (R). (Daniel Dugger 73)
- Recall that On ,→ GLn (R) is a deformation retraction, as a consequence of the Gram-Schmidt process (Daniel Dugger 73)
- When k > 2 any based map S k−1 → On must actually factor through the connected component of the identity, which is SOn . So we have Vectn (S k ) ∼ = ∼ = πk−1 GLn (R) = ∼ = πk−1 On ∼ ∼ = ∼ = πk−1 SOn (Daniel Dugger 73)
- 12.3. Vector bundles on S 1 . For k = 1 and n > 0 we have that Vectn (S 1 ) ∼ = ∼ = π0 GLn (R) = Z/2, and we have previously seen in Example 11.3 that the two isomorphism classes are represented by n and M ⊕ (n − 1) where M is the Möbius bundle. (Daniel Dugger 73)
- 12.4. Vector bundles on S 2 . Here we have Vectn (S 2 ) ∼ = ∼ = π1 SOn . Recall that SO2 ∼ = ∼ = S 1 , and so we get Vect2 (S 2 ) ∼ = ∼ = Z. We claim that for n > 2 one has π1 SOn ∼ = ∼ = Z/2, so that we have the following: (Daniel Dugger 73)
- Proposition 12.5. Vectn (S 2 ) ∼ = ∼ = π1 (SOn ) (Daniel Dugger 73)
- For n = 3 recall that SO3 ∼ = ∼ = RP 3 , so that π1 (SO3 ) ∼ = ∼ = Z/2. To see the homeomorphism use the model R P 3 ∼ = ∼ = D 3 /∼ where the equivalence relation has x ∼ −x for x ∈ ∂D3 . (Daniel Dugger 73)
- Proof. First of all SO1 = {1} and SO2 ∼ = ∼ = S 1 , so this takes care of n ≤ 2. For n = 3 recall that SO3 ∼ = ∼ = RP 3 , so (Daniel Dugger 73)
- Map D 3 → SO3 by sending a vector v to the rotation of R3 with axis hvi, through |v| · π radians, in the direction given by a right-hand-rule with the thumb pointed along v. Note that this makes sense even for v = 0, since the corresponding rotation is through 0 radians. For x ∈ ∂D3 this map sends x and −x to the same rotation, and so induces a map R P 3 → SO3 . This is clearly a continuous bijection, and therefore a homeomorphism since the spaces are compact and Hausdorff. (Daniel Dugger 74)
- For n ≥ 4 one can use the long exact sequence associated to the fibration SOn−1 ,→ SOn S n−1 to deduce that π1 (SOn ) ∼ = π1 (SOn−1 ). (Daniel Dugger 74)
- Definition 12.6. Let O(n) ∈ Vect2 (S 2 ) be the vector bundle Efn where fn : S 1 → SO2 is a map of degree n. Note that O(0) ∼ = 2. The bundles O(n), n ∈ Z , give a complete list of the rank 2 bundles on S 2 (Daniel Dugger 74)
- Putting all of this information together, the following table shows all the vector bundles on S 2 : n 1 2 3 4 5 6 Vectn (S 2 ) 1 O(n), n ∈ Z 3, O(1) ⊕ 1 4, O(1) ⊕ 2 5, O(1) ⊕ 3 ··· The operation (−) ⊕ 1 moves us from one column of the table to the next (Daniel Dugger 74)
- answer, namely what happens when one adds two rank 2 bundles (all other sums can be figured out once one knows how to do these): Theorem 12.7. O(j) ⊕ O(k) (Daniel Dugger 74)
- It is a standard fact in topology that the group structure on [S 1 , SO4 ]∗ given by (Daniel Dugger 74)
- pointwise multiplication agrees with the group structure given by concatenation of loops (this is true with SO4 replaced by any topological group). (Daniel Dugger 75)
- 12.8. Vector bundles on S 3 . Now we have to calculate π2 SOn . This is trivial for n ≤ 2 (easy), and for n = 3 it also trivial: use SO3 ∼ = ∼ = RP 3 and the fibration sequence Z /2 ,→ S 3 R P 3 . Finally, the fibration sequences SOn−1 ,→ SOn S n−1 now show that π2 SOn = 0 for all n. We have proven Proposition 12.9. Vectn (S 3 ) ∼ = ∼ = π2 (SOn ) ∼ = ∼ = 0. That is, every vector bundle on S 3 is trivializable. (Daniel Dugger 75)
- 12.10. Vector bundles on S 4 . Once again, we are reduced to calculating π3 SOn . Eventually one expects to get stuck here, but so far we have been getting lucky so let’s keep trying (Daniel Dugger 75)
- 12.12. Vector bundles on S k . Although we can not readily do the calculations for k > 4, at this point one sees the general pattern. One must calculate πk−1 SOn for each n, and these groups vary for a while but eventually stabilize. In fact, πi SOn ∼ = ∼ = πi SOn+1 for i + 1 < n. The calculation of these stable groups was an important problem back in the 1950s, that was eventually solved by Bott. (Daniel Dugger 75)
- The colimit of this sequence is denoted O and called the stable orthogonal group. The homotopy groups of O are the stable values that we encountered above. We computed the first few: π0 O = Z/2, π1 O = Z /2, π2 O = 0. And we stated, without proof, that π3 O = Z. Bott’s calculation showed the following: (Daniel Dugger 76)
- The pattern is 8-fold periodic: πi+8 O ∼ = ∼ = πi O for all i ≥ 0. One is supposed to remember the pattern of groups to the tune of “Twinkle, Twinkle, Little Star”: zee two zee two ze ro zee ze ro ze ro ze ro zee. (Daniel Dugger 76)
- 12.13. Complex vector bundles on spheres. One can repeat the above analysis for complex vector bundles on a sphere. One finds that VectC n n (S k ) ∼ = ∼ = πk−1 (GLn (C)) ∼ = ∼ = πk−1 (Un ), (Daniel Dugger 76)
- The stable value in the last row turns out to be 0, although one cannot figure this out without computing a connecting homomorphism in the long exact homotopy sequence (Daniel Dugger 76)
- We can write the stable value as πi U where U is the infinite unitary group defined as the colimit of U1 ,→ U2 ,→ U3 ,→ · · · Bott computed the homotopy groups of U to be 2-fold periodic, with πi U = ( Z if i is odd 0 if i is even. (Daniel Dugger 76)
- For a compact and Hausdorff space X, let KO(X) denote the Grothendieck group of real vector bundles over X. Swan’s Theorem gives that KO(X) ∼ = ∼ = Kalg (C(X)), where the latter denotes the Grothendieck group of finitely-generated projectives (Daniel Dugger 77)
- We can repeat this definition for both complex and quaternionic bundles, to define groups KU (X) and KSp(X), respectively. The group KU (X) is most commonly just written K(X) for brevity (Daniel Dugger 77)
- Remark 13.4. Both KO st (X) and g KO g (X, x) appear often in algebraic topology, and topologists are somewhat cavalier about mixing them up. (Daniel Dugger 77)
- Finally, here is a third description of KO st (X). Consider the chain of maps Vect0 (X) ⊕1 −→ Vect1 (X) ⊕1 −→ Vect2 (X) ⊕1 −→ · · · The colimit is clearly the set of equivalence classes described in the preceding paragraph, and therefore coincides with KO st (X). (Daniel Dugger 78)
- Let Gr∞ ( R ∞ ) denote the colimit of these maps Gr1 (R ∞ ) ⊕1 −→ Gr2 (R ∞ ) ⊕1 −→ Gr3 (R ∞ ) ⊕1 −→ · · · (we really want the homotopy colimit, if you know what that is, but in this case the colimit has the same homotopy type and is good enough). (Daniel Dugger 78)
- You might recall that Grn (R ∞ ) is also called BOn , and likewise Gr∞ (R ∞ ) is also called BO. (Daniel Dugger 78)
- So we have learned that KO st (X) ’ [X, BO]. (Daniel Dugger 78)
- The calculations of Bott therefore give us the values of g KO g (S KO k ). For k = 0 observe that KO(S 0 ) = KO(∗ t ∗) ∼ = ∼ g (S 0 ) ∼ = Z ⊕ Z, so we have KO = Z. This lets us fill out the table: Table 13.4. Reduced KO-theory of spheres k 0 1 2 3 4 5 6 7 8 9 10 11 ··· g KO (S g k ) Z Z /2 Z /2 0 Z 0 0 0 Z Z /2 Z /2 0 ··· (Daniel Dugger 78)
- Applying this in particular to X = S k we have that for k ≥ 1 KO g (S KO k ) ∼ = ∼ = KO st (S k ) ∼ = ∼ = [S k , BO] ∼ = ∼ = [S k , BO]∗ = πk (BO) = πk−1 (O). (Daniel Dugger 78)
- Theorem 13.5 (Bott Periodicity, Strong version). There is a weak equivalence of spaces Z × BO ’ Ω8 ( Z × BO). (Daniel Dugger 79)
- Using Bott Periodicity we can then calculate that for every pointed space X one has g KO g (Σ KO 8 X) = [Σ 8 X, Z × BO]∗ = [X, Ω 8 ( Z × BO)]∗ = [X, Z × BO]∗ = KO g KO g (X). Remark 13.6. In the complex case, Bott Periodicity gives the weak equivalence Z × BU ’ Ω2 ( Z × BU ). Consequently one obtains K e K(Σ 2 X) ∼ = ∼ = K e K(X) e for all pointed spaces X. (Daniel Dugger 79)
- Homotopy classes of maps into a fixed space Z always give rise to exact sequences: Proposition 13.8. Let X, Y be pointed spaces, and let f : X → Y be a pointed map. Consider the mapping cone Cf and the natural map p : Y → Cf . For any pointed space Z, the sequence of pointed sets [X, Z]∗ ← [Y, Z]∗ ← [Cf, Z]∗ is exact in the middle. (Daniel Dugger 79)
- Note that Cj0 ’ ΣX and Cj1 ’ ΣY (this is clear from the pictures). Up to sign the map Cj0 → Cj1 is just Σf , so that the sequence of spaces becomes periodic: X → Y → Cf → ΣX → ΣY → Σ(Cf ) → Σ 2 X → . . . This is called the Puppe sequence. Note that the composition of two subsequent maps is null-homotopic, and that every three successive terms form a cofiber sequence. (Daniel Dugger 80)
- Given f : X → Y we form the mapping cone Cf , which comes to us with an inclusion j0 : Y ,→ Cf . Next form the mapping cone on i, which comes with an inclusion j1 : Cf ,→ Cj0 . Keep doing this forever to get the sequence of spaces X → Y → Cf → Cj0 → Cj1 → · · · depicted below: Note that Cj0 ’ ΣX and Cj1 ’ ΣY (this is clear from the pictures). Up (Daniel Dugger 80)
- Now let Z be a fixed space and apply [−, Z]∗ to the Puppe sequence. We obtain the sequence of pointed sets [X, Z]∗ ← [Y, Z]∗ ← [Cf, Z]∗ ← [ΣX, Z]∗ ← [ΣY, Z]∗ ← [Σ(Cf ), Z]∗ ← . . . By Proposition 13.8 this sequence is exact at every spot where this makes sense (everywhere except at [X, Z]∗ ). At the left end this is just an exact sequence of pointed sets, but as one moves to the right at some point it becomes an exact sequence of groups (namely, at [ΣY, Z]∗ ). As one moves further to the right, it becomes an exact sequence of abelian groups by the time one gets to [Σ 2 Y, Z]∗ . (Daniel Dugger 80)
- Definition 13.9. An infinite loop space is a space Z0 together with spaces Z1 , Z2 , Z3 , . . . and weak homotopy equivalences Zn ’ ΩZn+1 for all n ≥ 0. Note that if Z is an infinite loop space then we really do get a long exact sequence—infinite in both directions—consisting entirely of abelian groups, having the form · · · ← [Cf, Zi+1 ] ← [X, Zi ]∗ ← [Y, Zi ]∗ ← [Cf, Zi ]∗ ← [X, Zi−1 ] ← · · · (Daniel Dugger 81)
- where it is convenient to use the indexing convention Z−n = Ω n Z for n > 0. This situation is very reminiscent of a long exact sequence in cohomology, so let us adopt the following notation: write E i (X) = [X+ , Zi ]∗ = ( [X+ , Zi ]∗ i ≥ 0, [Σ −i (X+ ), Z0 ]∗ i < 0. For an inclusion of subspaces j : A ,→ X write E i Z (X, A) = [Cj, Zi ]+ = ( [Cj, Zi ]∗ i ≥ 0, [Σ i (Cj), Z0 ]∗ i < 0. It is not hard to check that this is a generalized cohomology theory (Daniel Dugger 81)
- So we get a generalized cohomology theory whenever we have an infinite loop space. (You may know that it works the other way around, too: every generalized cohomology comes from an infinite loop space (Daniel Dugger 81)
- For us the importance of all of this is that by Bott’s theorem we have Z × BO ’ Ω 8 ( Z × BO) ’ Ω 16 ( Z × BO) ’ . . . . Thus, Z × BO is an infinite loop space and the above machinery applies. We obtain a cohomology theory KO ∗ . Moreover, periodicity gives us that KO i+8 (X, A) ∼ = ∼ = KO i (X, A), for any i. (Daniel Dugger 81)
- Let us try to compute KO( R P 2 ). (Daniel Dugger 81)
- The point of this section was to construct the cohomology theories KO and K, having the properties that when X is compact and Hausdorff the groups KO 0 (X) and K 0 (X) coincide with the Grothendieck groups of real and complex vector bundles over X. (Daniel Dugger 81)
- Next use the fact that R P 2 can be built by attaching a 2-cell to R P 1 = S 1 , where the attaching map wraps S 1 around itself twice. That is, R P 2 is the mapping cone for S 1 2 2 −→ S 1 . (Daniel Dugger 81)
- g 0 (S 1 ) = KOst (S 1 ) corresponds We have previously seen that the generator of KO to the Mobius bundle [M ] (Daniel Dugger 82)
- g (S 2 ) = KOst (S 2 ) is [O(1)], and the generator of KO the rank 2 bundle whose clutching map is the isomomorphism S 1 → SO(2). (Daniel Dugger 82)
- We happen to know one bundle on RP 2 , the tautological line bundle γ. (Daniel Dugger 82)
- We need g to decide if 2[γ] = 0 in KOst (RP 2 ); if it is, then KO g (RP KO 2 ) ∼ = (Z/2)2 and if it is g (RP not then KO ) 2 ∼ = Z/4. So the question becomes: is γ ⊕ γ stably trivial? (Daniel Dugger 82)
- The answer turns out to be that γ ⊕ γ is not stably trivial; this is an elementary exercise using characteristic classes (Stiefel-Whitney classes), (Daniel Dugger 82)
- Note that this calculation demonstrates an important principle to keep in mind: often the machinery of cohomology theories get you a long way, but not quite to the end, and one has to do some geometry to complete the calculation (Daniel Dugger 82)
- Recall that [E] in KOst (RP 2 ) corresponds to [E] − rank(E) in KO g KO g (RP 2 ); so the class we wrote as [γ] is [γ] − 1 in the shifted perspective, and we need to decide if 2([γ] − 1) = 0 in KO(RP 2 ). The element 1 − [γ] should be thought of as corresponding to a chain complex of vector bundles 0 → γ → 1 → 0, and thinking of it this way one finds that it plays the role of the K-theoretic fundamental class of the submanifold RP 1 ,→ RP 2 . Then (1 − [γ])2 represents the self-intersection product of RP 1 inside RP 2 , which we know is a point by the standard geometric argument (shown in the picture below, depicting an RP 1 and a small perturbation of it)): (Daniel Dugger 82)
- For the moment just get the idea that it is the intersection theory of submanifolds in RP g (RP that is ultimately forcing KO 2 ) to be Z/4 rather than (Z/2)2 (Daniel Dugger 83)
- Remark 13.12. It seems worth pointing out that in fact for every n one has n ) ∼ = Z/2k g (RP n ) ∼ KO = Z/2k for a certain value k depending on n. (Daniel Dugger 83)
- Exercise 13.13. It is a good idea for the reader to try his or her hand at similar calculations, to see how the machinery is working. Try calculating some of the groups below, at least for small values of n: • K(CP n ) (reasonably easy) • KO(CP n ) (a little harder) • K(RP n ) (even harder) • KO(RP n ) (hardest). (Daniel Dugger 83)
- It is a classical problem to determine how many independent vector fields one can construct on a given sphere S n . (Daniel Dugger 83)
- This problem was heavily studied throughout the 1940s and 1950s, and then finally solved by Adams in 1962 using K-theory. (Daniel Dugger 83)
- It is one of the great successes of generalized cohomology theories. (Daniel Dugger 83)
- 14.1. The vector field problem. Given a nonzero vector u = (x, y) in R 2 , there is a formula for producing a (nonzero) vector that is orthogonal to u: namely, (−y, x). However, there is no analog of this that works in R 3 . That is, there is no single formula that takes a vector in R 3 and produces a (nonzero) orthogonal vector. If such a formula existed then it would give a nonvanishing vector field on S 2 , and we know that such a thing does not exist by elementary topology. (Daniel Dugger 83)
- Let us next consider what happens in R4 . Given u = (x1 , x2 , x3 , x4 ), we can produce an orthogonal vector via the formula (−x2 , x1 , −x4 , x3 ). But of course this (Daniel Dugger 83)
- is not the only way to accomplish this: we can vary what pairs of coordinates we choose to flip. In fact, if we consider −x 2 −x3 −x4 x1 x4 −x3 v 1 = −x4 , v2 = x1 , v3 = x2 . −x 4 , v2 = x1 , v3 = x2 x 3 −x2 x1 then we find that v1 , v2 , and v3 are not only orthogonal to u but they are orthogonal to each other as well. In particular, at each point of S 3 we have given an orthogonal basis for the tangent space. (Daniel Dugger 84)
- Question 14.2. On S n , how many vectors fields v1 , v2 , . . . , vr can we find so that v 1 , v2 , . . . , vr are linearly independent for each x ∈ S n ? (Daniel Dugger 84)
- Note that by the Gram-Schmidt process we can replace “linearly independent” by “orthonormal.” If n is even, the answer is zero because there does not exist even a single nonvanishing vector field on an even sphere. (Daniel Dugger 84)
- Let u ∈ S 5 have the standard coordinates. We notice that the vector v1 = (−x 2 , x1 , −x4 , x3 , −x6 , x5 ) is orthogonal to v. However, a little legwork shows that no other pattern of switching coordinates will produce a vector that is orthogonal to both u and v1 . Of course this does not mean that there isn’t some more elaborate formula that would do the job, but it shows the limits of what we can do using our naive constructions. (Daniel Dugger 84)
- For v ∈ S 7 we can divide the coordinates into the top four and the bottom four. Take the construction that worked for S 3 and repeat it simultaneously in the top and bottom coordinates—this yields a set of three orthonormal vector fields on S 7 , given by the formulas (14.2) (−x2 , x1 , −x4 , x3 , −x6 , x5 , −x8 , x7 ), (−x 3 , x4 , x1 , −x2 , −x7 , x8 , x5 , −x6 ), (−x 4 , −x3 , x2 , x1 , −x8 , −x7 , x6 , x5 ). This idea generalizes at once to prove the following: (Daniel Dugger 84)
- Proposition 14.3. If there exist r (independent) vector fields on S n−1 , then there also exist r vector fields on S kn−1 for all k. (Daniel Dugger 84)
- For example, since there is one vector field on S 1 we also know that there is at least one vector field on S 2k−1 for every k. (Daniel Dugger 84)
- Likewise, since there are three vector fields on S 3 we know that there are at least three vector fields on S 4k−1 for every k. (Daniel Dugger 84)
- Recall that S 3 is a Lie group, being the unit quaternions inside of H. We can choose an orthonormal frame at the origin and then use the group structure to push this around to any point, (Daniel Dugger 84)
- hereby obtaining three independent vector fields; in other words, for any point x ∈ S 3 use the derivative of right-multiplication-by-x to transport our vectors in T 1 S 3 to Tx S 3 . The space S 7 is not quite a Lie group, but it still has a multiplication coming from being the set of unit octonions. The multiplication is not associative, but this is of no matter—the same argument works to construct 7 vector fields on S 7 . Note that this immediately gives us 7 vectors fields on S 15 , S 23 , etc. (Daniel Dugger 85)
- Based on the data so far, one would naturally guess that if n = 2r then there are n − 1 vector fields on S n−1 . However, this guess turns out to fail already when n = 16 (and thereafter). (Daniel Dugger 85)
- To give a sense of how the numbers grow, we give a chart showing the maximum number of vector fields known to exist on low-dimensional spheres: n 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 n−1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 v.f. on S n−1 1 3 1 7 1 3 1 8 1 3 1 7 1 3 1 9 (Daniel Dugger 85)
- Definition 14.4. If n = m · 2a+4b where m is odd, then the Hurwitz-Radon number for n is ρ(n) = 2a + 8b − 1. Theorem 14.5 (Hurwitz-Radon). There exist at least ρ(n) independent vector fields on S n−1 . (Daniel Dugger 85)
- We will prove the Hurwitz-Radon theorem by a slick, modern method using Clifford algebras. (Daniel Dugger 85)
- 14.6. Sums-of-squares formulas. Hurwitz and Radon were not actually thinking about vector fields on spheres. They were instead considering an algebraic question about the existence of certain kinds of “composition formulas” for quadratic forms. For example, the following identity is easily checked: (x21 + x22 ) · (y12 + y22 ) = (x1 y1 − x2 y2 )2 + (x1 y2 + x2 y1 )2 . Hurwitz and Radon were looking for more formulas such as this one, for larger numbers of variables: (Daniel Dugger 85)
- Definition 14.7. A sum-of-squares formulas of type [r, s, n] is an identity (x21 + x22 + . . . + x2r )(y12 + y22 + . . . + ys2 ) = z12 + z22 + . . . + zn2 in the polynomial ring R[x1 , . . . , xr , y1 , . . . , ys ], where each zi is a bilinear expression in x’s and y (Daniel Dugger 86)
- We will often just refer to an “[r, s, n]-formula”, for brevity. For what values of r, s, and n does such a formula exist? (Daniel Dugger 86)
- This is currently an open question. There are three formulas that are easily produced, coming from the normed algebras C, H, and O. The multiplication is a bilinear pairing, and the identity |xy|2 = |x|2 |y|2 is the required sums-of-squares formula. (Daniel Dugger 86)
- Perhaps surprisingly, most of what is known about the non-existence of sumsof-squares formulas comes from topology. To phrase the question differently, we are looking for a function φ : R r ⊗ R s → R n such that |φ(x, y)|2 = |x|2 · |y|2 for all x ∈ R r and y ∈ R s . The bilinear expressions z1 , . . . , zn are just the coordinates of φ(x, y). (Daniel Dugger 86)
- Corollary 14.9. If an [r, n, n]-formula exists, then there exist r − 1 independent vector fields on S n−1 . (Daniel Dugger 86)
- Therefore we have established that φ(e2 , −), φ(e 3 , −), . . . , φ(er , −) are orthonormal vector fields on S n−1 . (Daniel Dugger 87)
- 14.10. Clifford algebras. We have seen that we get r − 1 independent vector fields on S n−1 if we have a sums-of-squares formula of type [r, n, n]. Having such a formula amounts to producing matrices A2 , A3 , . . . , Ar ∈ On such that A2i = −I and Ai Aj + Aj Ai = 0 for i 6= j. If we disregard the condition that the matrices be orthogonal, we can encoded the latter two conditions by saying that we have a representation of a certain algebra: Definition 14.11. The Clifford algebra Clk is defined to be the quotient of the tensor algebra Rhe1 , . . . , ek i by the relations e2i = −1 and ei ej + ej ei = 0 for all i 6= j. (Daniel Dugger 87)
- The first few Clifford algebras are familiar: Cl0 = R, Cl1 = C, and Cl2 = H. After this things become less familiar: for example, it turns out that Cl3 = H × H (we will see why in just a moment). It is somewhat of a miracle that it is possible to write down a precise description of all of the Clifford algebras, and all of their modules. Before doing this, let us be clear about why we are doing it: Theorem 14.12. An [r, n, n]-formula exists if and only if there exists a Clr−1 module structure on R n . Consequently, if there is a Clr−1 -module structure on R n then there are r − 1 independent vector fields on S n−1 . (Daniel Dugger 87)
- The collection of monomials e i 1 · · · eir for 1 ≤ i1 < i2 < · · · < ir ≤ k give a vector space basis for Clk , which has size 2k (Daniel Dugger 87)
- But a miracle now occurs, in that we can analyze all the Clifford algebras by a simple trick. (Daniel Dugger 88)
- To do this part of the argument, we need a slight variant on our Clifford algebras. Given a real vector space V and a quadratic form q : V → R, define Cl(V, q) = TR (V )/hv ⊗ v = q(v) · 1 | v ∈ V i. For R with k q(x1 , . . . , xk ) = −(x21 + · · · + x2k ) this recovers the algebra Clk . For q(x 1 , . . . , xk ) = x21 + · · · + x2k this gives a new algebra we will call Cl − k . (Daniel Dugger 88)
- Proposition 14.14. There are isomorphisms of algebras Cl ± ∼ k = ∼ ± = Cl2 ∓ ⊗R Clk−2 . (Daniel Dugger 88)
- In the analysis that follows we will write A(n) for the algebra Mn×n (A), whenever A is an algebra. (Daniel Dugger 88)
- Table 14.15. Clifford algebras (Daniel Dugger 89)
- Lemma 14.16. There are isomorphisms H ⊗R C ∼ = C(2) and H ⊗R H ∼ = R(4). (Daniel Dugger 90)
- 14.17. Modules over Clifford algebras. Now that we know all the Clifford algebras, it is actually an easy process to determine all of their finitely-generated modules. We need three facts: • If A is a division algebra then all modules over A are free; • By Morita theory, the finitely-generated modules over A(n) are in bijective correspondence with the finitely-generated modules over A. The bijection sends an A-module M to the A(n)-module M n . • If R and S are algebras then modules over R × S can all be written as M × N where M is an R-module and N is an S-module (Daniel Dugger 91)
- Table 14.18. Dimensions of Clifford modules (Daniel Dugger 91)
- Recall that if Clr−1 acts on R n then there are r − 1 independent vector fields on S n−1 . (Daniel Dugger 91)
- Going down the rows of the above table, we make the following deductions: Cl 1 acts on R2 , therefore we have 1 vector field on S Cl 2 acts on R4 , therefore we have 2 vector field on S (Daniel Dugger 91)
- the smallest dimension of a nonzero module over Clr is 2σ(r) where σ(r) = #{s : 0 < s ≤ r and s ≡ 0, 1, 2, or 4 mod (8)} 2 σ(r) −1 −1 We know that we can construct r independent vector fields on S 2 . (Daniel Dugger 92)
- We will eventually see, following [ABS], that there is a very direct connection between the groups KO∗ and the module theory of the Clifford algebras. (Daniel Dugger 92)
- 14.20. Adams’s Theorem. So far we have done all this work just to construct collections of independent vector fields on spheres. The Hurwitz-Radon lower bound is classical, and was probably well-known in the 1940’s. The natural question is, can one do any better? Is there a different construction that would yield more vector fields than we have managed to produce? People were actively working on this problem throughout the 1950’s. Adams finally proved in 1962 [Ad2] that the Hurwitz-Radon bound was maximal, and he did this by using K-theory: Theorem 14.21 (Adams). There do not exist ρ(n) + 1 independent vector fields on S n−1 . (Daniel Dugger 92)
- Proposition 14.22. If there are r − 1 vector fields on S n−1 then the projection RP un−1 /RP un−r−1 → RP un−1 /RP un−2 ∼ = ∼ = S un−1 has a section in the homotopy 2k−2 2k−2 category, for every u > n . n (Daniel Dugger 93)
- Define V k (Rn ) = {(u1 , . . . , uk ) | ui ∈ R n and u1 , . . . , uk are orthonormal}. This is called the Stiefel manifold of k-frames in R n . (Daniel Dugger 93)
- Consider the map p 1 : Vk (Rn ) → S n−1 which sends (u1 , . . . , un ) 7→ u1 . There exist r vector fields on S n−1 if and only if there is a section of p1 : Vr+1 (Rn ) → S n−1 . (Daniel Dugger 93)
- We need a fact from basic topology, namely that there is a cell structure on V k (Rn ) where the cells look like e i 1 × · · · × eis with n − k ≤ i1 < i2 < · · · < is ≤ n − 1 and s is arbitrary. We will not prove this here: see Hatcher [Ha, Section 3.D] (Daniel Dugger 93)
- Proposition 14.23. If n + 2 > 2k then the n-skeleton of our cell structure on V k (Rn ) is homeomorphic to RP n−1 /RP n−k−1 . (Daniel Dugger 93)
- Exercise 14.24. Use singular cohomology to prove that RP n−1 /RP n−3 → S n−1 does not have a section when n is odd. Deduce that an even sphere does not have a non-vanishing vector field (which you already knew). (Daniel Dugger 93)
- At this point we have seen that there exist cohomology theories K ∗ (−) and KO∗ (−). We have not proven their existence, but we have seen that their existence falls out as a consequence of the Bott periodicity theorems Ω2 (Z × BU ) ’ Z × BU and Ω8 (Z × BO) ’ Z × BO. (Daniel Dugger 94)
- To some extent we have a “geometric” understanding of K 0 (−) and KO0 (−) in n terms of Grothendieck groups of vector bundles. We also know that any K n (−) (or n (−)) group can be shifted to a K 0 KOn (−)) group can be shifted to a K 0 (−) group using the suspension isomorphism and Bott periodicity (Daniel Dugger 94)
- . One often hears a slogan like “The geometry behind K-theory lies in vector bundles”. This slogan, however, doesn’t really say very much; our aim will be to do better. (Daniel Dugger 94)
- One way to encode geometry into a cohomology theory is via Thom classes for vector bundles. Such classes give rise to fundamental classes for submanifolds and a robust connection wth intersection theory. (Daniel Dugger 94)
- The theory of Thom classes begins with the cohomological approach to orientations. Recall that (Daniel Dugger 94)
- Moreover, an orientation on R n determines a generator for H n (Rn , Rn − 0) ∼ = ∼ = Z. (Daniel Dugger 94)
- Now consider a vector bundle p : E → B of rank n. Let ζ : B → E be the zero section, and write E −0 as shorthand for E −im(ζ). For any x ∈ B let Fx = p−1 (x). n (Fx , Fx − 0) ∼ = Then H n (Fx , Fx − 0) ∼ = Z, and an orientation of the fiber gives a generator. We wish to consider the problem of giving compatible orientations for all the fibers at once; this can be addressed through the cohomology of the pair (E, E − 0). (Daniel Dugger 94)
- Pick a generator UV ∈ H (EV , EV − 0) n ∼ = Z. (Daniel Dugger 94)
- For a neighborhood V of x, let EV = E|V = p−1 (V ). (Daniel Dugger 94)
- We think of UV as orienting all of the fibers simultaneously. (Daniel Dugger 95)
- Definition 15.1. Given a rank n bundle E → B, a Thom class for E is an element UE ∈ H n (E, E − 0) such that for all x ∈ B, jx∗ (UE ) is a generator in H n (Fx , Fx − 0). (Here jx : Fx ,→ E is the inclusion of the fiber). (Daniel Dugger 95)
- There is no guarantee that a bundle has a Thom class. Indeed, consider the following example: (Daniel Dugger 95)
- If a bundle E → B has a Thom class then the bundle is called orientable. Said differently, an orientation on a vector bundle E → B is simply a choice of Thom class in H n (E, E − 0; Z). (Daniel Dugger 96)
- One can also talk about Thom classes with respect to the cohomology theories H ∗ (−; R) for any ring R. Typically one only needs R = Z and R = Z/2, however. In the latter case, note that any n-dimensional real vector space V has a canonical orientation in H n (V, V − 0; Z/2). It follows that local Thom classes always patch together to give global Thom classes, and so every vector bundle has a Thom class ∗ in H ∗ (−; Z/2). (Daniel Dugger 96)
- Finally, note that we can repeat all that we have done for complex vector spaces and complex vector bundles. However, a complex vector space V of dimension n has a canonical orientation on its underlying real vector space, and therefore a canonical generator in H 2n (V, V − 0). Just as in the last paragraph, this implies that local Thom classes always patch together to give global Thom classes; so every complex vector bundle has a Thom class. (Daniel Dugger 96)
- Theorem 15.3. (a) Every complex bundle E → B of rank n has a Thom class in H 2n (E, E − 0). (b) Every real bundle E → B of rank n has a Thom class in H n (E, E − 0; Z/2). (Daniel Dugger 96)
- Theorem 15.4 (Thom Isomorphism Theorem). Suppose that p : E → B has a Thom class UE ∈ H ∗ (E, E − 0). Then the map H ∗ (B) → H ∗ (E, E − 0) given by z 7→ p∗ (z) ∪ UE is an isomorphism of graded abelian groups that increases degrees by n. (Daniel Dugger 96)
- 15.5. Thom spaces. The relative groups H ∗ (E, E − 0) coincide with the reduced cohomology groups of the mapping cone of the inclusion E − 0 ,→ E. This mapping cone is sometimes called the Thom space of the bundle E → B, (Daniel Dugger 97)
- Definition 15.6. Suppose that E → B is a bundle with an inner product. Define the disk bundle of E as D(E) = {v ∈ E | hv, vi ≤ 1}, and the sphere bundle of E as S(E) = {v ∈ E | hv, vi = 1}. (Daniel Dugger 97)
- This diagram shows that E − 0 ,→ E and S(E) ,→ D(E) have weakly equivalent mapping cones (Daniel Dugger 97)
- Unlike E −0 ,→ E, however, the map S(E) ,→ D(E) is a cofibration (under the mild condition that X is cofibrant, say): so the mapping cone is weakly equivalent to the quotient D(E)/S(E). This quotient is what is most commonly meant by the term ‘Thom space’: Definition 15.7. For a bundle E → B with inner product, the Thom space of E is Th E = D(E)/S(E). (Daniel Dugger 97)
- Note that if B is compact then Th E is homeomorphic to the one-point compactification of the space E. (Daniel Dugger 97)
- To see this it is useful to first compactify all the fibers separately, which amounts to forming the pushout of B ← S(E) → D(E). (Daniel Dugger 97)
- k ) ∼ = RP n+k /RP n−1 Example 15.9. We will show that Th(nL → RP k ) ∼ = RP n+k /RP n−1 , where L is the tautological line bundle. (Daniel Dugger 97)
- Proposition 15.13. For any real bundle E → X one has Th(E ⊕ n) ∼ = Σn Th(E). ∼ 2n = Σ For a complex bundle E → X one has Th(E ⊕ n) ∼ 2n = Σ Th(E). (Daniel Dugger 99)
- 15.12. Thom spaces for virtual bundles. Thom spaces behave in a very simple way in relation to adding on trivial bundles: (Daniel Dugger 99)
- From this one readily sees that D(E ⊕ n)/S(E ⊕ n) ∼ = [D(E)/S(E)] ∧ [Dn /S n−1 ] ∼ = Th(E) ∧ S n (Daniel Dugger 99)
- Proposition 15.13 allows one to make sense of Thom spaces for virtual bundles, provided that we use spectra. (Daniel Dugger 99)
- these Thom spectra play a large role in modern algebraic topology. (Daniel Dugger 99)
- 15.14. An application to stunted projective spaces. To demonstrate the usefulness of Thom spaces we give an application to periodicities amongst stunted projective spaces. This material will be needed later, in the solution of the vector fields on spheres problem. (Daniel Dugger 100)
- The natural question arises: fixing a and b, what values of r (if any) satisfy Σ r [RP a+b /RP a ] ’ RP a+b+r /RP a+r . (Daniel Dugger 100)
- One can use singular cohomology and Steenrod operations to produce some necessary conditions here. For example, integral singular homology easily yields that if if b ≥ 2 then r must be even. Use of Steenrod operations produces more stringent conditions ( (Daniel Dugger 100)
- We will use Thom spaces to provide some sufficient conditions for a stable homotopy equivalence between stunted projective spaces (Daniel Dugger 100)

Last Annotation: 04/04/2019

- Mapping come and cylinder diagrams (James Davis, Paul Kirk 147)

- The diﬀerential ∂ : Cq (X; R) → Cq−1 (X; R) can be deﬁned in two ways. The ﬁrst is purely algebraic, the second is geometric and involves the notion of the degree of a map f : S n → S n . If you don’t know what the degree of such a map is, look it up. If you know the deﬁnition of degree, then look up the diﬀerential-topological deﬁnition of degree for a smooth map f : S n → Sn. (James Davis, Paul Kirk 19)
- Deﬁnition 1.4. A cellular map f : X → Y is a continuous function between CW-complexes so that f (Xq ) ⊂ Yq for all q. A cellular map induces a chain map f∗ : C∗ (X; R) → C∗ (Y ; R), since f restricts to a map of pairs f : (Xq , Xq−1 ) → (Yq , Yq−1 ). Thus for every q, cellular homology is a functor (James Davis, Paul Kirk 20)
- So for example, the circle S 1 has a cell structure with one 0-cell and one 1-cell. The boundary map is trivial, so H1 (S 1 ) ∼ = ∼ = Z. A generator [S 1 ] ∈ H1 (S ) is speciﬁed by taking the 1-cell which parameterizes the circle 1 in a counterclockwise fashion. We can use this to deﬁne the Hurewicz map ρ : π1 (X, x0 ) → H1 (X; Z) (James Davis, Paul Kirk 20)
- Theorem 1.6. Suppose that X is path–connected. Then the Hurewicz map ρ : π1 (X, x0 ) → H1 (X; Z) is a surjection with kernel the commutator subgroup of π1 (X, x0 ). Hence H1 (X; Z) is isomorphic to the abelianization of π1 (X, x0 ). (James Davis, Paul Kirk 21)
- 1.1.4. Construction of the simplicial chain complex of a simplicial complex. Deﬁnition 1.7. An (abstract) simplicial complex K is a pair (V, S) where V is a set and S is a collection of non-empty ﬁnite subsets of V satisfying: 1. If v ∈ V then {v} ∈ S. 2. If τ ⊂ σ ∈ S and τ is non-empty, then τ ∈ S. Elements of V are called vertices. Elements of S are called simplices. A q-simplex is an element of S with q + 1 vertices. If σ ∈ S is a q-simplex we say dim (σ) = q. (James Davis, Paul Kirk 21)
- Deﬁnition 1.8. The geometric realization of a simplicial complex K is the quotient space |K| = σ∈S ∆ dim (σ) ∼ . In other words, we take a geometric q–simplex for each abstract q–simplex of K, and glue them together. (James Davis, Paul Kirk 22)
- The identiﬁcations are given as follows (James Davis, Paul Kirk 22)
- A triangulation of a topological space X is a homeomorphism from the geometric realization of a simplicial complex to X. (James Davis, Paul Kirk 22)
- It is not hard, using the acyclic models theorem, to show that the simplicial and cubical singular homology functors are naturally isomorphic (James Davis, Paul Kirk 22)
- Deﬁnition 1.9. The tensor product of A and B is the R-module A ⊗R B deﬁned as the quotient F (A × B) R(A × B) where F (A × B) is the free R-module with basis A × B and R(A × B) the submodule generated by 1. (a1 + a2 , b) − (a1 , b) − (a2 , b) 2. (a, b1 + b2 ) − (a, b1 ) − (a, b1 ) 3. r(a, b) − (ra, b) 4. r(a, b) − (a, rb). (James Davis, Paul Kirk 23)
- One denotes the image of a basis element (a, b) in A ⊗R B by a ⊗ b. Note that one has the relations 1. (a1 + a2 ) ⊗ b = a1 ⊗ b + a2 ⊗ b, 2. a ⊗ (b1 + b2 ) = a ⊗ b1 + a ⊗ b2 , 3. (ra ⊗ b) = r(a ⊗ b) = (a ⊗ rb). (James Davis, Paul Kirk 23)
- Any element of A ⊗ B can be expressed as a ﬁnite sum uct ai ⊗ bi , but it may not be possible to take n = 1, nor is the representation as a sum unique. (James Davis, Paul Kirk 23)
- The universal property of the tensor product is that this map is initial in the category of bilinear maps with domain A × B. (James Davis, Paul Kirk 23)
- Proposition 1.10. Given a R-bilinear map φ : A×B → M , there is unique R-module map φ̄ φ̄ : A ⊗R B → M so that φ̄ ◦ π = φ. (James Davis, Paul Kirk 23)
- Proposition 1.10 is useful for deﬁning maps out of tensor products, and the following exercise indicates that this is the deﬁning property of tensor products. (James Davis, Paul Kirk 24)
- . The basic properties of the tensor product are given by the next theorem. Theorem 1.11. 1. A ⊗ B ∼ = ∼ =B⊗A 2. R ⊗ B ∼ = ∼ =B 3. (A ⊗ B) ⊗ C ∼ = ∼ = A ⊗ (B ⊗ C) 4. (⊕α Aα ) ⊗ B ∼ = ∼ = ⊕ α (Aα ⊗ B) 5. Given R-module maps f : A → C and g : B → D, there is an Rmodule map f ⊗ g : A ⊗ B → C ⊗ D so that a ⊗ b → f (a) ⊗ g(b). 6. The functor − ⊗ M is right exact. That is, given an R-module M , and an exact sequence A f f − →B g g − → C → 0, the sequence (James Davis, Paul Kirk 24)
- the sequence A ⊗ M f ⊗Id f ⊗Id −−−→ B ⊗ M g⊗Id g⊗Id −−−→ C ⊗ M → 0 is exact. (James Davis, Paul Kirk 24)
- Example 1.12. Let M be an abelian group. Applying properties 5 and 2 of Theorem 1.11 we see that if we tensor the short exact sequence 0 → Z ×n ×n −−→ Z → Z/n → 0 by M we obtain the exact sequence M ×n ×n −−→ M → Z/n ⊗Z M → 0. Notice that Z/n ⊗Z M ∼ = ∼ = M/nM and that the sequence is not short exact if M has torsion whose order is not relatively prime to n. Thus − ⊗ M is not left exact. (James Davis, Paul Kirk 25)
- 1.2.2. Adjoint functors. Note that a R-bilinear map β : A × B → C is the same as an element of HomR (A, HomR (B, C)) (James Davis, Paul Kirk 25)
- Proposition 1.14 (Adjoint Property of Tensor Products). There is an isomorphism of R-modules HomR (A ⊗R B, C) ∼ = ∼ = HomR (A, HomR (B, C)), natural in A, B, C given by φ ↔ (a → (b → φ(a ⊗ b))). (James Davis, Paul Kirk 26)
- This is more elegant than the universal property for three reasons: It is a statement in terms of the category of R-modules, it gives a reason for the duality between tensor product and Hom, and it leads us to the notion of adjoint functor. Deﬁnition 1.15. (Covariant) functors F : C → D and G : D → C form an adjoint pair if there is a 1-1 correspondence MorD (F c, d) ←→ MorC (c, Gd), for all c ∈ Ob C and d ∈ Ob D, natural in c and d. The functor F is said to be the left adjoint of G and G is the right adjoint of F . (James Davis, Paul Kirk 26)
- The adjoint property says that for any R-module B, the functors − ⊗ R B : R-MOD → R-MOD and HomR (B, −) : R-MOD → R-MOD form an adjoint pair. Here R-MOD is the category whose objects are Rmodules and whose morphisms are R-maps. (James Davis, Paul Kirk 26)
- random functor may not have a left (or right) adjoint, but if it does, the adjoint is unique up to natural isomorphism (James Davis, Paul Kirk 26)
- Hence taking duals deﬁnes a contravariant functor from the category of R-modules to itself. (James Davis, Paul Kirk 26)
- The following computational facts may help with Exercise 8. 1. HomR (R, M ) ∼ = ∼ = M. 2. HomR (⊕α Aα , M ) ∼ = ∼ = α HomR (Aα , M ). 3. HomR (A, α Mα ) ∼ = ∼ = α HomR (A, Mα ). (James Davis, Paul Kirk 27)
- The starting observation is that the singular (or cellular) homology functor is a composite of two functors, the singular complex functor S∗ : { spaces, cts. maps } → { chain complexes, chain maps } and the homology functor H∗ :{chain complexes, chain maps}→{graded R-modules, homomorphisms}. The strategy is to place interesting algebraic constructions between S∗ and H∗ ; i.e. to use functors {Chain Complexes} → {Chain Complexes} to construct new homology invariants of spaces (James Davis, Paul Kirk 27)
- Deﬁnition 1.16. Taking the homology of C∗ ⊗ M yields the homology of C∗ with coeﬃcients in M : H∗ (C∗ ; M ) = ker ∂ : C∗ ⊗ M → C∗ ⊗ M Im ∂ : C∗ ⊗ M → C∗ ⊗ M . Applying this to the singular complex of a space leads to the following deﬁnition. Deﬁnition 1.17. The homology of S∗ (X; R) ⊗ M is called the singular homology of X with coeﬃcients in the R-module M and is denoted by H∗ (X; M ). (James Davis, Paul Kirk 28)
- Recall that for a chain complex (C∗ , ∂), a cycle is an element of ker ∂ and a boundary is an element of Im ∂. The terminology for cochain complexes is obtained by using the “co” preﬁx: Deﬁnition 1.20. A cocycle is an element in the kernel of δ and a coboundary is an element in the image of δ. (