Last Annotation: 10/07/2019

- The resulting theory [FQ] led quickly to a complete classifica tion of closed, simply connected topological 4-manifolds, and topological 4manifold theory now seems closely related to the theory of high-dimensional manifolds. (Robert Gompf, Andas Stipsicz 8)
- Using gauge theory (differential geometry and nonlinear analysis), Donaldson showed that smooth 4-manifolds are much different from their high-dimensional counterparts. In fact, the predictions made by the s-cobordism and surgery conjectures for smooth 4-manifolds failed miserably, resulting in a dramatic clash between the theorics of smooth and topological manifolds in this dimension. For example, this is the only dimension in which a fixed homeomorphism type of closed manifold is represented by infinitely many diffeomorphism types, or where there are manifolds homeomorphic but not diffeomorphic to R®. (In fact, there are uncountably many such “exotic R*’s”.) (Robert Gompf, Andas Stipsicz 8)
- Donaldson’s program of analyzing the self-dual Yang-Mills equations [DK] was central to smooth 4-manifold theory for 12 years, until it was superseded in 1994 (several revolutions later) by analysis of the Seiberg-Witten equations (Robert Gompf, Andas Stipsicz 8)
- That is, gauge theory proves the nonexistence of smooth manifolds satisfying various constraints, the nonexistence of connected-sum splittings, and the nonexistence of diffeomorphisms between pairs of manifolds. (Robert Gompf, Andas Stipsicz 8)
- perhaps the most powerful general technique for existence results (particularly for manifolds with small Betti numbers) is Kirby calculus. (Robert Gompf, Andas Stipsicz 8)
- This technique, which allows one to see the internal structure of a 4-manifold {or its boundary 3-manifold) without loss of information, was created and developed into a fine art in the late 1970°s by topologists such as Akbulut, Fenn, Harer, Kaplan, Kirby, Melvin, Rourke, Rolfsen and Stern. (Robert Gompf, Andas Stipsicz 8)
- Much time was spent on ambitious goals that gauge theory now shows are impossible. (Robert Gompf, Andas Stipsicz 8)
- surprising connections have emerged with affine complex analysis and contact topology [G13], [G14] since a discovery of Eliashberg led to a theory of Kirby diagrams for representing Stein surfaces (Robert Gompf, Andas Stipsicz 9)
- we have included Kirby diagrams representing all of the main types of closed, simply connected 4-manifolds (as viewed from the current perspective of the theory), namely complex surfaces of rational, elliptic and general type, a symplectic but noncomplex manifold and an irreducible nonsymplectic one. (Robert Gompf, Andas Stipsicz 9)
- There are many references for gauge theory as applied to 4-manifolds, notably [DK] (one of the most recent references from the viewpoint of the self-dual equations), and [KKM], [Mrl], [Sa] on Seiberg-Witten theory (Robert Gompf, Andas Stipsicz 9)
- imilarly, the theory of complex surfaces is covered in detail in [BPV], and symplectic topology is carefully treated in (Robert Gompf, Andas Stipsicz 9)
- Similarly, the theory of complex surfaces is covered in detail in [BPV], and symplect (Robert Gompf, Andas Stipsicz 9)
- [Mc81j, (Robert Gompf, Andas Stipsicz 10)
- For topological 4-manifolds, the reader is referred to [FQ] (Robert Gompf, Andas Stipsicz 10)
- Although we treat Rolfsen calculus in some detail, the reader is also referred to [Ro] for this 3-dimensional technique related to Kirby calculus. (Robert Gompf, Andas Stipsicz 10)
- We will use the terms principal G-bundle, tangent bundle of a manifold, associated vector bundle and section of a bundle without defining them. (Robert Gompf, Andas Stipsicz 15)
- {Detailed treatments of these topics can be found in, e.g., [GP], [MS] and [Sp].) D (Robert Gompf, Andas Stipsicz 15)
- A pair (Uy, ¢a) of such a neighborhood and homeomorphism is called a chart. A collection of charts {(Uy, do) | @ € A} is an atlas if it is a cover of X, that is, |J{Ua | @ € A} = X. The map ¢go¢,! (on ¢o(UaNUp)) is the transition function between the charts (Ua, $a) and (Us, $3). (Robert Gompf, Andas Stipsicz 15)
- orresponding to points in {(z1,...,2n) | Zn = 0} = R*1 CR" form a submanifold 0X of dimension n — 1, which is called the boundary of X. (Robert Gompf, Andas Stipsicz 16)
- The topological manifold X with an atlas {(Ua, ¢.) | @ € A} is a CT-manifold (r =1,2,3,...,00) if the transition functions ¢5 0 471 (a, 8 € A) are O"maps. In the case r = oo, X is called a smooth manifold. (Robert Gompf, Andas Stipsicz 16)
- Two C"structures on X are isotopic if the identity map idy is isotopic (homotopic through homeomorphisms) to a C-diffecomorphism between the structures. (Robert Gompf, Andas Stipsicz 16)
- We say that X is closed if it is compact and 0X = §. We call a space X a singular manifold if there is a finite subset Sing C X such that X — Sing is a smooth manifold. (Robert Gompf, Andas Stipsicz 16)
- An orientation of the Euclidean vector space B® is simply a choice of one orbit of the set of ordered bases under the action of the connected group GLT(n;R) = {A € GL(n;R) | det(A) > 0}. (Robert Gompf, Andas Stipsicz 16)
- An orientation can be given by fixing an ordering (up to even permutations) of a given basis. (Robert Gompf, Andas Stipsicz 16)
- A consistent choice of orientation of the tangent space at every point of X is called an orientation of X. (Robert Gompf, Andas Stipsicz 16)
- The standard convention of “outward normal first” provides an orientation of 8X induced by an orientation of X (Robert Gompf, Andas Stipsicz 16)
- At p € 0X the basis {v1,... ,un—1) of T,0X is positive if (v,v1,... ,vp—1) is a positive basis of T,X, where the vector » stands for the outward normal vector, which is tangent to X but not to 3X and pomts out of X. (Robert Gompf, Andas Stipsicz 16)
- It can be shown that an orientation specifies a class [X] in H,.(X,0X;Z)}, called the fundamental class (Robert Gompf, Andas Stipsicz 17)
- In this way, orientability can easily be extended to topological manifolds as well: An n-dimensional connected manifold X is orientable if Hn(X,8X;2Z) = Z, and an orientation of X is simply a choice of a generator of the group H,(X,8X;Z). (Robert Gompf, Andas Stipsicz 17)
- The induced orientation of 8X can be seen as the image of [X]| € H,(X,8X;Z) under the map Hn(X,0X;Z) — Hy,1(0X;Z) of the long exact sequence of the pair (X,8X); (Robert Gompf, Andas Stipsicz 17)
- The smooth ndimensional manifold X is orientable if the structure group of the tangent bundle TX — X (which is GL{n;R)}) can be reduced to its connected component GL’(n;R); by fixing a reduction we specify an orientation for X. (Robert Gompf, Andas Stipsicz 17)
- An atlas {{Uy, d.) | @ € Ag} on a (real) 2n-dimensional manifold X is a complex structure if each ¢, is a homeomorphism between U, and an open subset of C* (identified with B2"), and the transition functions ¢go¢,! are holomorphic. (Robert Gompf, Andas Stipsicz 17)
- Complex manifolds are canonically oriented. This is because the connected group GL(n;C) lies in GL (2n;R), so any complex n-dimensional vector space is canonically oriented -— by choosing a complex isomorphism with C" = C x ... x C, with C oriented as a real vector space by the ordered basis (1,3). (Robert Gompf, Andas Stipsicz 17)
- A smooth (resp. topological) isotopy between embceddings ¢g, 1: Y — X is a smooth (resp. topological) homotopy ¢;: Y — X (0 <t <1) through embeddings. (Robert Gompf, Andas Stipsicz 17)
- if is compact then any smooth isotopy ¥ — int X can be extended to an ambienl isotopy, an isotopy ®;: X — X through diffeomorphisms such that bg = idx and pp; = B; 0g for each t. (Robert Gompf, Andas Stipsicz 17)
- Theorem 1.1.6 demonstrates the fact that although the notion of C’-manifold is defined for every integer r (0 < r < oo), the r = 0 and r = 00 cases are the only interesting ones in terms of classification. (Robert Gompf, Andas Stipsicz 17)
- Theorem 1.1.6. ([Mu]) Suppose that X is a CT-manifold and 1 < r < k (including k = oo). Then there is a C*-atlas of X for which the induced (Robert Gompf, Andas Stipsicz 17)
- CT -structure is isotopic to the original C"-siructure of X. (Robert Gompf, Andas Stipsicz 18)
- consequently the C"-manifold X admits a unique induced C*-structure for every k > r (Robert Gompf, Andas Stipsicz 18)
- Our primary aim in manifold theory is to classify topological manifolds, i.e., to give a complete list of n-dimensional (closed) topological manifolds, and to find a way to tell which topological manifolds carry smooth structures (have C-atlases). (Robert Gompf, Andas Stipsicz 18)
- Furthermore, if there is one such atlas, we would like to determine the total number of these up to diffeomorphism. (Robert Gompf, Andas Stipsicz 18)
- In most dimensions this aim cannot be achieved for algebraic reasons (cf. Theorem 1.2.33 and Exercise 5.1.10(c}); in those cases we will impose further conditions (like . simple connectivity) for the manifolds at hand. (Robert Gompf, Andas Stipsicz 18)
- The classification problem is easy in dimension 1 and classical in dimension 2. Up to homeomorphism there is only one topological 1-manifold with the above properties, and this is the circle §! = {z € R? | ||z|| = 1}; it admits a unique isotopy class of smooth structures. For n = 2, the (oriented) topological 2manifolds are precisely the surfaces Xi with genus g¢ (9 = 0,1,2,...); in particular, £g is the sphere 5? and ¥; is the 2-dimensional torus T° = §1 x $1. (Robert Gompf, Andas Stipsicz 18)
- Assume that the manifolds we are working with are closed, connected and oriented. T (Robert Gompf, Andas Stipsicz 18)
- All these topological manifolds carry unique smooth structures (up to isotopy); actually these manifolds carry complex structures as well. (Robert Gompf, Andas Stipsicz 18)
- It is known that every topological 3-manifold admits a unique smooth structure [Mo]; (Robert Gompf, Andas Stipsicz 18)
- ; the classification problem of topological 3-manifolds is, however, still unsolved. (Robert Gompf, Andas Stipsicz 18)
- Understanding 3-manifolds homotopy equivalent to the 3-dimensional sphere S% would be a major step in this direction; (Robert Gompf, Andas Stipsicz 18)
- Conjecture 1.1.7. (Poincaré Conjecture) A simply connected closed 3manifold is homeomorphic to 5°. (Robert Gompf, Andas Stipsicz 18)
- For topological manifolds of dimension n > 5 there is sophisticated machinery for dealing with both the existence problem and the number of nonisotopic smooth structures on a given topological manifold. (Robert Gompf, Andas Stipsicz 18)
- Theorem 1.1.8. If X™ is a compact n-dimensional topological manifold andn > 6 (orn > 5 and 8X = B), then there are only finitely many smooth structures on X (up to isotopy). (Robert Gompf, Andas Stipsicz 19)
- In particular, there are smooth manifolds Yi,...,Y: homeomorphic to X such that any smooth manifold homeomorphic to X is diffeomorphic to some Y;. (Here k might be 0, meaning that there is no smooth structure on X.)} (Robert Gompf, Andas Stipsicz 19)
- Theorem 1.1.9. If X is homeomorphic to R® and n # 4 then X is diffeomorphic to R“. If n = 4, this statement is false, there are “exotic” R’s; such examples will be given in Sections 9.3 and 9.4. 0 In general, the term exotic smooth structure is used to refer to smooth structures not diffeomorphic to the given one on a smooth manifold X. (Robert Gompf, Andas Stipsicz 19)
- Remark 1.1.10. Another sort of structure frequently used by topologists is a piecewise linear (PL-) structure, which is defined by an atlas whose transition functions respect a suitable triangulation of R™ (e.g., [RS]). Any smooth structure determines a PL-structure, and the converse holds for n < 6 [HM], so for our purposes PL-structures are equivalent to smooth structures. (Robert Gompf, Andas Stipsicz 19)
- Definition 1.2.1. The symmetric bilinear form Qx: HY(X,0X;Z) x H4(X,0X;Z) — Z defined by Qx (a,b) = {(aUb, [X]}) = a-b eZ is called the intersection form of X. Since by Poincaré duality Ho(X;Z) = H%(X,0X;Z), Qx is defined on Ho{X;Z) x Hy(X; Z) as well. (Robert Gompf, Andas Stipsicz 19)
- Clearly, Qx{a,b}) = 0 if a or b is a torsion element, hence Qx descends to a pairing on homology mod torsion. By choosing a basis of Hy(X; Z)/Torsion, we can represent Qx by a matrix. The matrix M of Qx transforms under a basis transformation C as CTMC. Consequently the determinant det M is independent of the choice of the basis over Z: we sometimes denote this by det x. (Robert Gompf, Andas Stipsicz 19)
- (a) The above definition of @x can be extended to cohomology with arbitrary coefficient ring B. When R = Z,, the theory generalizes in the obvious way to nonorientable manifolds. (Robert Gompf, Andas Stipsicz 20)
- A class a € Ho(X™;Z) is represented by a closed, oriented surface T if there is an embedding i: ¥ «— X such that i.([X]) = a. (Again, [X] € Hyo(Z; Z) is the fundamental class of XI). (Robert Gompf, Andas Stipsicz 20)
- Proposition 1.2.3. Let X be a closed, oriented, smooth 4-manifold. Then every element of Hao(X;Z) can be represented by an embedded surface. (Robert Gompf, Andas Stipsicz 20)
- Elements of H?(X;Z) are in 1-1 correspondence with U(1)-bundles over X (Robert Gompf, Andas Stipsicz 20)
- For a € Hy(X; Z) take its Poincaré dual a = PD{a) € H%(X;Z) and denote the corresponding U(1)-bundle by Ly — X. The zero set of a generic section of the bundle L, — X will be a smooth surface representing a. (Robert Gompf, Andas Stipsicz 20)
- Note that if X is simply connected, then by the Hurewicz Theorem wa(X) = H3(X;Z). This implies that for a simply connected 4-manifold every second homology element can be represented by an immersed sphere. (Robert Gompf, Andas Stipsicz 20)
- Such an immersion will not be an embedding in general, but one can assume that an immersion §2 — X* intersects itself only in transverse double points. (Robert Gompf, Andas Stipsicz 20)
- Suppose that X*? is closed and oriented. For a,b € H?({X;Z) take surface representatives X, and 3g of the Poincaré duals « = PD{a} and § = PD(b). Suppose furthermore that X, and Xg have been chosen generically, so that their intersections are all transverse. The orientations of X, and Ig — (Robert Gompf, Andas Stipsicz 20)
- ow we are inh the position to give the geometric interpretation of x — this description explains the name of it. Proposition 1.2.5. For a,b € HX, Z) and a, € Ho(X;Z) as above, Qx (a,b) is the number of points in Ly, NXg, counted with sign. (Robert Gompf, Andas Stipsicz 21)
- together with the fixed orientation of the ambient 4-manifold X — assign a sign +1 to every intersection point of XZ, and I; in the following way [GP]. By concatenating positive bases of the tangent spaces Tpit, and TpXig at a point p € Ty NXg we get a basis of TpX. The sign of the intersection at p is positive if this basis is positive, and negative otherwise. (Robert Gompf, Andas Stipsicz 21)
- In arbitrary dimensions, the same method of counting intersections gives the intersection pairing (including relative versions) for any two homology classes of complementary dimension, and this is again dual to the cup product pairing [GH]. The only difference is that high-dimensional homology classes cannot always be represented by submanifolds, so one must allow smooth cycles with singularities. (Robert Gompf, Andas Stipsicz 21)
- Again, the above proposition applies if X* has boundary or is noncompact (Robert Gompf, Andas Stipsicz 21)
- and a similar statement holds over Z; without the orientability hypotheses. (Robert Gompf, Andas Stipsicz 21)
- An easy argument shows that the transverse intersection of complex submanifolds is always positive. (Robert Gompf, Andas Stipsicz 21)
- In particular, Qgs([C1],[C2]) => 0 if C1,C2 C 8 are transversely intersecting complex curves in a complex surface S. (Robert Gompf, Andas Stipsicz 21)
- bilinear form ¢ on the finitely generated free abelian group A (Robert Gompf, Andas Stipsicz 22)
- The rank rk(Q) of Q is the dimension of A. Extend and diagonalize Q over A ®z R. The number of +1’s (—1’s respectively) on the diagonal is denoted by 53 (resp. by }; the difference bf — by is the signature o(Q) of Q. Finally, Q is even if Qa, a) =0 (mod?2) for every a € A; @Q is odd otherwise (Robert Gompf, Andas Stipsicz 22)
- Definitions 1.2.8. (a) Q is positive (negative) definite if rk(Q) = o(Q) (rk(Q) = —(Q) resp.). Q is indefinite otherwise. (Robert Gompf, Andas Stipsicz 22)
- (b) The direct sum Q@ = @1 D @Q3 of the forms @; and Qs (given on Aj, Ay respectively) is defined on A; © A; in the following way. (Robert Gompf, Andas Stipsicz 22)
- (c) An clement = € A is called a characteristic element if Qa, x) = Qa, a) (mod 2) for all & € A. (Robert Gompf, Andas Stipsicz 22)
- An element a € A is primitive if we cannot write a as df (8 € A, d € Z) unless d = +1. (Robert Gompf, Andas Stipsicz 22)
- (d) Q is called unimodular if det @@ = £1. (Robert Gompf, Andas Stipsicz 22)
- For an elcment x € A define L, € A* by L(y} = Q(z, y). In this way we get a homomorphism L: A — A*. Lemma 1.2.9. The form Q is unimodular iff L is an isomorphism. (Robert Gompf, Andas Stipsicz 22)
- Remark 1.2.11. Exercise 1.2.10 can be extended to show that QQx is unimodular when 9X is a homology sphere. (A 3-manifold M is a homology sphere iff H,(M;Z) = H,.(5%Z), or equivalently, iff it is closed, orientable and connected with H1(M;Z) = 0.) (Robert Gompf, Andas Stipsicz 23)
- In fact, for a compact, oriented 4-manifold X with H1(X;Z) = 0, Qx is unimodular if and only if 2X is a disjoint union of homology spheres. (Robert Gompf, Andas Stipsicz 23)
- Lemma 1.2.12. Suppose that the restriction of the symmetric bilinear form Q to the subgroup A; C A is unimodular. Then (A, Q) can be split as the sum offorms (A, Q) = (A1, Q|A1) ® (AL, Q|AL), where Af ={y € A| Q(z,¥) = 0 for all x € A1}. Moreover, Q|As is unimodular iff Q is. (Robert Gompf, Andas Stipsicz 23)
- y 1.2.13. Suppose that dim A is equal to n and the determinant of the matriz (Robert Gompf, Andas Stipsicz 23)
- Corollary 1.2.13. Suppose that dim A is equal to n and the determinant of the matriz of Q on the set {ai,...,an} is £1. Then {a1,... ,a,} is a basis of the free group A. (Robert Gompf, Andas Stipsicz 23)
- At the same time, indefinite forms admit a very nice classification scheme. Theorem 1.2.14. If indefinite unimodular forms Q1, Q2 {defined on A1, Ag respe (Robert Gompf, Andas Stipsicz 23)
- pectively) have the same rank, signature and parity, then they are equivalent. (Robert Gompf, Andas Stipsicz 23)
- Theorem 1.2.15. If A # 0 and o(Q) = 0, then there exists a nonzero a € A with Q{a,a) =0. (Robert Gompf, Andas Stipsicz 23)
- Using the above result, we can easily classify intersection forms with signature 0. Lemma 1.2.16. If o(Q) = 0, then @Q is equivalent to kH if is even, and to 11) ® I{—1) if Q is odd (k,l € N). (Robert Gompf, Andas Stipsicz 24)
- Lemma 1.2.20. [If x € A is characteristic, then Q{z,z} = o(Q)} (mod8); in particular, if Q is even, then the signature o{Q) is divisible by 8. (Robert Gompf, Andas Stipsicz 25)
- Consider the matrix corresponding to the Dynkin diagram of the exceptional Lie algebra Fj (Robert Gompf, Andas Stipsicz 25)
- As the matrix of a bilinear form Q on Z8, Ej gives a positive definite, even, unimodular form with ¢{@Q) = 8. (Check these statements by diagonalizing Eg over Q; beware that the determinant is not an invariant over Q.) (Robert Gompf, Andas Stipsicz 25)
- Theorem 1.2.25. (Whitehead) The simply connected, closed, topological 4{-manifolds X, and Xy are homotopy equivalent iff Qx, = Qx,. (Robert Gompf, Andas Stipsicz 26)
- one can show that a simply connected topological 4-manifold X is homotopy equivalent to a CW-complex of the form VE, 5% uy, D* (Robert Gompf, Andas Stipsicz 26)
- g is the gluing map 5% — V¥_, 5? of the 4-cell, hence defines a class [g] € m3(; S7). (Robert Gompf, Andas Stipsicz 26)
- Define the matrix L([g]) = [Aii(9)] as Xij{g) = €k(g~ (zi), 97 (z;)) {and Xi(g) = k(g~ (xs), g~ (xh) for 2} close to z;), where ¢k(L1, Ls) denotes the linking number of the two oriented links L; and Lz. (Robert Gompf, Andas Stipsicz 27)
- It is not hard to see that L([g])} represents (x in an appropriate basis, hence if Qx, = Qx,, the gluing maps g; and go corresponding to X; and X5 are homotopic. (Robert Gompf, Andas Stipsicz 27)
- For a fixed homology element [X] € Hy(X; Z) the pair (X, [X]) is called a {(4-dimensional oriented) Poincaré duality space if the map a — aN [X] (with a € H*(X;Z)) defines an isomorphism H*(X;Z) — Hy;(X;Z) for i=0,...,4. (It can be shown that any oriented topological 4-manifold is homotopy equivalent to a finite CW complex, hence, to a Poincaré duality space.) (Robert Gompf, Andas Stipsicz 27)
- Theorem 1.2.27. (Freedman, [F|, [FQ]) For every unimodular symmetric bilinear form J there exists a simply connected, closed, topological f-manfold X such that Qx = Q. If Q is even, this manifold is unique (up to homeomorphism). If Q is odd, there are exactly two different homeomorphism types of manifolds with the given intersection form. At most one of these homeomorphism types carries a smooth structure. Consequently, simply connected, smooth J-manifolds are determined up to homeomorphism by their intersection forms. (Robert Gompf, Andas Stipsicz 27)
- Theorem 1.2.26. Two simply connected ({-dimensional) Poincaré duality spaces X; and Xo are homotopy equivalent iff Qx, = Qx,. Moreover, for each unimodular form Q there exists a (4-dimensional) Poincaré duality space X with Qx = Q. (Robert Gompf, Andas Stipsicz 27)
- The following theorem — due to M. Freedman — can be regarded as the topological strengthening of the above homotopy theoretic classification (Robert Gompf, Andas Stipsicz 27)
- Corollary 1.2.28. If a topological 4-manifold X is homotopy equivalent to 52, then X is homeomorphic to the J-sphere. (Robert Gompf, Andas Stipsicz 28)
- Regarding smooth structures, the two main questions (existence and uniqueness) can now be formulated as follows. oe Q1. Existence: Which simply connected topological manifolds (or equivalently, intersection forms) carry smooth structures? o (2. Uniqueness: If the intersection form ¢Q does carry a smooth structure, how many nondiffeomorphic smooth manifolds can be found with the same intersection form Q? (Robert Gompf, Andas Stipsicz 28)
- Theorem 1.2.29. (Rohlin, [R2]) If Qx is even, then the signature o(X) is divisible by 16. 0 This theorem tells us, for example, that the topological manifold corresponding to Eg does not carry any smooth structur (Robert Gompf, Andas Stipsicz 28)
- ure. Another constraint on the intersection form of a simply connected smooth 4-manifold was found by Donaldson, cf. also Corollary 2.4.29. Theorem 1.2.30. (Donaldson, [D1]) If the intersection form Qx of a smooth, simply connected, closed 4-manifold X is negative definite, then Qx is equivalent to n{—1). (Robert Gompf, Andas Stipsicz 28)
- Theorem 1.2.31. (Furuta, [Fur]} If X is a simply connected, closed, oriented, smooth 4-manifold and Qx is equivalent to 2kEs © IH, then we have [> 2k] + 1. 0 The 1 -Conjecture states that in the above theorem 1! > 3|k| should be the right answer — this conjecture, however, is still open. (Robert Gompf, Andas Stipsicz 28)
- The next result indicates how much we know about the answer of Q2. As a consequence of Theorem 1.2.27, the homeomorphism type of a smooth, simply connected, oriented, closed 4-manifold X is determined by the parity of @x and the two numerical invariants o(X) and by(X) = rk Hy (XZ). In contrast to Theorem 1.1.8, there is no finitencss result on the number of nondiffeomorphic smooth structures on a topological 4-manifold. (Robert Gompf, Andas Stipsicz 29)
- Throughout the last part of this section we always assumed that the 4manifolds we considered were simply connected. This assumption can be relaxed in some cases, but the general case (arbitrary fundamental group) is too difficult to study, since: Theorem 1.2.33. For every finitely presented group G there is a smooth, closed, oriented 4-manifold X with m(X) = G. (Robert Gompf, Andas Stipsicz 29)
- deduce a theorem of Markov {Exercise 5.1.10(c)) that there can be no algorithm for classifying closed 4-manifolds (or n-manifolds for any fixed n > 4). Thus, the difficulty of understanding finitely presented groups leads us to focus mainly on simply connected 4-manifolds (Robert Gompf, Andas Stipsicz 29)
- we will see some applications of the knowledge of the Seiberg-Witten function to the geometry of the underlying 4-manifold. In this way we will distinguish homcomorphic but nondiffcomorphic 4-manifolds (Robert Gompf, Andas Stipsicz 29)
- Using Kirby calculus one can {under favorable circumstances) prove that 4-manifolds defined by different constructions are actually diffeomorphic. (Robert Gompf, Andas Stipsicz 30)
- For our occasional use of characteristic classes, the reader is referred to Section 1.4 for an overview of background material or to [MS] for more details. (Robert Gompf, Andas Stipsicz 30)
- Definition 1.3.2. Let X;, Xs be oriented n-dimensional manifolds, and assume that Z; C 8X; (i = 1,2) are compact, codimension-zero submanifolds of the boundaries. Assume furthermore that ¢: Z; — Z2 is an orientationreversing diffeomorphism. By identifying Z; with Z5 via ¢ (and smoothing the corners) we get a new oriented manifold, denoted by X; Uy, X» (or by X1Uz Xp if Z=2, = Z, and p = idz). (Robert Gompf, Andas Stipsicz 31)
- The cartesian product CP! x CP! provides the next example of a simply connected 4manifold. (Robert Gompf, Andas Stipsicz 31)
- A special case of the construction of Definition 1.3.2 is the boundary sum — when we glue along the (n — 1)-dimensional ball Z; ~ Zp ~ D*~1, The result is denoted by Xi4X2 and is well-defined {independent of the embeddings of D™!} whenever each 8X; is connected. (Robert Gompf, Andas Stipsicz 31)
- Definition 1.3.4. For i = 1,2, let D? C X; be embedded disks, and let w: DF — DZ be an orientation-reversing diffeomorphism. The smooth manifold (X31 —int D1) Uap, (X2 —int D2) is called the connected sum X1#Xo of X; and Xj; (Robert Gompf, Andas Stipsicz 32)
- Note that, by definition, the boundary sum of X; and X; has the connected sum 8X1#3X, as boundary, so X1hXz2) = 8X,#0X, (Robert Gompf, Andas Stipsicz 32)
- The iterated application of the connected sum operation for CP2, CP? and 52 x 5% gives other examples of simply connected 4-manifolds (Robert Gompf, Andas Stipsicz 32)
- show that Qx, #X, = Q@x, SF Px, (Robert Gompf, Andas Stipsicz 32)
- namely that if X is a closed, smooth, simply connected 4manifold and Qx splits as 1 © Ja, then there are X31, Xo ¢ X such that X = X, Un X; giving the splitting ¢; © Q» for @x (as Q; = Qx,); moreover N is a (smoothly embedded) homology sphere. (Robert Gompf, Andas Stipsicz 32)
- Note also that the intersection form of §2 x S?#CP? is isomorphic to the intersection form of CP24#2CP? (cf. Exercise 1.2.17(c)). By Theo rem 1.2.27, this implies that these manifolds are homeomorphic. As we will sce later, 82 x S2#4CP2 is, in fact, diffeomorphic to CP2#2CP2. (Robert Gompf, Andas Stipsicz 32)
- #mX denotes the manifold we get by the connected sum of m (m > 0) copies of the same manifold X. (Robert Gompf, Andas Stipsicz 32)
- Polynomials in the variables {zp,... ,2,} are not well-dcfined functions on CP“, but for a homogeneous polynomial p of degree d, i.c., a polynomial satisfying p(Az) = A%(2) for all A € C and z € C**1, the zero set of p is well-defined. (If p vanishes on a point 2 € C”t! — {0} then it vanishes on its entire equivalence class [2] € CP®.) (Robert Gompf, Andas Stipsicz 32)
- It has becn proved [GH] that any complex projective manifold can be written as the zero set of a collection of homogeneous polynomials. Not every complex manifold, however, can be embedded in CP?, so not all complex manifolds are projective. (Robert Gompf, Andas Stipsicz 33)
- The Implicit Function Theorem shows that S$; C CP? is a smooth 4-manifold; the fact that m;(5g) = 1 follows from the Lefschetz Hyperplane Theorem 1.4.22 (see Exercise 8.1.1(b)). (Robert Gompf, Andas Stipsicz 33)
- For determining @g, we must compute its parity, rank and signature (Robert Gompf, Andas Stipsicz 33)
- Consider the hypersurface Sg={|zp:21:22: 23) € CP? | yz =0} c CP, (Robert Gompf, Andas Stipsicz 33)
- Note that Sy is a complex surface, hence it admits Chern classes (Robert Gompf, Andas Stipsicz 33)
- Recall that the total Chern class of CP3 is ¢(CP®) = (1+ g)* € H*(CP?;Z), where g denotes the generator of H2(CP3;Z) satisfying {g, [CP’]) = 1 [MS]. (Robert Gompf, Andas Stipsicz 33)
- Claim 1.3.11. If p1 and pe are lwo homogeneous polynomials with equal degree (and not powers of other polynomials) and the hypersurfaces F; = {P € CP" | pi(P) = 0} are smooth submanifolds of CP" (i = 1,2), then Fy is diffeomorphic to Fs. (Robert Gompf, Andas Stipsicz 34)
- it is easy to see that S; = CP2. (Robert Gompf, Andas Stipsicz 35)
- we see that S; = CP! x CP!. (Robert Gompf, Andas Stipsicz 35)
- We need additional tools to show that S3 = CP?#6CP? (Robert Gompf, Andas Stipsicz 35)
- The case d = 4 gives an example of a simply connected complex surface with ¢; = 0; such a surface is called a K3-surface. By algebraic geometric methods it can be shown that all K3-surfaces are diffeomorphic (cf. Theorem 3.4.9), so from the differential topological point of view we can call S53 the K3-surface. (Robert Gompf, Andas Stipsicz 35)
- More examples of simply connected 4-manifolds can be given by generalizing the above construction of S;. (Robert Gompf, Andas Stipsicz 35)
- In this appendix we will present the very basics of characteristic classes for a more detailed treatment sce [MS]. (Robert Gompf, Andas Stipsicz 36)
- At the end of the section we will give a quick review of spin structures and Dirac operators (Robert Gompf, Andas Stipsicz 36)
- 1.4.1. Characteristic classes. The set of isomorphism classes of U(1)bundles (O(1)-bundles resp.) over X will be denoted by Lx (Rx resp.). Obviously Lx and Rx admit group structurcs with the tensor product of linc bundles as multiplication. (Robert Gompf, Andas Stipsicz 36)
- Proposition 1.4.1. The groups Lx and H*(X;Z) are canonically isomorphic; similarly, Rx and H(X;Z;) are canonically isomorphic groups. (Robert Gompf, Andas Stipsicz 36)
- From algebraic topology we know that H2(X;Z) = |X, K(Z,2)], (Robert Gompf, Andas Stipsicz 37)
- Bundle theory tells us that Lx and [X, BU(1)] are isomorphic (where BU(1} is the classifying space for U{1)-bundles). An easy argument shows that both K(Z,2) and BU(1} are homotopy equivalent to CP, (Robert Gompf, Andas Stipsicz 37)
- The proof that Rx = H!(X;Z3) rests on the fact that both K{Zz,1) and BO(1) are homotopy equivalent to RP. (Robert Gompf, Andas Stipsicz 37)
- The isomorphism Cx — H2?(X;Z) (with suitably chosen sign) is usually called ¢;3, and ¢1(L) is the first Chern class of the complex line bundle L. Similarly, wi: Rx — H(X; Zs) is the isomorphism given by Proposition 1.4.1, and wy (R) is the first StiefelWhitney class of the real line bundle R — X. (Robert Gompf, Andas Stipsicz 37)
- An alternative obstruction theoretic — description of ¢; (and of un) can be given in the following way. {Compare with Section 5.6.) Suppose that X has a CW-decomposition and IL — X is a U(1)-bundle; note that for each cell f: D* — X, the bundle f*L over D* is canonically trivial. Obviously L is trivial on the O-skeleton of X, and since U{(1) is connected, such a trivialization can be extended over the 1-skeleton. Comparing this trivialization with the canonical trivialization over each 2-cell defines a map @: 0D? — U(1) = S? for every 2-cell D? hence associates a number (the degree of v) to every 2-cell. In this way we define a cochain c. Claim 1.4.2. The cochain c is a cocycle, and the class [¢] € H%*(X;Z) depends only on the bundle L — X (and is independent of the CW-decomposition and the trivialization). (Robert Gompf, Andas Stipsicz 37)
- Note that if [c] = 0, so the trivialization can be changed over the 1-skeleton in such a way that it extends over the 2-gkeleton, then I is trivial. This follows from the fact that when we want to extend the trivialization to higher dimensional cells, we do not find any more obstructions, since all maps 8D% — U(1) = §! are nullhomotopic once k > 2. Theorem 1.4.3. The above-defined class {¢] € H*(X:Z) of L — X coincides with the Chern class ¢1(L) defined by the isomorphism of Proposition 1.4.1. (Robert Gompf, Andas Stipsicz 37)
- Since each line bundle can be regarded as the pull-back of the tautological line bundle + — CP*°, Theorem 1.4.3 has to be proved only for 7. (Robert Gompf, Andas Stipsicz 37)
- Next we outline the definition of the other Chern classes (EF) in H®(X;Z) (i = 2,... ,n) for a complex n-plane bundle E. (Robert Gompf, Andas Stipsicz 38)
- fE=L1®...® Ly, is a sum of complex line bundles L;, take ¢(E) € H*(X;Z) (the total Chern class) to be the cup product e(E) = (1+ c1(L1))U...U(l+e1{Ly,)) € H*(X;Z). (Robert Gompf, Andas Stipsicz 38)
- The component of ¢(E) in H*#(X;Z) is called the i¥* Chern class ¢;(E) of E,so (BE) =1+c1(E) +... + cu(E). Hence ¢;(E) is the value of the i elementary symmetric polynomial of n variables evaluated on ¢;1{L1),... ,c1{Lyn). (Robert Gompf, Andas Stipsicz 38)
- Applying the Leray-Hirsch Theorem [Hu] (Robert Gompf, Andas Stipsicz 38)
- The real analogue of the above process defines Stiefel-Whitney classes w;(F) € H’(X;Zy) of a given R"-bundle F — X. For a real n-plane bundle F — X the Pontrjagin classes p;(F} € H¥(X;Z) can be defined by the formula p;(F) = (—1)c(F ®g C). The next proposition summarizes the most important properties of the Chern, Stiefel-Whitney and Pontrjagin classes. Let E — X denote a C*-bundle and F’ — X an arbitrary R™-bundle. (Robert Gompf, Andas Stipsicz 38)
- {They depend only on the bundles, not on the particular splitting (Robert Gompf, Andas Stipsicz 39)
- hese classes are natural with respect to continuous maps: ¢i(f*E) = FelB)), (Robert Gompf, Andas Stipsicz 39)
- (¢) (Whitney product formula) For direct sums of bundles we have the identities ¢(E ® E’}) = o(E) Uc(E), (Robert Gompf, Andas Stipsicz 39)
- There is one more characteristic class that will be used in our arguments, namely, the Euler class of an oriented real n-plane bundle. (Robert Gompf, Andas Stipsicz 39)
- Let s: X — F; be a generic smooth section of Fi} — X and Z = s71(0) be its zero set. (Robert Gompf, Andas Stipsicz 39)
- The fundamental class of the zero set Z defines a homology class [Z] € Hp—n(X;Z), so its Poincaré dual PD(|Z]} gives rise to an element in H* (XZ). Claim 1.4.7. The class ¢(F1) = PD([Z]) € H™(X;Z) depends only on the bundle F; — X, and by definition this cohomology class is the Euler class of F1 — X. (Robert Gompf, Andas Stipsicz 39)
- Fi —= X is a smooth, oriented R™-bundle. Let s: X — F; be a generic smooth section of Fi} — X and Z = s71(0 (Robert Gompf, Andas Stipsicz 39)
- The following proposition describes the most important relations among characteristic classes. Proposition 1.4.9. If E — X is an n-dimensional complex bundle, then the relations c,(E) = e(E) and ¢;(FE) = wu(E) (mod 2) hold for all i < n; moreover wi+1{E) = 0. For a smooth, closed, oriented n-dimensional manifold X we have e(X) = wa(X) (mod2), while {e(X), [X]) is equal to the Euler characteristic x(X) of X. (Robert Gompf, Andas Stipsicz 40)
- If X is a closed, complex n-dimensional manifold, and {i1,... ,4;} is a partition of n (that is, i; are positive integers and 4; +... + ix = n), then the product ci; (X)U...Ug;, (X) can be evaluated on the fundamental class [X], defining the Chern number corresponding to the partition {¢1,...,4¢}. (Robert Gompf, Andas Stipsicz 40)
- Theorem 1.4.12. (Hirzebruch signature theorem for 4-manifolds) If X is a smooth, closed, oriented {-dimensional manifold, then its signature o{X) is equal to 3 (p(X), (X]). (Robert Gompf, Andas Stipsicz 40)
- The next theorem gives a relation between the signature of X and the first Pontrjagin class of its tangent bundle TX (Robert Gompf, Andas Stipsicz 40)
- Theorem 1.4.13. (Noether formula) For a complex surface S the integer A218] +e2[8] = 3(a(S) + x(X)) is divisible by 12, or equivalently, 1 —b;(S) + b (8S) is even. In particular, if S is a simply connected complex surface then bf (5) is odd. (Robert Gompf, Andas Stipsicz 41)
- However, a C"-structure on the fibers (defining multiplication by # fiberwise) can be defined for a much wider class of manifolds. Such a structure on a manifold is called an almnost-complez structure. (Robert Gompf, Andas Stipsicz 41)
- Definition 1.4.14. An almost-complez siructure on the bundle TX — X is a smooth, fberwise linear map J: TX — TX covering idx such that J? = —idrx. (Robert Gompf, Andas Stipsicz 41)
- An almost-complex structure defines a natural orientation on the smooth manifold X, since the choice of .J reduces the structure group of the tangent bundle to GL(n;C) ¢ GLT(2n:R). (Robert Gompf, Andas Stipsicz 41)
- Once an almost-complex structure is specified, Chern classes ¢; make sense. (Robert Gompf, Andas Stipsicz 41)
- The following theorem provides a neces sary and sufficient condition for the existence of an almost-complex structure on the 4-manifold X. (Robert Gompf, Andas Stipsicz 41)
- Theorem 1.4.15. (Wu {Wu], see also [HH]|) For a given 4-manifold X and almost-complez structure J on X we have c3(X, J) = ¢(X) € HYXZ), a(X,J) = we(X) (mod2) and AX, J] = 30(X) + 2x{(X). Conversely, if for h € H¥{(X;Z) the equation h® = 30(X) + 2x(X) and the congruence h = wa(X) (mod2) hold, then there is an almost-complez structure J on TX with h = cy (X, J). (Robert Gompf, Andas Stipsicz 41)
- (a) Prove that §¢, (52 x §%)#(5% x 5?) and CP?#CP2 do not admit any almost-complex structure. (Robert Gompf, Andas Stipsicz 42)
- More generally, prove (using Theorem 1.4.15) that a simply connected, smooth, closed 4-manifold X admits an almost-complex structure iff 53 (X) is odd. (Robert Gompf, Andas Stipsicz 42)
- For a € H2(X;Z;) one has {uo(X),a} = Qx(a, a). This equation is called the Wu formula (Robert Gompf, Andas Stipsicz 42)
- There exists a manifold X (e.g., the Enriques surface, cf. Section 3.4) with nontrivial wo(X), @x = (—Es) @ H and 71 (X) = Zy. (Robert Gompf, Andas Stipsicz 43)
- the first Chern class of an almost-complex structure is a characteristic element, as we needed when proving Theorem 1.4.13. (Robert Gompf, Andas Stipsicz 43)
- To demonstrate how useful characteristic classes are, we describe the classification of U/(2), SU(2), SO(3) and SO(4)-bundles over a 4-manifold X. (Robert Gompf, Andas Stipsicz 43)
- Two U(2)-bundles Ej and Ey on X are isomorphic iff c1(Ey) = c1(Es) and c2(F1) = c2(Ez). (Robert Gompf, Andas Stipsicz 43)
- a U(2)-bundle E can be reduced to an SU(2)-bundle iff e1(E) = 0. (Robert Gompf, Andas Stipsicz 43)
- Consequently, two SU(2}-bundles E, and Fy are isomorphic iff c2(Er) = c2(Ea). (Robert Gompf, Andas Stipsicz 43)
- Two SO(4)-bundles Fy and Fy are isomorphic iff wo(F) = we(Fy), 71 (F1) = p1(F%) and e(F) = e(Fy). (Robert Gompf, Andas Stipsicz 43)
- Two SO(3)-bundles F1,Fy — X are isomorphic iff wa(F1) = wa(F2) and p1(F1) = pi(F). (Robert Gompf, Andas Stipsicz 43)
- In the above theorem, the map P: H%(X; Zs) — H*(X;Z4) denotes the Pontrjagin square. (Robert Gompf, Andas Stipsicz 43)
- We close this subsection by quoting the Lefschetz Hyperplane Theorem used in various places in the text. (The proof appears in [M2], see also Exercise 11.2.3(b).) Theorem 1.4.22. (Lefschetz Hyperplane Theorem) Let X be a compact, complex n-dimensional submanifold of CPY. If H is a hyperplane in CPV, then the homomorphisms mi(X NH} — m(X) and Hi(X NH) — H;(X) are isomorphisms for i <n — 1 and surjections fori =n — 1. (Robert Gompf, Andas Stipsicz 44)
- For a given real n-plane bundle FF — X one can always reduce the structure group GL(n;R) to O(n) by introducing a Riemannian metric on F. (Robert Gompf, Andas Stipsicz 44)
- The Lie group O(n) is not connected, however, and the possibility of a further reduction of the structure group to SO(n) (a connected component of O(n)) depends on a characteristic class of F’. (Robert Gompf, Andas Stipsicz 44)
- Lemma 1.4.23. The bundle F — X is orientable iff wy (F) € HY(X;Z,) vanishes; the orientations are parametrized by H°(X;Zy) (which is isomorphic to Zo if X is connecled). (Robert Gompf, Andas Stipsicz 44)
- Although SO(n) is connected,it is not a simply connected group; for n => 3 we have m1 (SO(n) = Z,. (Robert Gompf, Andas Stipsicz 44)
- The universal (double) cover of SO(n) is the spin group Spin(n). (Robert Gompf, Andas Stipsicz 44)
- . Let F — X be an oriented Riemannian (i.e., §0(n)-) bundle; the corresponding principal frame bundle will be denoted by Pgg(ny — X. (Robert Gompf, Andas Stipsicz 44)
- The bundle F — X is spinnable if Pgpny — X can be covered by a principal Spin(n)-bundle Pgp;pn(ny — X such that the double covering Pgpininy — Pgo(n) is the universal cover p: Spin(n) — SO(n) fiberwise, (Robert Gompf, Andas Stipsicz 44)
- Fixing such a cover of Pgpyy — a spin (Robert Gompf, Andas Stipsicz 44)
- structure — realizes F as a spin bundle. A spin structure on F = TX turns X into a spin manifold. (Robert Gompf, Andas Stipsicz 45)
- From spectral sequences we get an exact sequence 0 — H’(X;Z2) — H(Psogny Ze) — HY (SO(n); Zo) ~~ H*(X; Zs), (Robert Gompf, Andas Stipsicz 45)
- Proposition 1.4.25. The SO{n)-bundle 7: F — X is spinnable iff its second StiefelWhitney class wo(F) € H?*(X;Zy) vanishes. (Robert Gompf, Andas Stipsicz 45)
- Show that the double covers of a manifold X are in 1-1 correspondence with elements of H!(X;Z,). (Robert Gompf, Andas Stipsicz 45)
- Use the fact that for any Lie group, my(G) = 0. (Robert Gompf, Andas Stipsicz 45)
- For a 3manifold X* we have w?(X) = wo(X) [MS], so an orientable 3-manifold always has a spin structure. (In fact, if X3 is orientable, then its tangent bundle is trivial, since w2(850(3)) = 0.) (Robert Gompf, Andas Stipsicz 45)
- A simply connected {not necessarily closed} 4-manifold X is spin iff (2x is even (since both statements are equivalent to wo(X) vanishing). If H1{X;Z) has 2-torsion, this equivalence no longer holds. (Robert Gompf, Andas Stipsicz 45)
- Theorem 1.4.28. (Rohlin, [R2]} IfX is a smooth, closed, spin {-manifold, then o(X) = 0 (mod 16). (Robert Gompf, Andas Stipsicz 46)
- The importance of these constructions in dimension four becomes clear once we generalize the notion of spin structures to spin® structures and list the spectacular results based on that theory (Robert Gompf, Andas Stipsicz 46)
- The group Spin(n) can be constructed as a subgroup of a 2“-dimensional real algebra, the Clifford algebra Cl,. By definition Cl, = T(R”®)/I(R“), where T(R™) is the tensor algebra @(R”)®* and I(R“) is the ideal generated by elements of the form v ® v + {v,v}1 € T(R”), v € R®. (Robert Gompf, Andas Stipsicz 46)
- In this way we get two complex 2F-dimensional representations of Spin(n) C Cl,; (Robert Gompf, Andas Stipsicz 46)
- Ifn = 2k+1 is odd, then the complex Clifford algebra Cl,, is isomorphic to the direct sum of two isomorphic matrix algebras: Cl, = Moi. (CY Moi (C) (Robert Gompf, Andas Stipsicz 46)
- If n = 2k is even, then Cl, = Ma (TC); the corresponding complex 2% -dimensional representation Sn of Spin(n) C Cl, splits into two (nonisomorphic) irreducible representations (Robert Gompf, Andas Stipsicz 46)
- these are irreducible and isomorphic (Robert Gompf, Andas Stipsicz 46)
- Check that for the real Clifford algebras we have Cl; = C, Cla = H, Clz = M5(C) and Cly = M(H) (Robert Gompf, Andas Stipsicz 46)
- Using the above complex representation Sp, of Spin(n), one can associate the vector bundle § — X to Pgpin(n) — X. Sections of § — X are the spinors over X. (Robert Gompf, Andas Stipsicz 46)
- There is a canonical object associated to the metric g on X, which is the Levi-Civita connection Vg: T(X; TX) » IX; TX @T*X). (As usual, I’(X; F) denotes the vector space of C®-sections of the vector bundle F’ — X.} This connection can be pulled back to the Spin(n)-bundle Pg, — X, defining a covariant differentiation V: I’(X;S) — I’(X;S ® 7" X) on the associated bundle S§ — X (Robert Gompf, Andas Stipsicz 47)
- Remark 1.4.31. It is a standard fact of differential geometry (cf. [DK]) that a covariant differentiation on the associated bundle F determines a Lie(G)-valued 1-form on the principal G-bundle Pg corresponding to F, and vice versa. (Here Lie{(F) denotes the Lie algebra of the Lie group (G.) (Robert Gompf, Andas Stipsicz 47)
- Hence the Levi-Civita connection V, determines a Lie(SO(n})-valued 1-form on Pgo(n), which can be pulled back to Pgpin(y) (Robert Gompf, Andas Stipsicz 47)
- Since T*X is a subbundle of the Clifford bundle CI(X}, T*X acts on the spinor bundle §. Hence one can define a map (the Clifford multiplication) C:T(X; TX) =TI’(X;9). (Robert Gompf, Andas Stipsicz 47)
- Definition 1.4.32. For a given Riemannian manifold X with a fixed spin structure Pgp,n(,) — X, the composition @=CoV:I(X;S) —-T(X;5) is called the Dirac operator of the spin manifold X. (Robert Gompf, Andas Stipsicz 47)
- Spin(4) = SU(2)xSU(2) (Robert Gompf, Andas Stipsicz 47)
- Sp(1) = SU(2) — SO(3) =~ RP3. (Robert Gompf, Andas Stipsicz 47)
- We define a spin structure for V as a pair of 1-dimensional quaternionic vector spaces V1,V~ with hermitian metrics and a fixed isomorphism ~v: V — Homyg (V+, V™) compatible with the metrics (Robert Gompf, Andas Stipsicz 47)
- Applying the above definition fiberwise, we get an alternative definition of spin structures over a 4-manifold: A spin structure for the 4-dimensional (oriented) Riemannian manifold X* is a pair of SU(2)-bundles $* — X and an isomorphism : TX — Homyg(S*, 5™) compatible with the metrics. (Robert Gompf, Andas Stipsicz 48)
- It is known that n = 4 is the unique dimension in which Lie(SO(n)) splits as a Lie algebra. The splitting of Lie(S0{4)) as Lie(SO(3))® Lie(S0O(3)) will be exploited in the definition of the 4-manifold invariants (Robert Gompf, Andas Stipsicz 48)
- understanding the genus function (7 should lead us to a better understanding of the smooth structure of 4manifolds. The genus function G is defined on Ho (X;Z) as follows: For a € Ho(X;Z), consider G(a) = min{genus(X) | ¥ C X represents a, i.e., [2] = a}, (Robert Gompf, Andas Stipsicz 49)
- where 2: ranges over closed, connected, oriented surfaces smoothly embedded in the 4-manifold X. (Robert Gompf, Andas Stipsicz 49)
- The first section of this chapter is devoted to the description of G for CP2. Later on we will show certain techniques (removing singularities and the blow-up process} which give partial information about G for other manifolds as well. (Robert Gompf, Andas Stipsicz 49)
- In the following, we will present a method for constructing smooth surfaces representing various homology classes, by starting with polynomials defining singular subsets of CP? and then “resolving” the singular points. (Robert Gompf, Andas Stipsicz 50)
- surfaces ¥; and %y intersect each other transversally in P € CP?. Although £; U E, is not a smooth surface (at P it fails to be a manifold), it still defines a homology class, which is equal to [Zi] + [Z2] € H2(CP% Z). (Robert Gompf, Andas Stipsicz 50)
- the union ¥; UY; is modeled {up to reversing the ambient orientation) on F = {(2;, 22) € C2 | z120 = 0, 21]? + | 222 < 1}: two 2-dimensional disks intersecting each other in one point in the 4-ball. (Robert Gompf, Andas Stipsicz 50)
- We cut out the pair (D, F) and replace it with (D, R), where R C D is obtained by perturbing R={(zn,2)€C®lan=¢ af +|2f <1} 0<lg <1) to achieve that 8F = OR C 8D. (Robert Gompf, Andas Stipsicz 50)
- By replacing (D, F} with (D, R) we eliminate the singular point P, but we do not change either the ambient manifold CP? (since we cut out D and then glue it back) or the homology class of 3; UX, (since the subsets F and R are homologous in (D,0D}) (Robert Gompf, Andas Stipsicz 50)
- Remark 2.1.2. Note that 8F C 8D = 8° is the Hopf link (see Figure 2.1 and Section 4.6) and R is a Seifert surface on the Hopf link (Robert Gompf, Andas Stipsicz 50)
- the surface R is the optimal choice because it is the unique minimal genus Seifert surface for the Hopf link. (Robert Gompf, Andas Stipsicz 50)
- In a 4-ball neighborhood D of the intersection point P, the union ¥; UY; is (Robert Gompf, Andas Stipsicz 50)
- The polynomial defining the d distinct lines in the above exercise is a product of d linear polynomials, and so does not satisfy the hypothesis of the Implicit Function Theorem everywhere. (Robert Gompf, Andas Stipsicz 51)
- in this way one changes the surface defined by the polynomial everywhere, but obtains a complex submanifold. The desingularization process described above changes 2; U 22 only in a small neighborhood of P (so it is a local process), but we lose the property that the resulting manifold is a complex submanifold. (Robert Gompf, Andas Stipsicz 51)
- However, the two processes coincide in a small neighborhood of P, and the resulting smooth surfaces are related by a small smooth isotopy of CP2. (Robert Gompf, Andas Stipsicz 51)
- Proposition 2.1.4. The class 3h € Hy(CP? Z) cannot be represented by a smoothly embedded sphere. (Robert Gompf, Andas Stipsicz 51)
- This is because for complex surfaces the evenness of b1(S) is equivalent to the Kahler condition (cf. Theorem 10.1.4), and this latter implies that the manifold is symplectic. (Robert Gompf, Andas Stipsicz 52)
- This conjecture (frequently attributed to Thom) was recently proved using Seiberg-Witten theory: Theorem 2.1.5. ((KM1], [MSzT]|) A smooth surface representing dh € H,(CP?%,Z) (d € N) has genus at least 1(d — 1)(d — 2) (the genus of the smooth complex curve representing the given homology class). Thus, among the smooth surfaces representing dh, the complex submanifolds have minimal genus. (Robert Gompf, Andas Stipsicz 52)
- Theorem 2.1.6 can be extended verbatim to 2-dimensional symplectic submanifolds of symplectic 4-manifolds. (Robert Gompf, Andas Stipsicz 52)
- Using Theorem 2.1.6 and other arguments involving the existence of orientation-reversing diffeomorphisms, one can determine & for $2 x 5% and CP?#CP? (Robert Gompf, Andas Stipsicz 53)
- The examples above are very special 4-manifolds {CP?, §% x $? and CP24CP?). This is the rcason we have so much information about the genus function GG in these cases. (Robert Gompf, Andas Stipsicz 53)
- This fibration my: 7 — CP! is called the tautological bundle over CP! (and can be generalized with a similar formula to any CP? or RP"). (Robert Gompf, Andas Stipsicz 53)
- Taker = {{I,p) e CP x C*? |pel} = {([u: v),(z,y)) €e CPI x C? | xv = yu} C CPL x CZ. The space 7 can be projected to the first or to the second factor of CP! x C2. It is easy to prove that the projection m1: 7 — CP! gives a complex line bundle structure to 7. (Trivialize 7 over CP! — [0 : 1] and over CP! — [1 : 0].) (Robert Gompf, Andas Stipsicz 53)
- Removing UU and replacing it with nyLy) C 7, we get a new complex manifold §’. (Robert Gompf, Andas Stipsicz 55)
- Definition 2.2.6. The surface §’ is called the blow-up of § at P. (Robert Gompf, Andas Stipsicz 55)
- Informally, when we blow up S at P, we replace the point P with the space of all lines going through P, which is a copy of CP. (Robert Gompf, Andas Stipsicz 55)
- Definition2.2.7. For a smooth, oriented manifold X, theconnected sum X’ = X#CP? is called the blow-up of X. (Robert Gompf, Andas Stipsicz 55)
- In algebraic geometry, a complex surface is called minimal if it does not contain any rational —1-curve, so it is not the blow-up of another complex surfa (Robert Gompf, Andas Stipsicz 58)
- arting from S and blowing down the rational —1-curves, we get a new surface S,,,,, called a minimal model of §. (Robert Gompf, Andas Stipsicz 58)
- We close this chapter by introducing the SeibergWitten invariants of a smooth, closed, oriented, simply connected 4-manifold X. (Robert Gompf, Andas Stipsicz 63)
- Let X be a smooth, closed, oriented, simply connected 4-manifold with bF(X) > land odd. Let Cx = {K € H%(X;2Z) | K = we(X) (mod 2}} be the set of characteristic elements; recall that for KX € Cx and a € Hy(X;Z) this means {K, a} = o? = Qx (a, a) (mod (Robert Gompf, Andas Stipsicz 63)
- Definition 2.4.2. The SeibergWitten invariant SWyx : Cx — Z of the simply connected, smooth 4-manifold X (with bf (X) > 1 and odd) on X € Cx (Robert Gompf, Andas Stipsicz 63)
- is defined as SWx(K) = (u™, [M5 (9)]), where dim MS (g) = 2m. If dim M4, (g) < 0 then SW(K) = 0 by definition (Robert Gompf, Andas Stipsicz 64)
- Theorem 2.4.3. The Seiberg-Witten function SW: Cx — Z is a diffeomorphism invariant of the smooth {-manifold X, i.e., SWyx does not depend on the chosen metric g or perturbation 6. For an orienlation-preserving difJeomorphism f: X — X’ we have SW(K) = £SWx(f*K). (Robert Gompf, Andas Stipsicz 64)
- Recall that a 2-form w is a symplectic form on X if it is nondegenerate (w Aw > 0) and dw = 0. (Robert Gompf, Andas Stipsicz 65)
- Every symplectic manifold admits an almostcomplex structure, hence simply connected symplectic manifolds have odd by. (Robert Gompf, Andas Stipsicz 65)
- Theorem 2.4.8. (Generalized adjunction formula) Assume that £ C X is an embedded, oriented, connected surface of genus g(X) with self-intersection [Z]? > 0 (and |Z] £ 0). Then for every SeibergWitten basic class K € Basx we have 2g(X) — 2 > [EP + |K{[ED)]. If X is of simple type and g(X) > 0, then the same inequality holds for © C X with arbitrary square [Z]%. O (Robert Gompf, Andas Stipsicz 65)
- The blow-up formula has been proved for every 4-manifold ( (Robert Gompf, Andas Stipsicz 66)
- Note, for example, that CP? admits both a complex structure and a metric with positive scalar curvature — s (Robert Gompf, Andas Stipsicz 66)
- The group S! x §O(4) admits three different nontrivial double covers. (This follows from the fact that H(S! x SO(4); Zs) = Zy & Za, cf. also Exercise 1.4.26.) It is easy to see that these double covers are S x Spin(4), St x SO(4) (where S! — St is a double cover) and Spin©(4). Note that since S1 2 SO(2), the group S! x SO{4) can be embedded in SO(6). (Robert Gompf, Andas Stipsicz 68)
- One can identify the space of 2-forms A%2(R*) on R* with the Lie algebra Lie(SO(4)) = Lie(SU(2) x SU(2)) (or more generally A2(R"™) with Lie(SO(n))) by the following definition: (Robert Gompf, Andas Stipsicz 71)

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: 11/26/2020

- 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: 03/17/2019

- Full expression of Taylor Expansion (Patrick Fitzpatrick 219)
- The remainder theorem for Taylor polynomials (Patrick Fitzpatrick 222)
- Integral expression for the Taylor remainder (Patrick Fitzpatrick 235)
- Extended Binomial formula (Patrick Fitzpatrick 236)
- We can make the denominator as small as we want (Patrick Fitzpatrick 265)
- Essentially the last theorem, just replacing with Cauchy condition (Patrick Fitzpatrick 271)

- Definition Let I be a neighborhood of the point x 0 • Two functions f: I –+ lR and g: I –+ lR are said to have contact of order 0 at x 0 provided that f(x 0 ) = g(x0 ). (Patrick Fitzpatrick 218)
- For a natural number n, the functions f and g are said to ha (Patrick Fitzpatrick 218)
- ve contact of order n at x 0 provided that f : I –+ lR and g : I –+ lR have n derivatives and for 0::::: k::::: n. (Patrick Fitzpatrick 218)
- What is really surprising is that frequently it happens that lim [f(x)Pn(x)] = 0, n-+oo even when the point x is far away from x 0 • As we will show in Section 8.6, it can also happen that the Taylor polynomials for certain functions do not provide good approx1 imations at any point x other than x 0 , no matter how large the index n (Patrick Fitzpatrick 221)
- The sequence {( -1 )n} shows that, in general, it is not true that any bounded sequence converges (Patrick Fitzpatrick 247)
- Definition A sequence of numbers {an} is said to be a Cauchy sequence (Patrick Fitzpatrick 247)
- We will prove that a sequence of numbers converges if and only if it is a Cauchy sequence. (Patrick Fitzpatrick 247)
- Proposition 9.5 Suppose that the series 2:::: an converges. Then lim an = 0. (Patrick Fitzpatrick 250)
- As we have already seen in Chapter 2, the Harmonic Series 00 1 00 1 I:-n n n=1 does not converge despite the fact that limn~oo 1/n = 0. (Patrick Fitzpatrick 250)
- We have two principal general criteria for a sequence of numbers to converge, namely, the Monotone Convergence Theorem and the Cauchy Convergence Criterion. (Patrick Fitzpatrick 251)
- Theorem 9.7 Suppose that {ak} is a sequence of nonnegative numbers. Then the series 2:::~ 1 ak converges if and only if the sequence of partial sums is bounded; that is, there is a positive number M such that for every index n. (Patrick Fitzpatrick 251)
- The Monotone Convergence Theorem asserts that the sequence of partial sums converges if and only if the sequence of partial sums is bounded. (Patrick Fitzpatrick 251)
- Corollary 9.8 The Comparison Test Suppose that {ad and {bd are sequences of numbers such that for index k, i. The series 2:: 1 ak converges if the series 2:: bk converges. ii. The series 2:: bk diverges if the series 2:: ak diverges (Patrick Fitzpatrick 252)
- Then the series 2:~ 1 ak is convergent if and only if the sequence of integrals {j n f (x) dx} is bounded. (Patrick Fitzpatrick 252)
- f : [ 1, oo) ~ IR is continuous and monotonically (Patrick Fitzpatrick 252)
- Since the function f is continuous, its restriction to each bounded interval is integrable (Patrick Fitzpatrick 253)
- inequalities imply that the sequence of partial sums for the series 2:~ 1 ak is bounded if and only if the sequence {ft f (x) dx} is bounded. (Patrick Fitzpatrick 253)
- Therefore, in view of Theorem 9. 7, it follows that the series 2:~ 1 ak is convergent if and only if the sequence {j1n f (x) dx} is bounded. (Patrick Fitzpatrick 253)
- Corollary 9.13 The pTest For a positive numb er p, the series 00 1 LkP converges if and only if p > 1 (Patrick Fitzpatrick 254)
- When the terms of a series fail to be of orie sign, it is not possible to directly invoke the Monotone Convergence Theorem (Patrick Fitzpatrick 254)
- ically deTheorem 9.15 The Alternating Series Test Suppose that {ak} is a monoton creasing sequenc e of nonnega tive numbers that converges to 0. Then the series converges. (Patrick Fitzpatrick 255)
- By the Monoton e Converg ence Theorem , the sequence {s2n} converges (Patrick Fitzpatrick 255)
- show first that the subseque nce {s2n} converge s. (Patrick Fitzpatrick 255)
- For series whose terms are neither of one sign nor alternating in sign, it is natural to apply the Cauchy Convergence Criterion for Sequences to the sequence of partial sums (Patrick Fitzpatrick 256)
- It is sometimes useful, particularly when considering series, to restate the definition of a Cauchy sequence as follows: A sequence {sn} is a Cauchy sequence provided that for each positive number E there is an index N such that for each index n ::=::: N and any natural number k, (Patrick Fitzpatrick 256)
- Theorem 9.17 The Cauchy Convergence Criterion for Series The series 2:~ 1 ak converges if and only if for each positive number E there is an index N such that for all indices n ::=::: N and all natural numbers k. (Patrick Fitzpatrick 256)
- Corollary 9.18 The Absolute Convergence Test An absolutely convergent series converges; that is, the series 2:~ 1 ak converges if the series 2:~ 1 lak I converges. (Patrick Fitzpatrick 256)
- Theorem 9.20 For the series .Z::::~ 1 ab suppose that there is a number r with 0 =::: r < 1 and an index N such that for all indices n ~ N. (9.7) Then the series .Z::::~ 1 ak is absolutely convergent. (Patrick Fitzpatrick 257)
- Corollary 9.21 The Ratio Test for Series For the series 2:::~ 1 ab suppose that lim ian+! I = f.. n–+oo ian I i. If,£ < 1, the series 2:::: 1 an converges absolutely. ii. If,£ > 1, the series 2:::: 1 an diverges. (Patrick Fitzpatrick 258)
- Definition Given a function f: D -+ IR and a sequence of functions {fn: D -+ JR}, we say that the sequence {fn: D-+ IR} converges pointwise to f: D -+ IR, or that {fn} converges pointwise on D to f, provided that for each point x in D, lim fn(x) = f(x). (Patrick Fitzpatrick 260)
- Observe that this is an example of a sequence of continuous functions that converges pointwise to a discontinuous function. (Patrick Fitzpatrick 260)
- Observe that this is an example of a sequence of functions, each of which is differentiable on IR, that converges pointwise on IR to a function that is not differentiable atx =0. (Patrick Fitzpatrick 261)
- This is an example of a sequence of integrable functions that converges pointwis e on a closed bounded interval to a function that is not integrable (Exercise 5). (Patrick Fitzpatrick 262)
- Thus, the sequence of functions {fn} converges pointwise on the interval [0, 1] to 0 (by this we mean to the function that is identically equal to 0 on [0, 1]). Observe 1 (by th J that 0 0 1 f = 0, while for each index n, J 0 fn = 1. (Patrick Fitzpatrick 262)
- Question A Suppose that each function fn : D –+ lR is continuous. Is the limit function f : D –+ lR also continuous ? Answer: No. Example 9.22 describes a sequence of polynomials that converges pointwise on the interval [0, 1] to a discontinuous function. (Patrick Fitzpatrick 264)
- Question B If D = I is an open interval and each function fn : I –+ lR is differentiable, is the limit function f : I –+ lR also differentiable? If it is, is nction f : I –+ lR also diff . [dfn hm l -(x) = -(x)? df n-+00 dx dx Answer: No. Example 9.23 describes a sequence of exponential functions that converges pointwise on lR to a nondifferentiable function. (Patrick Fitzpatrick 264)
- Question C If D =[a, b] and eachfuncti on fn: [a, b]–+ lR is integrable, is the limit function f: [a, b]–+ lR also integrable? !fit is, is }~ [ [ fn] = [ f? Answer: No. Example 9.24 describes a sequence of step functions that converges pointwise on the interval [0, 1] to a nonintegrable function. Moreover, as Example 9.25 shows, even if the limit function is integrable, it is not necessarily the case that the limit of the integrals equals the integral of the limit. (Patrick Fitzpatrick 264)
- Definition Given a function f: D –+ lR and a sequence of functions {fn: D –+ lR}, the sequence {fn: D –+ lR} is said to converge uniformly to f: D –+ lR, or {fn} is said to converge uniformly on D to f, provided that for each positive number E there is an index N such that lf(x)fn(x)J < E for all indices n ::::: N and all points x in D. (Patrick Fitzpatrick 264)
- Indeed, forE = 1/2 (Patrick Fitzpatrick 265)
- taking x = (3/4) 1/(N+l), (Patrick Fitzpatrick 265)
- Definition The sequence of functions {fn : D —+ IR} is said to be uniformly Cauchy, or {fn} is said to be uniformly Cauchy on D, provided that for each positive number E there is an index N such that lfn+k(x)fn(x)l < E (9.11) for every index n ~ N, every natural number k, and every point x in D. (Patrick Fitzpatrick 266)
- Theorem 9.29 The Weierstrass Uniform Convergence Criterion The sequence of functions {fn: D —+ JR.} converges uniformly to a function f: D —+ IR if and only if the sequence {fn: D —+ JR.} is uniformly Cauchy (Patrick Fitzpatrick 266)
- = lfn+k(x)f(x) + f(x)fn(x)l (Patrick Fitzpatrick 266)
- Theorem 9.31 Suppose that {fn: D —+ JR} is a sequence of continuo us functions that converges uniformly to the function f : D —+ IR.. Then the limit function f : D —+ JR also is continuous. (Patrick Fitzpatrick 268)
- Theorem 9.32 Suppose that {fn: [a, b] —-+ JR} is a sequence of integrable functions that converges uniformly to the function f: [a, b] —-+ JR. Then the limit function f : [a, b] —-+ JR also is integrable. Moreover, nil,~ [[ fn] = [f. (Patrick Fitzpatrick 269)
- The uniform limit of differentiable functions need not be differentiable (Exercise 1). (Patrick Fitzpatrick 271)
- A function f: I -+ JR, defined on an open interval, is called continuously differentiable provided that it is differentiable and its derivative is continuous. (Patrick Fitzpatrick 271)
- Theorem 9.33 Let I be an open interval. Suppose that {fn: I -+ JR} is a sequence of continuously differentiable functions that has the following two properties: i. The sequence {fn} converges pointwise on I to the function f, and ii. The derived sequence {f~} converges uniformly on I to the function g. Then the function f :I -+ lR is continuously differentiable, and f’(x) = g(x) for all x in I. (Patrick Fitzpatrick 271)
- Theorem 9.34 Let I be an open interval. Suppose that {fn: I -+ JR} is a sequence of continuously differentiable functions that has the following two properties: i. The sequence {fn} converges pointwise on I to the function f, and ii. The derived sequence {f~} is uniformly Cauchy on I. (Patrick Fitzpatrick 271)
- Then the function f :I -+ IR. is continuously differentiable, and for each x in I, lim f~(x) = f’(x) . (Patrick Fitzpatrick 272)
- The Weierstrass Uniform Convergence Criterion (Patrick Fitzpatrick 272)
- The principal objective of this section is to show that if the function f : (r, r) -+ JR. is defined by the power series expansion for lxl < r, then f: (-r, r) -+ JR. is differentiable, and moreover, (Patrick Fitzpatrick 275)
- The above computation is known as term-by-term differentiation of a series expansion (Patrick Fitzpatrick 275)
- The series .Z::~o ckxk is said to be convergen t uniformly on the set A provided that the sequence of partial sums {sn} converges uniformly on A to the function f. (Patrick Fitzpatrick 276)
- Assume the following: There is a positive number M and a number a with 0 ~ a < 1 such that ~· for all indices k and all x in A . (9.30) Then the power series .Z::~o ckxk is uniformly convergent on A. (Patrick Fitzpatrick 276)
- However, the Weierstrass Uniform Convergence Criterion asserts that a sequence of functions converges uniformly if and only if the sequence is uniformly Cauchy (Patrick Fitzpatrick 276)
- Moreover, each of the power series and converges uniformly on the interval [ -r, r ] (Patrick Fitzpatrick 277)
- Suppose that the nonzero number x 0 is in the domain of convergence of the power series ’E~o ckxk. Let r be any positive number less than lxo 1. (Patrick Fitzpatrick 277)
- The general result follows by induction since, according to Theorem 9 .40, the derived series of any power series that converges on ( -r, r) is another power series that also converges on ( -r, r). (Patrick Fitzpatrick 279)
- Then the function f: (-r, r) –+ lR has derivatives of all orders (Patrick Fitzpatrick 279)
- Since the series 2:::~ 0 ckxk converges at each point between R and r, according to Theorem 9.40, each of the series and converges uniformly on the interval [R, R]. (Patrick Fitzpatrick 279)
- The above theorem implies that a function defined by a power series expansion on the interval ( -r, r) coincides with its Taylor series expansion about 0; this is a uniqueness result for the coefficients of a power series expansion (Patrick Fitzpatrick 280)
- Weierstrass presented the first example of a continuous function f : IR -+ IR that has the remarkable property that there is no point at which it is differentiable (Patrick Fitzpatrick 283)
- We will prove that the sequence of functions {fn} is uniformly Cauchy on IR. Once this is proven, it follows from the Weierstrass Uniform Convergence Criterion that {fn} converges uniformly on IR. Then, by Theorem 9.31, we can conclude that the limit function f, being the uniform limit of a sequence of continuous functions, is continuous (Patrick Fitzpatrick 283)

Last Annotation: 02/29/2020

- Another version of the Modularity Theorem says that every complex elliptic curve with a rational j-value is the holomorphic homomorphic image of a Jacobian, (Fred Diamond & Jerry Shurman 9)

Last Annotation: 09/25/2019

- Roughly speaking, a representation is a vector space equipped with a linear action of the algebraic structure (Gruson 7)
- If G is a finite group, a representation V of G is a complex vector space V together with a morphism of groups q : G ! GL(V). One says V is irreducible if (1) there exists no non-zero proper subspace W V such that W is stable under all qðgÞ; g 2 G and (2) V 6¼ f0g. (Gruson 7)
- the character of V is the complex-valued function g 2 G 7! TrðqðgÞÞ where Tr is the trace of the endomorphism (Gruson 7)
- These characters form a basis of the complex-valued functions on G that are invariant under conjugation (Gruson 7)
- In both cases, every finite dimensional representation of the group is a direct sum of irreducible representations (we say that the representations are completely reducible). (Gruson 7)
- We will see that any irreducible representation over Q is absolutely irreducible, in other words Q is a splitting field for Sn . (Gruson 116)
- We will realize the irreducible representations of Sn as minimal left ideals in the group algebra Q(Sn ). (Gruson 116)
- Given a Young tableau t(λ), we denote by Pt(λ) the subgroup of Sn preserving the rows of t(λ) and by Qt(λ) the subgroup of permutations preserving the columns. (Gruson 117)
- Definition 3.1. This set of data is called a Hopf algebra if the following property holds: (H): the map m∗ : A → A ⊗ A is a homomorphism of Z-algebras. Moreover, if the antipode axiom is missing, then we call it a bialgebra. (Gruson 126)
- Example 3.4. If M is a Z-module, then the symmetric algebra S • (M ) has a Hopf algebra structure, for the comultiplication m∗ defined by: if ∆ denotes the diagonal map M → M ⊕ M , then m∗ : S • (M ⊕ M ) = S • (M ) ⊗ S • (M ) is the canonical morphism of Z-algebras induced by ∆. (Gruson 127)

Last Annotation: 11/26/2020

- 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: 10/17/2016

- Then the number of walks of length ` in Cn between u and v is given by (Richard Stanley 26)
- determine the probability of being at a given vertex after a given number ` of steps (Richard Stanley 31)
- Muv is just the probability that if one starts at u, then the next step will be to v. (Richard Stanley 31)
- What is the probability p
`that after`

steps one is again at the origin? (Richard Stanley 32) - We say that the graph G is regular of degree d if each du D d , i.e., each vertex is incident to d edges (Richard Stanley 32)
- Deﬁne the access time or hitting time H.u; v/ to be the expected number of steps that a random walk (as deﬁned above) starting at u takes to reach v for the ﬁrst time. (Richard Stanley 33)
- We say that two posets P and Q are isomorphic if there is a bijection (one-to-one and onto function) ‘W P ! Q such that x y in P if and only if’.x/ ’.y/ in Q. Thus one can think that two posets are isomorphic if they differ only in the names of their elements. (Richard Stanley 42)
- A chain C in a poset is a totally ordered subset of P , i.e., if x; y 2 C then either x y or y x in P (Richard Stanley 42)
- We say that a ﬁnite poset is graded of rank n if every maximal chain has length n (Richard Stanley 42)

Last Annotation: 11/06/2020

- We define affine n-space over k, denoted Aj or simply A”, to be the set of all n-tuples of elements of k. (Robin Hartshorne 18)
- 4 = &{x,.....x,| b (Robin Hartshorne 19)
- nterpret the semen of A as functions from the affine n-space to k, by defining f(P) = f(a, ....qa,), where f € A and Pe A". (Robin Hartshorne 19)
- Z(f)= [PeA"f(P) = 0]. (Robin Hartshorne 19)
- IV oY IC ¥ hie aeal ar 4 gane 3 Eg . 3PNY § ¥ He Sy mESE ody ee 41S CIA Of Aa EONCrai DY §, IRCn Li 4) = (Robin Hartshorne 19)
- A 1s a noetherian ring, (Robin Hartshorne 19)
- Z(T) can be expressed as the common zeros of the finite set of polynomials f,,..../,. (Robin Hartshorne 19)
- A SEY ME 1 MEERANC EY INE F, ¥ J NE FA COAT; AT £3 se ¥ ey I Yok TAside SA seadyiva SR I : fier cerry Nod rifoad FE there TET) ES NER e YX oo ‘ W hg Nt TE§ 8 { ee — A nd2X SEY MEERANC INE F ¥ J NE FA AT £3 se ¥ ey I TAside SA seadyiva SR I fier cerry Nod FE TET) ES NER YX ‘ W Nt 8 { — Definition. AUSUUSTD FO A ES GRP ONG Lr her BL LIL C$ ist § a Su SE L008 MM Lr 1 PY — FTO in3 yf : x7 ¥ && 84&8 3 CY > ¥ { fR—— S48 y ¥ && 84&8 3 CY > ¥ SU Cf oii { 4 : mn fR—— S48 Loy 8Fd F y (Robin Hartshorne 19)
- Definition. We define the Zariski topology on A" by taking the open subsets to be the complements of the algebraic sets. (Robin Hartshorne 19)
- Every ideal in A = k[ x] is principal, (Robin Hartshorne 19)
- he open sets are the empty set and the complements of finite subsets. (Robin Hartshorne 19)
- Definition. A nonempty subset Y of a topological space X is irreducible if it cannot be expressed as the union Y = Y, u Y, of two proper subsets, each one of which is closed in Y. (Robin Hartshorne 20)
- k is algebraically closed, hence infinite). (Robin Hartshorne 20)
- ny nonempty open subset of an irreducible space 1s irreducible and dense. (Robin Hartshorne 20)
- Definition. An affine algebraic variety (or simply affine variety) is an irreducible closed subset of A” (with the induced topology). An open subset of an affine variety 1s a quasi-affine variety. (Robin Hartshorne 20)
- So for any subset Y = A”, let us define the ideal of Y in A by I(Y)={feA|f(P) =0forall Pe Y]. (Robin Hartshorne 20)
- Now we have a function Z which maps subsets of A to algebraic sets, and a function I which maps subsets of A” to ideals. (Robin Hartshorne 20)
- va=(feA fe aforsomer > 0]. (Robin Hartshorne 20)
- Theoren:t 1.3A (Hilbert’s Nullstellensatz). Let k be an algebraically closed field, let a be an ideal in A = k[ xy... .. , |. and let fe A be a polynomial which vanishes at all points of Z{a). Then "ea for some integer r > 0. (Robin Hartshorne 21)
- Corollary 1.4. There is «a one-to-one inclusion-reversing correspondence hetween algebraic sets in A" and radical ideals (1.e., ideals which are equal to their own radical) in A, given by Y + I(Y) and a +— Z(a). Furthermore, an algebraic set is irreducible if and only if its ideal is a prime ideal. (Robin Hartshorne 21)
- If Y is irreducible, we show that I(Y) is prime. (Robin Hartshorne 21)
- Let / be an irreducible polynomial in 4 = k[ x.y]. Then f generates a prime ideal in 4, since 4 1s a unique factorization domain. so the zero set Y = Z(f) is irreducible. We call it the affine curve defined by the equation f(x.yv) = 0. (Robin Hartshorne 21)
- More generally, if f 1s an irreducible polynomial im 4 = NET v, |. we obtain an affine variety Y = Z(f), which is called a surface ifn = 3. or a hypersurface ifn > 3. (Robin Hartshorne 21)
- This shows that every maximal ideal of 4 is of the form N= (Ny — dye... «,). for some «ay, ....u, ek. (Robin Hartshorne 21)
- the curve x? + 2 + 1 = 0in Ag has no points. (Robin Hartshorne 21)
- Definition. If Y © A” is an affine algebraic set, we define the uffine coordinate ring ACY) of Y.to be A I(Y). (Robin Hartshorne 21)
- Definition. In a ring 4. the /icight of a prime ideal p 1s the supremum of all integers 1 such that there exists a chain p, cp, =... cp, =p of distinct prime ideals. We define the dimension (or Krull dimension) of 4 to be the supremum of the heights of all prime ideals. (Robin Hartshorne 23)
- Now if X is a scheme of finite type over C, we define the associated complex analytic space X, as follows. Cover X with open affine subsets Y; = Spec A;. Each A; is an algebra of finite type over C, so we can write it as A; = Clxy,....x,)/(f1,....f,). Here fi, ... f arepolynomialsin x, ... x,. We can regard them as holomorphic functions on C”, so that their set of common zeros is a complex analytic subspace (Y;), & C". The scheme X 1s obtained by glueing the open sets Y;, so we can use the same glueing data to glue the analytic spaces (Y;), into an analytic space X,. This is the associated complex analytic space of X. (Robin Hartshorne 456)
- If X 1s a compact complex manifold, then one can show that a scheme X such that X, = X, if it exists, 1s unique. So if such an X exists, we will simply say X is algebraic. (Robin Hartshorne 458)

Last Annotation: 11/29/2017

- (6.3) Theorem. The ideal class group elK = JK I PK is finite. Its order (Jürgen Neukirch 52)
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- This absolute value R is called the regulator of the field K. (Jürgen Neukirch 59)

Last Annotation: 08/04/2020

- Example 0.5. Since RP n is obtained from RPn−1 by attaching an n cell, the infinite union RP∞ can view RP ∞ = S = n RPn becomes a cell complex with one cell in each S ∞ as the space of lines through the origin in R∞ = n R n R (Allen Hatcher 15)
- The existence of a retracting homomorphism ρ : G→H is quite a strong condition on H . If H is a normal subgroup, it implies that G is the direct product of H and the kernel of ρ . If H is not normal, then G is what is called in group theory the semi-direct product of H and the kernel of ρ . (Allen Hatcher 45)
- Lemma 1.19. If ϕt : X →Y is a homotopy and h is the path ϕt (x0 ) formed by the images of a basepoint x0 ∈ X , then the three maps in the diagram at the right satisfy ϕ0∗ = βh ϕ1∗ . (Allen Hatcher 46)
- Φ : Aα Theorem 1.20. If X is the union of path-connected open sets A α each containing the basepoint x0 ∈ X and if each intersection Aα ∩ Aβ is path-connected, then the h homomorphism morphism Φ : ∗α π1 (Aα )→π Aβ ∩ Aγ is path-connected, )→π1 (X) is surjective. If in addition each intersection homomorphism Φ : ∗α π1 (Aα )→π1 (X) is surjective. If in addition each intersection Aα ∩ Aβ ∩ Aγ is path-connected, then the kernel of Φ is the normal subgroup N generated by all elements of the form iαβ (ω)iβα (ω)−1 for ω ∈ π1 (Aα ∩ Aβ ) , and hence Φ induces an isomorphism π1 (X) ≈ ∗α π1 (Aα )/N . Example 1.21: Wedge Sums. In Chapter 0 we defined the (Allen Hatcher 52)
- a free product of nontrivial groups has trivial center. (Allen Hatcher 57)
- (a) If Y is obtained from X by attaching 2 cells as described above, then the inclusion X ֓ Y induces a surjection π1 (X, x0 )→π1 (Y , x0 ) whose kernel is N . Thus π1 (Y ) ≈ π1 (X)/N . (b) If Y is obtained from X by attaching n cells for a fixed n > 2 , then the inclusion X ֓ Y induces an isomorphism π1 (X, x0 ) ≈ π1 (Y , x0 ) . (c) For a path-connected cell complex X the inclusion of the 2 skeleton X 2 ֓ X induces an isomorphism π1 (X 2 , x0 ) ≈ π1 (X, x0 ) . (Allen Hatcher 59)
- First we have the homotopy lifting property, also known as the covering homotopy property: (Allen Hatcher 69)
- Proposition tion 1.30. Given a covering space p p : Xe →X , a homotopy ft : Y →X , and a a unique homotopy fet : Y →X e of fe0 that Propos map fe0 lifts ft . ition 1.30. Given a covering space p : Xe →X , a homoto : Y →Xe lifting f0 , then there exists a unique homotopy f (Allen Hatcher 69)
- Taking Y to be a point gives the path lifting property for a covering space p:Xe →X , which says that for each path f : I →X and each lift x e point f (0) = x there is a unique path fe : I →X 0 e lifting f starting at x e 0 of t x e . 0 (Allen Hatcher 69)
- Proposition f : (Y , y0 )→ y0 )→(X, 1.33. Suppose given a covering space p : ( X, x0 ) with Y path-connected and locally p : (X, e xe 0 )→(X, x0 ) ally path-connected. e 0 ) x thand a map Then a lift (Allen Hatcher 70)
- : (Y , y0 When )→(X, e xe we say a x e 0 ) of f exists iff f∗ π1 (Y , y0 ) a space has a certain property ⊂ p∗ π1 (X, e x e0 locally, such as x e0 ) . as being . ing locally pathconnected, we usually mean that each point has arbitrarily small open neighborhoods with this property. (Allen Hatcher 70)
- Thus for Y to be locally path-connected means that for each point (Allen Hatcher 70)
- Proposition with X and X and X e p For a loop 1.32. The numb e path-connected X op g in X based 2. The number of s ath-connected equals g in X based at x0 , sheets of a covering space p : ls the index of p∗ π1 (X, e xe 0 ) i 0 , let g e x p : (X, e 0 ) in π1 (X, )→(X, x0 ) X, x0 ) . (Allen Hatcher 70)
- Proposition p : (X, e x p : (X, e consists e x (X, e nsists e 0 ) x s of ion 1.31. The map p∗ : π1 (X, e 0 )→ e x )→(X, x0 ) is injective. The image f the homotopy classes of loops in 0 )→π1 (X, x0 ) in age subgroup p∗ in X based at x0 p x 0 induced b p∗ π1 (X, e x0 whose X e s by y a c x e 0 ) lifts a covering space in π1 (X, x0 ) s to X at x e xe 0 )→(X, x0 ) is injective. The image subgroup p∗ π1 (X, e xe 0 ) in ists of the homotopy classes of loops in X based at x0 whose lifts to (Allen Hatcher 70)
- y ∈ Y and each neighborhood U of y there is an open neighborhood V ⊂ U of y that is path-connected. (Allen Hatcher 71)
- Each point x ∈ X has a neighborhood U such that the inclusion-induced map π1 (U, x)→π1 (X, x) is trivial; one says X is semilocally simply-connected (Allen Hatcher 72)
- A necessary condition for X to have a simply-connected covering space (Allen Hatcher 72)
- A locally simply-connected space is certainly semilocally simply-connected. (Allen Hatcher 72)
- CW complexes have the much stronger property of being locally contractible, (Allen Hatcher 72)
- An example of a space that is not semilocally simplyconnected is the shrinking wedge of circles, the subspace X ⊂ R 2 consisting of the circles of radius 1/n centered at the point (1/n , 0) for n = 1, 2, ··· , (Allen Hatcher 72)
- if we take the cone CX = (X × I)/(X × {0}) on the shrinking wedge of circles, this is semilocally simply-connected since it is contractible, but it is not locally simply-connected. (Allen Hatcher 72)
- e is path-connected. In particular, only dentity deck transformation can fix a point of Xe. e →X is called normal if for each x ∈ X and each pair of lifts A covering space p : X e. e →X is called normal if for each x A covering space p : X ′ x, e xe of x there is a deck transformation taking x e to xe ′. F (Allen Hatcher 79)
- e x π1 (X, e 0 ) ⊂ This covering x e 0 ) ove ⊂ π1 (X, x0 ) . Then : ring space is normal iff H is a normal subgroup of π1 (X, x0 ) . (Allen Hatcher 80)
- If X is a space and A is a nonempty closed subspace that is a deformation retract of some neighborhood in X (Allen Hatcher 123)
- e n (X/A) can be represented by a chain α in X with ∂α a cycle in A se homology class is ∂x ∈ He n−1 (A) . Pairs of spaces (X, A) satisfying the hypothesis of the theorem will be called good pairs (Allen Hatcher 123)
- For example, if X is a CW complex and A is a nonempty subcomplex, then (X, A) is a good pair by Proposition A.5 in the Appendix. (Allen Hatcher 123)
- If f : S n →S n has no fixed points then deg f = (−1)n+1 (Allen Hatcher 143)
- If f ≃ g then deg f = deg g since f∗ = g∗ . The converse statement, that f ≃ g if deg f = deg g , is a fundamental theorem of Hopf from around 1925 which we prove in Corollary 4.25. (Allen Hatcher 143)
- deg f = −1 if f is a reflection of S n , fixing the points in a subsphere S n and interchanging the two complementary hemispheres (Allen Hatcher 143)
- Note that the antipodal map has no fixed points (Allen Hatcher 144)
- In the case of S 1 , the map f (z) = z k , where we view S 1 as the unit circle in C , has degree k (Allen Hatcher 146)
- Proposition 2.33. deg Sf = deg f , where Sf : S n+1 →S n+1 is the suspension of the map f : S n →S n . (Allen Hatcher 146)
- A rotation is a homeomorphism so its local degree at any point equals its global degree, (Allen Hatcher 146)
- 0 → → Cn (A ∩ B) —-ϕ ϕ → Cn (A) ⊕ Cn (B) —-ψ ψ → Cn (A + B) → 0 (Allen Hatcher 158)
- if A∩B is path-connected, the H1 terms of the reduced Mayer–Vietoris sequence yield an isomorphism H1 (X) ≈ H1 (A) statement of the van Kampen theorem, ⊕ H1 (B) , and H B) H / Im Φ 1 is the Φ . This is exactly the abelianized he abelianization of π1 for pathconnected spaces (Allen Hatcher 159)
- ϕ(x) = (x, −x) (Allen Hatcher 159)
- ψ(x, y) = x + y . (Allen Hatcher 159)
- The exactness of this short exact sequence can be checked as follows (Allen Hatcher 159)
- . (Allen Hatcher 159)
- ∂ : Hn (X)→Hn−1 (A ∩ B) (Allen Hatcher 159)
- α ∈ Hn (X) is represented by a cycle z (Allen Hatcher 159)
- barycentric subdivision (Allen Hatcher 159)
- choose z to be a sum x +y of chains in A and B (Allen Hatcher 159)
- ∂x = −∂y since ∂(x + y) = 0 , (Allen Hatcher 159)
- X = S n with A and B the northern and southern hemispheres, (Allen Hatcher 159)
- the Klein bottle K as the union of two Möbius bands A and B (Allen Hatcher 159)
- The map Φ is twice around Z→Z ⊕ Z , 1 ֏ (Allen Hatcher 160)
- 2, −2) , since the boundary circle of a Möbius band wraps (Allen Hatcher 160)
- If only one of f and g , say f , is the identity map, then Z is homeomorphic to what is called the mapping torus of g , the quotient space of X × I under the identifications (x, 0) ∼ (g(x), 1) (Allen Hatcher 160)
- C n (X, A; G) = Cn (X; G)/Cn (A; G) (Allen Hatcher 162)
- To each category C there is associated a ∆ complex B C called the classifying space of C , whose n simplices are the strings X0 →X1 → ··· →Xn of morphisms in C . (Allen Hatcher 174)
- A natural transformation from a functor F to a functor G induces a homotopy between the induced maps of classifying spaces. (Allen Hatcher 174)
- By regarding loops as singular 1 cycles (Allen Hatcher 175)
- the Lefschetz P number τ(f ) is defined to be n (−1)n tr f∗ : Hn (X)→Hn (X) . In particular, if f is the identity, or is homotopic to the identity, then τ(f ) is the Euler characteristic (Allen Hatcher 188)
- χ (X) since the trace of the n× n identity matrix is n . (Allen Hatcher 188)
- Theorem 2C.3. If X is a finite simplicial complex, or more generally a retract of a finite simplicial complex, and f : X →X is a map with τ(f ) ≠ 0 , then f has a fixed point. (Allen Hatcher 188)

/home/zack/Dropbox/Library/Serge Lang/Algebra (484)/Algebra - Serge Lang.pdf

Last Annotation: 03/21/2011

Last Annotation: 02/27/2019

- Hochschild and Cyclic Homology (Charles A. Weibel 312)

/home/zack/Dropbox/Library/MoonReader/attachments/Primer on MOD.pdf

Last Annotation: 02/18/2020

- It is defined to be the group of isotopy classes of orientationpreserving diffeomorphisms of S (that restrict to the identity on ∂S if ∂S = ∅): Mod(S) = Diﬀ + (S, ∂S)/ Diﬀ 0 (S, ∂S). (Benson Farb, Dan Margalit 18)
- Here Diﬀ 0 (S, ∂S) is the subgroup of Diﬀ + (S, ∂S) consisting of elements that are isotopic to the identity. (Benson Farb, Dan Margalit 18)
- M(S) = Teich(S)/ Mod(S) is the moduli space of Riemann surfaces homeomorphic to S (Benson Farb, Dan Margalit 18)
- Teich(S) = HypMet(S)/ Diﬀ 0 (S). The space Teich(S) is a metric space homeomorphic to an open ball. The group Diﬀ + (S) acts on HypMet(S) by pullback. This action descends to an action of Mod(S) on Teich(S (Benson Farb, Dan Margalit 18)
- mathematics (Benson Farb, Dan Margalit 18)
- Mod(S) encodes most of the topological features of M(S). (Benson Farb, Dan Margalit 18)
- any closed curve α on a flat torus is homotopic to a geodesic: one simply lifts α to R2 and performs a straight-line homotopy. Note that the corresponding geodesic is not unique. (Benson Farb, Dan Margalit 41)
- Proposition 1.10 Let α and β be two essential simple closed curves in a surface S. Then α is isotopic to β if and only if α is homotopic to β. (Benson Farb, Dan Margalit 51)
- Given an isotopy between two simple closed curves in S, it will often be useful to promote this to an isotopy of S, which we call an ambient isotopy of S. (Benson Farb, Dan Margalit 51)
- in order to prove a topological statement about an arbitrary nonseparating simple closed curve, we can prove it for any specific simple closed curve. (Benson Farb, Dan Margalit 53)
- a homotopy of homeomorphisms can be improved to an isotopy; a homeomorphism of a surface can be promoted to a diffeomorphism; and Homeo0 (S) is contractible, so in particular any isotopy from the identity homeomorphism to itself is homotopic to the constant isotopy. (Benson Farb, Dan Margalit 58)
- T HEOREM 1.12 Let S be any compact surface and let f and g be homotopic homeomorphisms of S. Then f and g are isotopic unless they are one of the two examples described above (on S = D 2 and S = A). In particular, if f and g are orientation-preserving, then they are isotopic. (Benson Farb, Dan Margalit 58)
- It turns out that these two examples are the only examples of homotopic homeomorphisms that are not isotopic (Benson Farb, Dan Margalit 58)
- T HEOREM 1.13 Let S be a compact surface. Then every homeomorphism of S is isotopic to a diffeomorphism of S. (Benson Farb, Dan Margalit 59)
- It is a general fact that any homeomorphism of a smooth manifold can be approximated arbitrarily well by a smooth map. By taking a close enough approximation, the resulting smooth map is homotopic to the original homeomorphism. However, this general fact, which is easy to prove, is much weaker than Theorem 1.13 because the resulting smooth map might not be smoothly invertible; indeed, it might not be invertible at all. (Benson Farb, Dan Margalit 59)
- T HEOREM 1.14 Let S be a compact surface, possibly minus a finite number of points from the interior. Assume that S is not homeomorphic to S 2 , R2 , D 2 , T 2 , the closed annulus, the once-punctured disk, or the oncepunctured plane. Then the space Homeo0 (S) is contractible. (Benson Farb, Dan Margalit 60)
- It answers the fundamental question: how can one prove that a homeomorphism is or is not homotopically trivial? Equivalently, how can one decide when two homeomorphisms are homotopic or not? (Benson Farb, Dan Margalit 61)
- As a general rule, the term “mapping class group” refers to the group of homotopy classes of homeomorphisms of a surface, (Benson Farb, Dan Margalit 62)
- that there are four surfaces for which homotopy is not the same as isotopy: the disk D 2 , the annulus A, the once-punctured sphere S0,1 , and the twice-punctured sphere S0,2 (Benson Farb, Dan Margalit 63)

Last Annotation: 06/09/2019

- The key is that when you partition this way, the intersection of the two subsets is no empty. This fails for n=2. (Miklos Bona 44)
- Useful table (Miklos Bona 67)
- Summary table (Miklos Bona 119)
- Good example of product of generating functions (Miklos Bona 173)
- ODE arising from generating functions (Miklos Bona 184)

- Theorem 1.1. [Pigeon-hole Principle] Let n and k be positive integers, and let n > k. Suppose we have to place n identical balls into k identical boxes, where n > k. Then there will be at least one box in which we place at least two balls. Proof. (Miklos Bona 21)
- E x a m p l e 1.2. There is an element in the sequence 7,77,777,7777, •• • , that is divisible by 2003. Solution. (Miklos Bona 22)
- Theorem 1.4. [Pigeon-hole Principle, general version] Let n,m and r be positive integers so that n > rm. Let us distribute n identical balls into m identical boxes. Then there will be at least one box into which we place at least r + 1 balls (Miklos Bona 23)
- Example 1.3. A chess tournament has n participants, and any two players play one game against each other. Then it is true that in any given point of time, there are two players who have finished the same number of games. Solution (Miklos Bona 23)
- Example 1.5. Ten points are given within a square of unit size. Then there are two of them that are closer to each other than 0.48, and there are three of them that can be covered by a disk of radius 0.5. Solution. (Miklos Bona 24)
- Example 1.6. During the last 1000 years, the reader had an ancestor A such that there was a person P who was an ancestor of both the father and (Miklos Bona 24)
- the mother of A. (Miklos Bona 25)
- Example 1.7. Mr. and Mrs. Smith invited four couples to their home. Some guests were friends of Mr. Smith, and some others were friends of Mrs. Smith. When the guests arrived, people who knew each other beforehand shook hands, those who did not know each other just greeted each other. After all this took place, the observant Mr. Smith said “How interesting. If you disregard me, there are no two people present who shook hands the same number of times”. How many times did Mrs. Smith shake hands? (Miklos Bona 26)
- An obvious reformulation of that Example shows that it is simply impossible to have a party at which no two people shake hands the same number of times (Miklos Bona 29)
- Exercises (Miklos Bona 29)
- If we can complete both of these steps, then we will have proved our statement for all natural values of m. Indeed, suppose not, that is, that we have completed the two steps described above, but still there are some positive integers for which our statement is not true (Miklos Bona 39)
- Example 2.1. For all positive integers n, ,9 „2 9 m(m + l)(2m+1) l 2 + 2 2 + + m 2 = —^ ^ ^—L. (2.1) (Miklos Bona 40)
- Example 2.2. Let f(m) be the maximum number of domains into which m straight lines can divide the plane. Then / ( m ) = "H™ + ’ + 1. (Miklos Bona 41)
- Example 2.3. Let ao = 1, and let a n +i = 3a n + 1, for all positive integers n > 1. Find an explicit formula for am. (Miklos Bona 42)
- discussion: that of the first n positive integers, that is, the set {1,2,3, ••• , n}. As this set will be our canonical example, we introduce the notation [n] = {1,2,3, • • • , n) for this set. (Miklos Bona 43)
- T h e o r e m 2.4. For all positive integers n, the number of all subsets of [n] is2 n . Proof. (Miklos Bona 43)
- One common pitfall is to omit a careful proof of the Initial Step, then “prove” a faulty statement by a correct Induction Step. For example, we could “prove” the faulty statement that all positive integers of the form In +1 are divisible by 2, if (Miklos Bona 43)
- We claim that all horses have the same color. As the number of all horses in the world is certainly finite, we can restate our claim as follows. For any positive integer n, any n horses always have the same color (Miklos Bona 43)
- These two sets have no intersection, so nothing forces the color of the horse in the first set to be the same as that of the horse in the second one! (Miklos Bona 44)
- Example 2.5. Let the sequence {an} be defined by the relations ao = 0, and an+ — a0 + ai + a 2 + • • • + an + n + 1 if n > 1. Prove that for all positive integers n, the equality a„ = 2" 1 holds (Miklos Bona 44)
- Definition 3.1. The arrangement of different objects into a linear order using each object exactly once is called a permutation of these objects. The number n • (n — 1) • (n — 2) • • • 2 • 1 of all permutations of n objects is called n factorial, and is denoted by n!. (Miklos Bona 58)
- You may wonder how large this number is, in terms of n. (Miklos Bona 58)
- n! ~ v ^ (-) . (3.1) The symbol n! ~ z(n) sign means that limn_>.oo jfcx = 1Relation (3.1) is called Stirling’s formula, and we will use it in several later chapters. (Miklos Bona 58)
- The simplicity of the answer to the previous question was due to several factors: we used each of our objects exactly once, the order of the objects mattered, and the objects were all different. In the rest of this section we will study problems without one or more of these simplifying factors. (Miklos Bona 59)
- This problem differs from the previous one in only one aspect: the objects are not all different. The collection of the five red, three yellow, and two white flowers is often called a multiset. A linear order that contains all the elements of a multiset exactly once is called a multiset permutation. (Miklos Bona 59)
- Theorem 3.5. Let n, k,01,02, •• • , a* be non-negative integers satisfying Oi + 02 + • • • + Ofc = n. Consider a multiset of n objects, in which ai objects are of type i, for all i 6 [k]. Then the number of ways to linearly order these objects is n! ai! -a 2 ! a*!’ (Miklos Bona 60)
- Theorem 3.6. The number of k-digit strings one can form over an nelement alphabet is nk. (Miklos Bona 60)
- Definition 3.8. Let X and Y be two finite sets, and let / : X —> Y be a function so that (1) if f(a) = f(b), then a = b, and (2) for all y £ Y there is an x £ X so that f(x) — y, then we say that / is a bijection from X onto Y. Equivalently, / is a bijection if for all y £ Y, there exists a unique x € X so that f(x) = y (Miklos Bona 61)
- Definition 3.9. Let / : X —> Y be a function. If / satisfies criterion (1) of Definition 3.8, then we say that / is one-to-one or injective, or is an injection. If / satisfies criterion (2) of Definition 3.8, then we say that / is onto or surjective, or is a surjection. (Miklos Bona 61)
- Theorem 3.13. Let n and k be positive integers satisfying n > k. Then the number of k-digit strings over an n-element alphabet in which no letter is used more than once is „(„_!)...(n_fc + l)= _ J _ (Miklos Bona 63)
- The number n(n — 1) • • • (n — k + 1) is sometimes denoted {n) (Miklos Bona 63)
- Definition 3.15. The number of fc-element subsets of [n] is denoted (?) and is read “n choose fc”. The numbers (£) are often called binomial coefficients, for reasons that will become clear in Chapter 4. T h e o r e m 3.16. For all non-negative integers k < n, fn _ n! = (n)k fc!(n-fc)! ~ k ’ (Miklos Bona 64)
- E x a m p l e 3.19. A medical student has to work in a hospital for five days in January. However, he is not allowed to work two consecutive days in the hospital. In how many different ways can he choose the five days he will work in the hospital? Solution. The difficulty here is to make sure that we do not choose two consecutive days. This can be assured by the following trick. Let o-i, &i, 03; 0,4,05 be the dates of the five days of January that the student will spend in the hospital, in increasing order. Note that the requirement that there are no two consecutive numbers among the aj, and 1 < a,{ < 31 for all % is equivalent to the requirement that l < a i < a?, — < a^ — 2 a — 3 < a5 — 4 < 27. In other words, there is an obvious bijection between the set of 5-element subsets of [31] containing no two consecutive elements and the set of 5-element subsets of [27]. (Miklos Bona 65)
- Theorem 3.21. The number of k-element multisets whose elements all belong to [n] is n+ k-l (Miklos Bona 66)
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- A high-level survey (using commutative algebra) of results concerning magic squares can be found in “Combinatorics and Commutative Algebra” [35] by Richard Stanley, while a survey intended for undergraduate and starting graduate students is presented in Chapter 9 of “Introduction to Enumerative Combinatorics” [6] by the present author. (Miklos Bona 67)
- (Miklos Bona 68)
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- 14) (Miklos Bona 68)
- 14) (Miklos Bona 68)
- (Miklos Bona 69)
- (23) (Miklos Bona 70)
- 23) (Miklos Bona 70)
- Theorem 4.1. (Binomial theorem) For all non-negative integers n, (x + y)n = it(nf)xkyn~kfc=0 ^ ’ (4-1) (Miklos Bona 85)
- (Miklos Bona 94)
- (4) L (Miklos Bona 94)
- 4) (Miklos Bona 94)
- (5) (Miklos Bona 94)
- (5) (Miklos Bona 94)
- 2 , • • • , a/t) of integers fulfilling a* > 0 for all i, and (a + a% + • • • + a*) = n is called a weak composition of n. If, in addition, the aj are positive for all i € [k], then the sequence (a, 02, • • • , a/t) is called a composition of n. (Miklos Bona 109)
- Definition 5.1. A sequence (ai,a 2 , • • • , a/t) o (Miklos Bona 109)
- Let us assume we want to give away twenty identical balls to four children, Alice, Bob, Charlie and Denise. As the balls are identical, what matters is how many balls each child will get. (Miklos Bona 109)
- Clearly, the order of the integers will matter, that is, 1 + 6 + 8 + 5 does not correspond to the same way of distributing the balls as 6 + 1 + 5 + (Miklos Bona 109)
- Theorem 5.2. For all positive integers n and k, the number of weak compositions of n into k parts is (Miklos Bona 110)
- roof. The problem is certainly equivalent to counting the number of ways of putting n identical balls into k different boxes. (Miklos Bona 110)
- Corollary 5.3. For all positive integers n and k, the number of compositions of n into k parts is (£Zj)- (Miklos Bona 110)
- Place the k boxes in a line, then place the balls in them in some way and align them in the middle of the boxes. This creates a long line consisting of n balls and k — 1 walls separating the k boxes from each other. Note that simply knowing in which order the n identical balls and k — 1 separating walls follow each other is the same as knowing the number of balls in each box. So our task is reduced to finding the number of ways to permute the multiset consisting of n balls and k — 1 walls. T (Miklos Bona 110)
- ber of ways to permute the multiset consisting of n balls and k — 1 walls. Theorem 3.21 tells us that this number is (Miklos Bona 110)
- How about the number of all compositions, that is, the number of compositions of n into any number of parts? Clearly, this question only makes real sense for compositions, not for weak compositions. (Miklos Bona 110)
- Corollary 5.4. For all positive integers n, the number of all compositions ofn is2n~1. (Miklos Bona 110)
- Now assume that the balls are different, but the boxes are not (Miklos Bona 111)
- Definition 5.5. The number of partitions of [n] into k nonempty parts is denoted by S(n,k). The numbers S(n,k) are called the Stirling numbers of the second kind. (Miklos Bona 111)
- The parts of a partition of [n] are also called the blocks of that partit (Miklos Bona 111)
- The parts of a partition of [n] are also called the blocks of that partition. (Miklos Bona 111)
- E x a m p l e 5.6. For all n > 1, we have S(n, 1) = 5(n, n) = 1. For all n > 2, the equality S(n, n — 1) = (£) holds as a partition of [n] into n — 1 blocks must consist of one doubleton and n — 2 singletons. (Miklos Bona 111)
- However, there exists no closed formula for S(n,k). T (Miklos Bona 112)
- There is a formula for S(n, k) that contains one summation sign, and we will prove it in Chapter 7 as we need the sieve formula to obtain i (Miklos Bona 112)
- Theorem 5.8. For all positive integers k <n, S(n, k) = S{n-l,k-l) + kS(n 1, k). (Miklos Bona 112)
- Proof. As before, we can obtain a combinatorial proof by taking a close (Miklos Bona 112)
- look at one particular element, say the maximum element n. (Miklos Bona 112)
- If this element forms a singleton block, then the remaining n— 1 elements have S(n— 1, k — 1) ways to complete the partition. These partitions are enumerated by the first member of the right-hand side. If, on the other hand, the element n does not form a block by itself, then the remaining n — 1 elements must form a partition with k blocks in one of S(n — 1, k) ways. Then we can add n into any of the k blocks formed by this partition, multiplying the number of all our possibilities by k. These partitions are enumerated by the second member of the right-hand side. As the left-hand side enumerates all partitions of [n] into k blocks, the claim is proved. (Miklos Bona 112)
- If we have to put n different balls into k different boxes then the number of ways to do this is k • S(n,k). Indeed, first we can partition [n] into k non-distinguishable parts in S(n, k) ways, then we can label the k parts with labels 1,2, • • • , k in k different ways. (Miklos Bona 112)
- Corollary 5.9. The number of all surjective functions / : [n] —> [k] is k-S(n,k). (Miklos Bona 112)
- Both sides are polynomials of x of degree n. So if we can show that they agree for more than n values of x, we will be done (Miklos Bona 113)
- Corollary 5.10. For all real numbers x, and all non-negative integers n, n z" = ^S(n,fc)(:r) f c . (5. (Miklos Bona 113)
- Definition 5.11. The number of all set partitions of [n] into nonempty parts is denoted by B(n), and is called the nth Bell number. We also set B(0) = 1. So B(n) — Y^i=o S(n’ *)• The Bell numbers also satisfy a nice recurrence relation. Theorem 5.12. For all non-negative integers n, 2?(n+l) = £ ( " W ) . (5 (Miklos Bona 113)
- 5.3 Integer Partitions Now assume that both the balls and the boxes are indistinguishable, so when we distribute the balls into the boxes, the only thing that matter is their numbers. (Miklos Bona 114)
- Definition 5.13. Let ai > a?. > • • • > a,k > 1 be integers so that a± + (Miklos Bona 114)
- Definition 5.13. Let ai > a?. > • • • > a,k > 1 be integers so that a± + a-i + • • • + Ofc = n. Then the sequence (01,02, • • • , Ofc) is called a partition of the integer n. The number of all partitions of n is denoted by p(n). The number of partitions of n into exactly k parts is denoted by Pk(n). (Miklos Bona 114)
- The problem of finding an exact formula for p(n) is even harder than that of finding an exact formula for S(n, k). If we know p(n — 1), or, for that matter, p(i) for all i < n, we still cannot directly compute p(n) (Miklos Bona 114)
- A Ferrers shape of a partition p = (xi, X2, • • • ,Xk) is a set of n square boxes with horizontal and vertical sides so that in the ith row we have X{ boxes (Miklos Bona 115)
- If we reflect a Ferrers shape of a partition p with respect to its main diagonal, we get another shape, representing the conjugate partition of p. (Miklos Bona 115)
- Theorem 5.17. The number of partitions ofn into at most k parts is equal to that of partitions ofn into parts not larger than k. (Miklos Bona 116)
- Theorem 5.18. The number of partitions ofn into distinct odd parts is equal to that of all self-conjugate partitions ofn. (Miklos Bona 116)
- The following Table summarizes our results from this chapter when no empty boxes are allowed. (Miklos Bona 118)
- We would like to avoid the danger of confusion caused by the phenomenon we have just described. Therefore, we will write our permutations in canonical cycle form. That is, each cycle will be written with its largest element first, and the cycles will be written in increasing order of their first elements. (Miklos Bona 131)
- Theorem 6.9. Let a,, a 2 , • • • ,an he nonnegative integers so that the equality 5Z™=1 i-ai = n holds. Then the number of n-permutations with a* cycles of length i where i € [n], is (Miklos Bona 132)
- If an n-permutation p has a, cycles of length i, for i = 1,2, • • • , n, then we say that (ai,a 2 , • • • ,a„) is the type of (Miklos Bona 133)
- Definition 6.11. The number of n-permutations with k cycles is called the (n, k) signless Stirling number of the first kind, and is denoted by c(n,k). The number s(n, k) = (—l) n ~ k c(n, k) is called the (n, k) Stirling number of the first kind. (Miklos Bona 133)
- Theorem 6.12. Let n and k be positive integers satisfying n > k. Then c(n, k) = c(n 1, k 1) + (n l)c(n 1, A;). (Miklos Bona 133)
- The reader is probably wondering whether there is some strong connection between the Stirling numbers of the first kind and the Stirling numbers of the second kind that justifies the similar names. The following Lemma is our main tool in establishing that connection. Lemma 6.13. Let n be a fixed positive integer. Then n J2 c{n, k)xk = x(x + l)—(x + n-l). k=0 (6.3) (Miklos Bona 134)
- In other words, we proved that two polynomials were identical. The only way that can happen is when the coefficients of the corresponding terms agree in the two polynomials (Miklos Bona 134)
- (6.4) fc=0 n ~ k Now the reader can see why we included the term (—\) n ~ k in the definition of s(n, k). Comparing this equation to (5.2), that stated n x n = Y,S(n,k)(x)k, we see that the Stirling numbers of the first kind have the “inverse effect” of the Stirling numbers of the second kind. (Miklos Bona 135)
- Y J s{n,k)x k = {x)n. (Miklos Bona 135)
- Lemma 6.15. [Transition Lemma] Let p : [n] -> [n] be a permutation written in canonical cycle notation. Let g(p) be the permutation obtained from p (Miklos Bona 135)
- A surprising application of Lemma 6.15 is the following. Proposition 6.18. Let i and j be two elements of [n]. Then i and j are in the same cycle in exactly half of all n-permutations. (Miklos Bona 136)
- by omitting the parentheses and reading the entries as a permutation in the one-line notation. Then g is a bijection from the set Sn of all permutations on [n] onto Sn. (Miklos Bona 136)
- In other words, we have to show that there is exactly one way to insert parentheses into the string q — q<i • • • qn so that we get a permutation in canonical cycle form. (Miklos Bona 136)
- Let ODD(m), resp. EVEN{m) be the set of m-permutations with all cycle lengths odd, resp. even. Lemma 6.20. For all positive integers m, the equality (2m) = (2m) holds. (Miklos Bona 137)
- The following surprising result shows that the likelihood that a given entry i is part of a fc-cycle is independent of k. In fact, it is 1/n. Lemma 6.19. Let j 6 [n]. Then for all k € [n], there are exactly (n — 1)! permutations of length n in which the cycle containing i is of length k. (Miklos Bona 137)
- Would you have thought that the number of these permutations is always a perfect square? Theorem 6.24. For all positive integers m, (2m) = (2m) = l 2 • 3 2 • 5 2 • • • (2m l ) 2 . (6.5) (Miklos Bona 138)
- Before proving this quintessential theorem, we would like to stress that the seemingly complicated expression on the right-hand side refers in fact to a conceptually simple sum: the alternating sum of the sizes of all j-fold intersections. (Miklos Bona 153)
- I, in the second example, the sum on the right-hand side was + + nA2 1r 2nA3 + nA2f]A3\. (Miklos Bona 153)
- In a more mathematical formulation: how many permutations of the set [n] have no fixed points, that is, have the element i in the ith position for no i? Such permutations are called derangements. (Miklos Bona 154)
- To that end, let us multiply both sides of (8.1) by xn+1, then sum over all n > 0. (Miklos Bona 166)
- To avoid such a waste of time and energy, it is best to find an explicit formula for an. That is, we would like to deduce a formula for an that does not contain a„_i, or any other elements of the sequence; a formula that depends only on n, and is therefore directly computable. (Miklos Bona 166)
- Definition 8.1. Let {/ n } n >o be a sequence of real numbers. Then the formal power series F(x) = Y^n>o fnxn is called the ordinary generating function of the sequence {fn}n>o- (Miklos Bona 166)
- Let us summarize the technique we have just learned to turn recursive formulae into explicit ones. (1) Define the ordinary generating function G(x) of the sequence {fln}n>o(2) Transform the recursive formula into an equation in G(x). This can usually be done by multiplying both sides of the recursion by xn, or x n+1 , sometimes xn+k, and summing for all non-negative n. (3) Solve for G{x). (4) Find the coefficient of xn in G(x). As this coefficient is an, this will provide an explicit formula for an. (Miklos Bona 169)
- n In other words, the coefficient of x in A{x)B{x) is c„ = J27=oai^n-i- (Miklos Bona 172)
- Theorem 8.5. [The Product formula] Let an be the number of ways to build a certain structure on an n-element set, and let bn be the number of way to build another structure on an n-element set. Let cn be the number of ways to separate n into the intervals S = {1,2, • • • , i} and T — {i +1, i + 2, • • • , n (Miklos Bona 172)
- (the intervals S and T are allowed to be empty), then to build a structure of the first kind on S, and a structure of the second kind on T. Let A(x), B(x), and C(x) be the respective generating functions of the sequences {an}, {&„}, and {cn}. Then A{x)B{x) = C(x). (Miklos Bona 173)
- Example 8.9. If p</t(n) denotes the number of partitions of the integer n into parts of size at most k, then oo k .. n>0 i=l (Miklos Bona 175)
- i=l = (l + a; + a; 2 +a; 3 + –)(l + a; 2 +x 4 +a; 6 + • • • ) • • (Miklos Bona 175)
- Example 8.9. If p</t(n) denotes the number of partitions of the integer n into parts of size at most k, then oo k .. n>0 i=l = (l + a; + a; 2 +a; 3 + –)(l + a; 2 +x 4 +a; 6 + • • • ) • • • (l + xk+x2k+x3k + •••). (Miklos Bona 175)
- e obtain a partition of n into the sum of parts that are at most fc. (Miklos Bona 175)
- x a m p l e 8.10. If p(n) denotes the number of partitions of the integer n, then oo oo 1 Erf*)*"=n IT^T < 8 12 ) (Miklos Bona 176)
- E x a m p l e 8.11. The number p0dd(n) of partitions of n into odd parts is equal to the number Pd{n) of partitions of n into all distinct parts. Solution. The crucial idea is this. It suffices to show that the generating functions of the two sequences are equal. (Miklos Bona 176)
- We proved in Chapter 5 that p<k{n) is also the number of partitions of S a ^ s o t n e n into at most k parts (Miklos Bona 176)
- The following theorem is a major application of compositions of generating functions. T h e o r e m 8.13. Let an be the number of ways to build a certain structure on an n-element set, and let us assume that ao = 0. Let hn be the number of ways to split the set [n] into an unspecified number of disjoint nonempty intervals, then build a structure of the given kind on each of these intervals. Set ho — 1. Denote A{x) = ^2n>Qanxn, and H(x) = J2n>o hnxn• Then l Hlx) = —-. v ; -A{x) (Miklos Bona 177)
- ned totally within the kingdom of integers, will involve powers of a = / (3 + /5)/2, and /3 = (3 v / 5)/2? (Miklos Bona 179)
- Would you have guessed that our answer to this problem, that was defined totally within the kingdom (Miklos Bona 179)
- In Theorem 8.13, we first split [n] into nonempty intervals, then we take a structure of the same kind on each of these intervals. However, we do not take a structure on the set of the intervals. (Miklos Bona 179)
- Theorem 8.15. [The Compositional formula] Let an be the number of ways to build a certain structure on an n-element set, and assume ao = 0. Let b n be the number of ways to build a second structure on an n-element set, and let bo = 1. Let gn be the number of ways to split the set [n] into an unspecified number of nonempty intervals, build a structure of the given kind on each of these intervals, and then build a structure of the second kind on the set of the intervals. Set go = 1. Denote by A(x), B(x), and G(x) the generating functions of the sequences {an}, {bn}, and {<?„}. Then G(x) = B(A(x)). (Miklos Bona 179)
- Not all recurrence relations can be turned into a closed formula by using an ordinary generating function. Sometimes, a closed formula may not exist. Some other times, it could be that we have to use a different kind of generating function. (Miklos Bona 180)
- Definition 8.18. Let {fn}n>o be a sequence of real numbers. Then the 1S c a u e formal power series F(x) = 2 « > o /« 7T 1S c a u e d the exponential generating function of the sequence {/n}n>o- (Miklos Bona 180)
- Note that while Theorems 8.5 and 8.21 sound very similar, they apply in different circumstances. Theorem 8.5 applies when [n] is split into two parts so that one part is [i]. That is, [n] is split into intervals. Theorem 8.21 applies when [n] is split into two parts with no restrictions. In other words, the first theorem applies when our objects are linearly ordered (like days in a calendar, or people in a line), and we cut that linear order somewhere to get two subsets. The second theorem applies when we are free to choose our two subsets, that is, they do not have to be consecutive objects in a line. (Miklos Bona 182)
- A particularly useful property of exponential generating functions is that their derivatives are very easy to describe. Indeed, ( Z+u\) = ^r> and therefore (Miklos Bona 183)
- Example 8.23. Let B(x) be the exponential generating function of the e " _ 1 Bell numbers B(n). Prove that B(x) = e e " _ 1 . (Miklos Bona 184)
- Theorem 8.24. [The Exponential formula] Let an be the number of ways to build a certain structure on an n-element set, and assume ao = 0. Let hn be the number of ways to partition the set [n] into an unspecified number of nonempty subsets, then build a structure of the given kind on each of these subsets. Set ho = 1. Denote by A(x) and H{x) the exponential generating functions of these sequences. Then H{x)=e A{x (Miklos Bona 184)
- Example 8.25. In how many different ways can we arrange n people into groups, and then have each group sit at a circular table? (Miklos Bona 185)
- There are (k — 1)! ways for a /c-member group to sit at a circular table (Miklos Bona 185)
- Example 8.26. Find the exponential generating function F{x) for the sequence {/„} that denotes the number of partitions of [n] into blocks of size 3, 4, and 9 (Miklos Bona 185)
- number of ways an n-element set can form a block of size 3. Obviously, £3 = 1, and tn = 0 if n ^ 3 (Miklos Bona 186)
- Theorem 8.27. [The Compositional formula for Exponential Generating functions] Let an be the number of ways to build a certain structure on an n-element set, and assume 00 = 0. Let bn be the number of ways to build a second structure on an n-element set, and let bo = 1. Let gn be the number of ways to partition the set [n] into an unspecified number of nonempty subsets, then build a structure of the first given kind on each of these subsets, then build a structure of the second kind on the set of the subsets. Denote by A(x), B(x), and G(x) the generating functions of the sequences {an}, {bn}, and {gn}Then G(x) =B(A(x)). (Miklos Bona 186)
- Let gn be the number of ways to partition the set [n] into an unspecified number of nonempty subsets, (Miklos Bona 186)
- Definition 16.1. Let P be a set, and let < be a relation on P so that (a) < is reflexive, that is, x < x, for all x G P, (b) < is transitive, that is, if x < y and y < z, then x < z, (c) < is antisymmetric, that is, if x < y and y < x, then x = y. Then we say that P< = (P, <) is a partially ordered set, or poset. We also say that < is a partial ordering of P. (Miklos Bona 396)
- If x G P is such that there is no y G P for which x < y, then we say that a; is a maximal element of P. If for all z € P, z < x holds, then we say that x is the maximum element of P. Minimal and minimum elements are defined accordingly. The reader should verify that all finite posets have minimal and maximal elements. Not all finite posets have minimum or maximum elements, however. (Miklos Bona 397)
- If x < y in a poset P, but there is no element z € P so that x < z < y, then we say that y covers x. This notion enables us to formally define Hasse diagrams, the kind of diagrams we informally used in our introductory example (Miklos Bona 397)
- The Hasse diagram of a finite poset P is a graph whose vertices represent the elements of the poset. If x < y in P, then the vertex corresponding to y is above that corresponding to x. If, in addition, y covers x, then there is an edge between x and y. (Miklos Bona 397)
- The dual notion is that of antichains. If the subset S C P contains no two comparable elements, then we say that S is an antichain. For example, {{2,3}, {1,3}, {3,4}, {2,4}} forms an antichain in B4 as none of these four sets contains another one. (Miklos Bona 398)
- A chain cover of a poset is a collection of disjoint chains whose union is the poset itself. It seems plausible that if a poset has a large antichain, then it cannot be covered with just a few chains, and vice versa. The following classic theorem shows the precise connection between the sizes of antichains and chain covers of a poset. (Miklos Bona 399)
- Just as for matchings, a chain, (resp. antichain) X of P is called maximum if P has no larger chain (resp. antichain) than X, and X is called maximal if it cannot be extended. That is, no element can be added to X without destroying the chain (resp. antichain) property of X. (Miklos Bona 399)
- Theorem 16.8. [Dilworth’s Theorem] In a finite partially ordered set P, the size of any largest antichain is equal to the number of chains in any smallest chain cover. (Miklos Bona 399)
- If x < y are elements of P, then the set of all elements z satisfying x < z < y is called the closed interval between x and y, and is denoted by [x,y]. If all intervals of P are finite, then P is called locally finite. Note that this does not necessarily mean that P itself is finite. The set of all positive integers with the usual ordering provides a counterexample. (Miklos Bona 401)
- A set of elements I C P is called an ideal if x E I and y < x imply y £ I. If an ideal is generated by one element, that is, I = {y : y < x}, then / is called a principal ideal. (Miklos Bona 401)
- Let Int(P) be the set of all intervals of P. Definition 16.10. Let P be a locally finite poset. Then the incidence algebra I(P) of P is the set of all functions / : Int(P) -> R. (Miklos Bona 401)
- The following element of I{P) is also a simply defined zero-one function. It is surprisingly useful, however. Definition 16.11. Let P be a locally finite poset. Let £ 6 I(P) be denned by C(x, y) = 1 if x < y. Then £ is called the zeta-function of P. (Miklos Bona 402)
- Proposition 16.12. Let x < y be elements of the locally finite poset P. Then the number of multichains x = XQ < x < x-x < • • • < Xk = y is equal to( k (x,y). (Miklos Bona 402)
- A multichain in a poset is a multiset of elements a, a%, • • • ,am satisfying ai < a,2 < • • • < am. Note that the inequalities are not strict, unlike in the definition of chains. (Miklos Bona 402)
- Definition 16.14. The inverse of the zeta function of P is called the Mobius function of P, and is denoted by /i = /ip. (Miklos Bona 403)
- It turns out that the inverse of the zeta function of P is even more important than the zeta function itself. Therefore, it has its own name. (Miklos Bona 403)
- The major application of the Mobius function, the Mobius Inversion Formula, will generalize this idea for any locally finite poset P . Theorem 16.21. [Mobius Inversion Formula] Let P be a poset in which each principal ideal is finite, and let f : P —> R be a function. Let g : P —> R be defined by Then <?(</) = £ / ( * ) • x<y f(v) = ^9(x)fJ,(x,y). x<y (Miklos Bona 407)
- There is a natural class of partial ordered sets called lattices for which additional techniques to compute the values of the Mobius functions are available. Let P be a poset, and let x 6 P. If a; <p a, then we say that a is an upper bound for x. If b <p x, then we say that b is a lower bound for x. (Miklos Bona 408)
- Definition 16.25. A poset L is called a lattice if any two elements x and y of L have a minimum common upper bound a, and a maximum common lower bound b. (Miklos Bona 408)
- In this case, a is called the join of x and y, and b is called the meet of x and y. We denote these relations by x V y = a, and x Ay = b. (Miklos Bona 409)
- A finite lattice always has a minimum and a maximum element, as we show in Exercise 7. This is not necessarily true in infinite lattices. For example, the lattice of all finite subsets of N does not have a maximum element, or even a maximal element, for that matter. (Miklos Bona 409)
- Proposition 16.29. If x, y, and t are elements of a lattice L, and x < t, and y < t, then xVy<t also holds. Similarly, if r € L, and x > r, and y > r, then x Ay > r. (Miklos Bona 410)

Last Annotation: 07/21/2015

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Last Annotation: 12/23/2015

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Last Annotation: 02/09/2019

- Definition of smooth or C^(John Milnor & James D. Stasheff 7)
- Definition of a smooth manifold (John Milnor & James D. Stasheff 8)
- Expression of tangent space as the span of partial derivatives (John Milnor & James D. Stasheff 10)
- The derivative is an endofunctor on the category of smooth manifolds (John Milnor & James D. Stasheff 12)
- Definition of a vector bundle (John Milnor & James D. Stasheff 17)
- Local triviality condition (John Milnor & James D. Stasheff 17)
- Definition: Local coordinate system (John Milnor & James D. Stasheff 17)
- Definition: Isomorphism of vector bundles (John Milnor & James D. Stasheff 18)
- Definition: the normal bundle (John Milnor & James D. Stasheff 19)
- Definition of the canonical line bundle in RP^n (John Milnor & James D. Stasheff 19)
- Theorem: The canonical line bundle is nontrivial for n >= 1 (John Milnor & James D. Stasheff 20)
- Definition: Section of a vector bundle (John Milnor & James D. Stasheff 20)
- Definition: a nowhere zero section (John Milnor & James D. Stasheff 20)
- The canonical line bundle on RP^n does not have a nowhere zero section (John Milnor & James D. Stasheff 20)
- Definition: Independent sections (John Milnor & James D. Stasheff 22)
- An isomorphism of total spaces that isomorphically maps fibers to fibers is an isomorphism of bundles (John Milnor & James D. Stasheff 22)
- Proof that S^3 is parallelizable. (John Milnor & James D. Stasheff 24)
- Definition: Quadratic function (John Milnor & James D. Stasheff 25)
- Deriving an inner product from a quadratic map (John Milnor & James D. Stasheff 25)
- Trivial bundle iff there exist n independent orthonormal sections (John Milnor & James D. Stasheff 26)
- Definition of a map of bundles (John Milnor & James D. Stasheff 30)
- Definition: Whitney Sums (John Milnor & James D. Stasheff 31)
- Definition: subbundle (John Milnor & James D. Stasheff 31)
- Bundles split as the Whitney sum of any subbundle and its perp (John Milnor & James D. Stasheff 32)
- Definition: Immersion (John Milnor & James D. Stasheff 34)
- Big list of ways to combine vector spaces (John Milnor & James D. Stasheff 35)
- Definition of a continuous functor (John Milnor & James D. Stasheff 36)
- Definition: The tensor product of bundles (John Milnor & James D. Stasheff 37)
- Definition: Submersion (John Milnor & James D. Stasheff 39)
- Axioms for Stiefel-Whitney classes (John Milnor & James D. Stasheff 41)
- Definition: The Stiefel-Whitney numbers (John Milnor & James D. Stasheff 54)
- How to compute a Stiefel-Whitney number (John Milnor & James D. Stasheff 55)
- Stiefel-Whitney numbers classify manifolds up to cobordism (John Milnor & James D. Stasheff 57)
- Theorem: The cohomology ring of the infinite Grassmanian in Z/2Z coefficients is generated by the Stiefel-Whitney classes (John Milnor & James D. Stasheff 85)
- Definition: The Thom Isomorphism (Needed to define the Stiefel-Whitney class) (John Milnor & James D. Stasheff 92)
- Definition of the fundamental class (John Milnor & James D. Stasheff 92)
- Definition: Stiefel-Whitney Class (Depends on Thom’s identity) (John Milnor & James D. Stasheff 93)
- Definition: Orientation of a bundle (John Milnor & James D. Stasheff 98)
- Definition: The Euler class of an n-plane bundle (John Milnor & James D. Stasheff 99)

- DEFINITION. A real vector bundle £ over B consists of the following: 1) a topological space E = E(£) called the total space, 2) a (continuous) map 7: E » B called the projection map, and 3) for each b ¢ B the structure of a vector space” over the real numbers in the set 7~1(b). (John Milnor & James D. Stasheff 17)
- These must satisfy the following restriction: Condition of local triviality. For each point b of B there should exist a neighborhood U C B, an integer n> 0, and a homeomorphism h:Ux RP > 71) so that, for each b ¢ U, the correspondence x + h(b, x) defines an isomorphism between the vector space R™ and the vector space 7 1(b). (John Milnor & James D. Stasheff 17)
- Such a pair (U,h) will be called a local coordinate system for & about b. If it is possible to choose U equal to the entire base space, then & will be called a trivial bundle. (John Milnor & James D. Stasheff 17)
- In Steenrod’s terminology an R™-bundle is a fiber bundle with fiber R™ and with the full linear group GL(R) in n variables as structural group. (John Milnor & James D. Stasheff 18)
- Example 2. The tangent bundle ry; of a smooth manifold M. (John Milnor & James D. Stasheff 18)
- If ry is a trivial bundle, then the manifold M is called parallelizable. (John Milnor & James D. Stasheff 19)
- THEOREM 2.1. The bundle Ve over P" is not trivial, for n> 1. (John Milnor & James D. Stasheff 20)
- (A cross-section of the tangent bundle of a smooth manifold M is usually called a vector field on M.) (John Milnor & James D. Stasheff 20)
- THEOREM 2.2. An R™-bundle ¢ is trivial if and only if & admits n cross-sections s,,...,S, n which are nowhere dependent. n (John Milnor & James D. Stasheff 22)
- Hence S° is parallelizable. (John Milnor & James D. Stasheff 24)
- DEFINITION. A Euclidean vector bundle is a real vector bundle ¢& together with a continuous function §: E() > R such that the restriction of pu to each fiber of ¢ is positive definite and quadratic. The function pu itself will be called a Euclidean metric on the vector bundle ¢£. (John Milnor & James D. Stasheff 25)
- In the case of the tangent bundle ry; of a smooth manifold, a Euclidean metric yp: DM -> R (John Milnor & James D. Stasheff 25)
- is called a Riemannian metric, and M together with pu is called a Riemannian manifold. (John Milnor & James D. Stasheff 26)
- ote. In Steenrod’s terminology a Euclidean metric on & gives-rise to a reduction of the structural group of £ from the full linear group to the orthogonal group. (John Milnor & James D. Stasheff 26)
- A priori there appear to be two different concepts of triviality for Euclidean vector bundles; (John Milnor & James D. Stasheff 26)
- Show that the unit sphere S" admits a vector field which is nowhere zero, providing that n is odd. Show that the normal bundle of S$ c R™! is trivial for all n. (John Milnor & James D. Stasheff 27)
- If S" admits a vector field which is nowhere zero, show that the identity map of S" is homotopic to the antipodal map. (John Milnor & James D. Stasheff 27)
- For n even show that the antipodal map of S™ is homotopic to the reflection {CSP Xn41) = (=X, Xy, 000, Xp) (John Milnor & James D. Stasheff 27)
- and therefore has degree —1. (John Milnor & James D. Stasheff 27)
- show that S? is not parallelizable for n even, n> 2. (John Milnor & James D. Stasheff 27)
- More generally a smooth map f:M -» N between smooth manifolds is called an immersion if the Jacobian Df, : DM, -> DN¢(x) maps the tangent space DM, injectively (i.e., with kernel zero) for each x ¢ M. [It follows from the implicit function theorem that an immersion is locally an embedding of M in N, but in the large there may be selfintersections. (John Milnor & James D. Stasheff 34)
- if M C N with normal bundle v, where N is a smooth Riemannian manifold, then the ‘second fundamental form’’ can be defined as a smooth symmetric cross-section of the bundle Hom (ry; ® ney) (John Milnor & James D. Stasheff 39)
- A module is projective if it is a direct summand of a free module. (John Milnor & James D. Stasheff 40)
- THEOREM 4.9 [Pontrjagin]. If B is a smooth compact (n+1)dimensional manifold with boundary equal to M (compare §17), then the Stiefel-Whitney numbers of M are all zero. (John Milnor & James D. Stasheff 56)
- THEOREM 4.10 [Thom]. If all of the Stiefel-Whitney numbers of M are zero, then M can be realized as the boundary of some smooth compact manifold. (John Milnor & James D. Stasheff 57)
- DEFINITION. Two smooth closed n-manifolds M; and M, belong to the same unoriented cobordism class iff their disjoint union M; UM, is the boundary of a smooth compact (n+l)-dimensional manifold. (John Milnor & James D. Stasheff 57)
- DEFINITION. The Grassmann manifold G(Ro+K) is the set of all n-dimensional planes through the origin of the coordinate space ROK, (John Milnor & James D. Stasheff 60)
- An n-frame in R™K is an n-tuple of linearly independent vectors of R™K, (John Milnor & James D. Stasheff 60)
- The collection of all n-frames in R™¥ forms an open subset of the n-fold Cartesian product ROHK X eee X RO+K called the Stiefel manifold v(ROK), (John Milnor & James D. Stasheff 60)
- THEOREM 7.1. The cohomology ring H*G,; 2/2) is a polynomial algebra over 7/2 freely generated by the Stiefel-Whitney classes w, (YD, ery w,(y™). (John Milnor & James D. Stasheff 85)
- PROPERTY 9.7. If the oriented vector bundle ¢ possesses a nowhere zero cross-section, then the Euler class e(£) must be zero. (John Milnor & James D. Stasheff 103)
- LEMMA 14.1. If w is a complex vector bundle, then the underlying real vector bundle wg has a canonical preferred orientation. (John Milnor & James D. Stasheff 154)
- Applying this lemma to the special case of a tangent bundle, it follows that any complex manifold has a canonical preferred orientation. (John Milnor & James D. Stasheff 154)
- every orientation for the tangent bundle of a manifold gives rise to a unique orientation of the manifold. (John Milnor & James D. Stasheff 154)
- But we already know from Section 14 that any characteristic class for complex vector bundles can be expressed as a polynomial in the Chern classes. (John Milnor & James D. Stasheff 294)

/home/zack/Dropbox/Library/Milne/Class Field Theory (690)/Class Field Theory - Milne.pdf

Last Annotation: 05/26/2012

- A more recent version of these notes is available at www.jmilne.org/math/ (Milne 1)

Last Annotation: 09/02/2020

- Proposition 1.1. There is a one-to-one order-preserving, correspondence between the ideals b of A which contain a, and the ideals b of A/a, given by b = cp-l(b). (M. F. Atiyah, I. G. MacDonald 11)
- Proposition 1.2. Let A be a ring ’# o. Then the following are equivalent: i) A is a field; ii) the only ideals in A are 0 and (1); iii) every homomorphism of A into a non-zero ring B is injective. (M. F. Atiyah, I. G. MacDonald 12)
- Proof i) => ii). Let a ‘# 0 be an ideal in A. Then a contains a non-zero element x; x is a unit, hence a ;;2 (x) = (1), hence a = (1). ii) => iii). Let 4>: A –+ B. be a ring homomorphism. Then Ker (4)) is an ideal’# (1) in A, hence Ker (4)) = 0, hence 4> is injective. iii) => i). Let x be an element of A which is not a unit. Then (x) ’# (1), hence B = A/(x) is not the zero ring. Let 4>: A -?B be the natural homomorphism of A onto B, with kernel (x). By hypothesis, 4> is injective, hence (x) = 0, hence x = O. • (M. F. Atiyah, I. G. MacDonald 12)
- Theorem 1.3. Every ring A ’# 0 has at least one maximal ideal. (Remember that “ring” means commutative ring with 1.) (M. F. Atiyah, I. G. MacDonald 12)
- Corollary 1.5. Every non-unit of A is contained in a maximal ideal. (M. F. Atiyah, I. G. MacDonald 13)
- i) Let A be a ring and m :f. (1) an ideal of A such that every x E A m is a unit in A. Then A is a local ring and m its maximal ideal. (M. F. Atiyah, I. G. MacDonald 13)
- 1) A = k[Xb ’ . " x n ], k a field. Let f E A be an irreducible polynomial. By unique factorization, the ideal (f) is prime. (M. F. Atiyah, I. G. MacDonald 13)
- The ideal m of all polynomials in A = k[Xl, ’ , " x n ] with zero constant term is maximal (M. F. Atiyah, I. G. MacDonald 13)
- 2) There exist rings with exactly one maximal ideal, for example fields. A ring A with exactly one maximal ideal m is called a local ring. The field k = A/m is called the residue field of A. (M. F. Atiyah, I. G. MacDonald 13)
- Proposition 1.7. The set 91 of all nilpotent elements in a ring A is an ideal, and A/91 has no nilpotent element ’# O. (M. F. Atiyah, I. G. MacDonald 14)
- Proposition 1.8. The nilradical of A is the intersection of all the prime ideals ofA. (M. F. Atiyah, I. G. MacDonald 14)
- fm E P + (x), rEp + (y) for some m, n. It follows thatfm+n E p + (xy), hence the ideal p + (xy) is not in ~ and therefore xy ¢ p. Hence we have a prime ideal p such thatf ¢ p, so that f¢ 91’. • (M. F. Atiyah, I. G. MacDonald 14)
- The product of two ideals a, 0 in A is the ideal ao generated by all products xy, where x E a and YEO. It is the set of all finite sums 2: XIYI where each XI E a and each YI EO (M. F. Atiyah, I. G. MacDonald 15)
- an (n > 0) is the ideal generated by all products XiX:!· •• Xn in which each factor XI belongs to a. (M. F. Atiyah, I. G. MacDonald 15)
- 2) A = k[Xl, ••. , x n ], a = (Xl, . .. , Xn) = ideal generated by Xl>"" X n• Then am is the set of all polynomials with no terms of degree < m. (M. F. Atiyah, I. G. MacDonald 15)
- Two ideals a, 0 are said to be coprime (or comaximal) if a + b = (I). Thus for coprime ideals we have a (") b = ao (M. F. Atiyah, I. G. MacDonald 16)
- Clearly two ideals a, b are coprime if and only if there exist x E a and YEO such that x + Y = 1. (M. F. Atiyah, I. G. MacDonald 16)
- Proof i) by induction on n. The case n = 2 is dealt with above. Suppose n > 2 and the result true for al> ... , an l> and let 0 = n~;l a j = n~.;1 a j • Since a j + an = (1) (1 .;;:; i .;;:; n 1) we have equations Xj + Yj = 1 (Xj E aj, Yj E an) and therefore n n-1 Xj = n n-1 (1 Yj) == 1 (mod an). j= 1 j =1 Hence an + 0 = (I) and so (M. F. Atiyah, I. G. MacDonald 16)
- The union a u 0 of ideals is not in general an ideal. (M. F. Atiyah, I. G. MacDonald 16)
- Their direct product n A =nAj i= 1 is the set of all sequences x = (Xl>’ .. , x n ) with Xj E Aj (1 .;;:; i .;;:; n) and componentwise addition and m. (M. F. Atiyah, I. G. MacDonald 16)
- If a, 0 are ideals in a rIng A, their Ideal quotient is (a:o) = {xEA:xo S; a} (M. F. Atiyah, I. G. MacDonald 17)
- Example. If A = Z, a = (m), 0 = (n), where say m = Dp pltp, n = Dp pVP, then (a:o) = (q)whereq = DppYp and yp = max (J.Lp lip, 0) = J.Lp min (J.Lp, lip). Hence q = m/(m, n), where (m, n) is the h.c.f. of m and n. (M. F. Atiyah, I. G. MacDonald 17)
- If a is any ideal of A, the radical of a is rea) = {x E A :x n E a for some n > O}. (M. F. Atiyah, I. G. MacDonald 17)
- Exercise 1.13. i) rea) ~ a ii) r(r(a)) = rea) iii) r(ab) = rea n b) = rea) n reb) iv) rea) = (1) <0> a = {I) v) rea + b) = r(r(a) + reb»~ vi) if p is prime, r(pn) = pior all n > O. (M. F. Atiyah, I. G. MacDonald 18)
- Proposition 1.14. The radical of an ideal a is the intersection of the prime ideals which contain a. (M. F. Atiyah, I. G. MacDonald 18)
- Proposition 1.15. D = set of zero-divisors of A = UX,.o r(Ann (x». (M. F. Atiyah, I. G. MacDonald 18)
- Proposition 1.16. Let a, b be ideals in a ring A such that r(a), reb) are coprime. Then a, b are coprime. (M. F. Atiyah, I. G. MacDonald 18)
- We define the extension a e of a to be the ideal Bf(a) generated by f(a) in B: explicitly, a e is the set of all sums 2: yJ(Xi) where Xi E a, Yi E B. (M. F. Atiyah, I. G. MacDonald 18)
- If a is an ideal in A, the setf(a) is not necessarily an ideal in B (M. F. Atiyah, I. G. MacDonald 18)
- Proposition 1.17. i) a s; aec , b ;:> bC’ ; ii) b C = b C’ c , a e = aece ; iii) If C is the set of contracted ideals in A and if E is the set of extended ideals in B, then C = {a/ aee = a}, E = {b / bee = b}, and a f-+ ae is a bijective map of C onto E, whose inverse is b f-+ be. (M. F. Atiyah, I. G. MacDonald 19)
- Proof i) is trivial, and ii) follows from i). (M. F. Atiyah, I. G. MacDonald 19)
- i) (2)" = (1 + i)2), the square of a prime ideal in Z[i]; ii) If p == 1 (mod 4) then (p)e is the product of two distinct prime ideals (for example, (5)e = (2 + i)(2 i)); iii) If p == 3 (mod 4) then (p)e is prime in Z[i]. (M. F. Atiyah, I. G. MacDonald 19)
- A = k[x] where k is a field; an A-module is a k-vector space with a linear transformation. (M. F. Atiyah, I. G. MacDonald 26)
- Homomorphisms u: M’ ~ M and v: N ~ N" induce mappings u: Hom (M, N) ~ Hom (M’, N) and v: Hom (M, N) ~ Hom (M, N") defined as follows: u(f) =f 0 u, v(f) = v 0 f (M. F. Atiyah, I. G. MacDonald 27)
- For any module M there is a natural isomorphism Hom (A, M) ~ M: any A-module homomorphism f: A ~ M is uniquely determined by f{l), which can be any element of M. (M. F. Atiyah, I. G. MacDonald 27)
- A-module homomorphism (or is A-linear) if f(x + y) = f(x) + fey) f(ax) = aI(x) (M. F. Atiyah, I. G. MacDonald 27)
- If M’ is a submodule of M such that M’ ~ Ker (f), thenfgives rise to a homomorphism]: M/M’ -+ N, defined as follows: if x E M/M’ is the image of x EM; then](x) = f(x). The kernel of]is Ker (f)/M’. The homomorphism] is said to be induced by f In particular, taking M’ = Ker (I), we have an isomorphism of A-modules M/Ker (f) ~ 1m (I). (M. F. Atiyah, I. G. MacDonald 28)
- Proposition 2.1. i) If L :2 M :2 N are A-modules, then (L/ N)/(M/ N) ~ L/ M. ii) If Mh M2 are submodules of M, then (Ml + M 2 )/M1 ~ M 2/(M 1 (‘) M2)’ (M. F. Atiyah, I. G. MacDonald 28)
- The composite homomorphism M2 -+ Ml + M2 -+ (Ml + M 2)/M1 is surjective, and its kernel is Ml (’) M 2 ; hence (ii). (M. F. Atiyah, I. G. MacDonald 28)
- Define 0: L/ N -+ L/ M by O(x + N) = x + M. (M. F. Atiyah, I. G. MacDonald 28)
- We cannot in general define the product of two submodules, but we can define the product aM, where a is an ideal and M an A-module (M. F. Atiyah, I. G. MacDonald 28)
- The cokernel off is Coker (f) = N/Im (J) (M. F. Atiyah, I. G. MacDonald 28)
- we define (N:P) to be the set of all a E A such that aP ~ N; it is an ideal of A (M. F. Atiyah, I. G. MacDonald 28)
- (0: M) is the set of all a E A such that aM = 0; this ideal is called the annihilator of M and is also denoted by Ann (M). (M. F. Atiyah, I. G. MacDonald 28)
- An A-module is faithful if Ann (M)=O (M. F. Atiyah, I. G. MacDonald 29)
- we can define their direct sum EEM j ; its elements are families (Xi)ieI such that Xj E Mi for each i E I and almost all XI are 0. If we drop the restriction on the number of non-zero x’s we have the direct product OjeI Mi (M. F. Atiyah, I. G. MacDonald 29)
- Proposition 2.4. Let M be afinitely generated A-module, let a be an ideal of A, and let ep be an A-module endomorphism of M such that ep(M) S; a M. Then ep satisfies an equation of the form where the aj are in a. (M. F. Atiyah, I. G. MacDonald 30)
- Proposition 2.3. M is a finitely generated A-module ~ M is isomorphic to a quotient of An for some integer n > O. (M. F. Atiyah, I. G. MacDonald 30)
- Proof (M. F. Atiyah, I. G. MacDonald 30)
- Corollary 2.5. Let M be a finitely generated A-module and let a be an ideal of A such that aM = M. Then there exists X == l(mod a) such that xM = O. (M. F. Atiyah, I. G. MacDonald 30)
- Proposition 2.6. (Nakayama’s lemma). Let M be a finitely generated A-module and a an ideal of A contained in the Jacobson radical 9t of A. Then aM = M implies M = O. (M. F. Atiyah, I. G. MacDonald 30)
- MlmM is annihilated by m, hence is naturally an Aim-module, (M. F. Atiyah, I. G. MacDonald 31)
- Proposition 2.8. Let x, (1 ~ i ~ n) be elements of M whose images in MlmM form a basis of this vector space. Then the x, generate M. (M. F. Atiyah, I. G. MacDonald 31)
- o –+ M’ .4. M is exact <:> f is injective; (M. F. Atiyah, I. G. MacDonald 31)
- M ~ M" –+ 0 is exact <:> g is surjective; (M. F. Atiyah, I. G. MacDonald 31)
- 0–+ M’ .4. M 4 M" –+ 0 is exact <:> f is injective, g is surjective and g induces an isomorphism of Coker if) = Mlf(M’) onto M". (M. F. Atiyah, I. G. MacDonald 31)
- <=> for all A-modules N, the sequence 0–+ Hom (M", N) ~ Hom (M, N) ~ Hom (M’, N) (M. F. Atiyah, I. G. MacDonald 31)
- <=> for all A~modules M, the sequence (5) Then the sequence (5) is 0–+ Hom (M, N’) ~ Hom (M, N) ~ Hom (M, N") (M. F. Atiyah, I. G. MacDonald 32)
- The boundary homomorphism d is defined as follows (M. F. Atiyah, I. G. MacDonald 32)
- The function A is additive if, for each short exact sequence (3) in which all the terms belong to C, we have A(M’) ’\(M) + A(M") = O. (M. F. Atiyah, I. G. MacDonald 32)
- Proposition 2.11. Let 0 ~ Mo ~ Ml
_{"’}Mn ~ 0 be an exact sequence of A-modules in which all the modules M, and the kernels of all the homomorphisms belong to C. Then for any additive function A on C we have n L (-l)lA(M,) = O. (M. F. Atiyah, I. G. MacDonald 33) - A-bilinear if for each x EM the mappingy 1–+ f(x, y) of N into P is A-linear, and for each yEN the mapping x 1–+ f(x, y) of Minto P is A-linear. (M. F. Atiyah, I. G. MacDonald 33)
- construct an A-module T, called the tensor product (M. F. Atiyah, I. G. MacDonald 33)
- Proof (M. F. Atiyah, I. G. MacDonald 33)
- In particular, if M and N are finitely generated, so is M 181 N. (M. F. Atiyah, I. G. MacDonald 34)
- these three conditions determine the ring S -1 A up to isomorphism. (M. F. Atiyah, I. G. MacDonald 46)
- s E S => g(s) is a unit in B; (M. F. Atiyah, I. G. MacDonald 46)
- g: A -+ B (M. F. Atiyah, I. G. MacDonald 46)
- Corollary 3.2. (M. F. Atiyah, I. G. MacDonald 46)
- g(a) = 0 => as = ofor some SE S; (M. F. Atiyah, I. G. MacDonald 46)
- The process of passing from A to All is called localization at .p. (M. F. Atiyah, I. G. MacDonald 47)
- Hence m is the only maximal ideal in All; in other words, All is a local ring. (M. F. Atiyah, I. G. MacDonald 47)
- 8 = A .p is multiplicatively closed (M. F. Atiyah, I. G. MacDonald 47)
- (in fact A P is multiplicatively closed ~ .p is prime). (M. F. Atiyah, I. G. MacDonald 47)
- Every element of B is of the form g(a)g(s) -1 (M. F. Atiyah, I. G. MacDonald 47)
- hen there is a unique isomorphism h: 81 A ~ B such that g = h 0 f (M. F. Atiyah, I. G. MacDonald 47)
- We write All for 8 -lA (M. F. Atiyah, I. G. MacDonald 47)
- 81 A ess of passing from A to ° is the zero ring ~ E 8 (M. F. Atiyah, I. G. MacDonald 47)
- 8 = {rn},,~o. We write A, for 81A (M. F. Atiyah, I. G. MacDonald 47)
- Let Kbe a field. A discrete valuation on Kis a mapping v of K* onto Z (where K* = K {O} is the multiplicative group of K) such that 1) v(xy) = vex) + v(y), i.e., v is a homomorphism; 2) vex + y) ~ min(v(x),v(y»). (M. F. Atiyah, I. G. MacDonald 103)
- An integral domain A is a discrete valuation ring if there is a discrete valuation v of its field of fractions K such that A is the valuation ring of v. (M. F. Atiyah, I. G. MacDonald 103)
- The set consisting of 0 and all x E K* such that vex) ~ 0 is a ring, called the valuation ring of v. It is a valuation ring of the field K (M. F. Atiyah, I. G. MacDonald 103)
- A ring satisfying the conditions of (9.3) is called a Dedekind domain. (M. F. Atiyah, I. G. MacDonald 104)
- i) A is integrally closed; ii) Every primary ideal in A is a prime power; iii) Every local ring All (.):> of 0) is a discrete valuation ring. (M. F. Atiyah, I. G. MacDonald 104)
- Theorem 9.3. Let A be a Noetherian domain of dimension one. Then the following are equivalent: (M. F. Atiyah, I. G. MacDonald 104)
- Corollary 9.4. In a Dedekind domain every non-zero ideal has a unique factorization as a product of prime ideals. (M. F. Atiyah, I. G. MacDonald 104)
- Theorem 9.5. The ring of integers in an algebraic number field K is a Dedekind domain. (M. F. Atiyah, I. G. MacDonald 105)
- An A-submodule M of K °in A. In particular, the (M. F. Atiyah, I. G. MacDonald 105)
- Let A be an integral domain, K its field of fractions. is a fractional ideal of A if xM s;; A for some x “# ons. An A”# ° in A (M. F. Atiyah, I. G. MacDonald 105)
- Any element U E K generates a fractional ideal, denoted by (u) or Au, and called principal. If M is a fractional ideal, the set of all x E K such that x M s;; A is denoted by (A: M). (M. F. Atiyah, I. G. MacDonald 105)

Last Annotation: 02/06/2020

- en B = A @ C. (Reason: If a map 7 with the desired property exists, then im 7, the image of 7, is disjoint from the image of a, and together they generate B, so B = a(A) ® 7(C). But a(A) 2 A and 7(C) = C.) (David Eisenbud & Professor David Eisenbud 28)
- An R-module M is Noetherian if every submodule of N is finitely generated (David Eisenbud & Professor David Eisenbud 38)
- An R-module M is Noetherian if every submodule of N is finitely generated. (David Eisenbud & Professor David Eisenbud 38)
- An R-module M is Noetherian if every submodule of N is finitely generated. (David Eisenbud & Professor David Eisenbud 38)

Last Annotation: 07/22/2020

- conformal self-maps of the open unit disk have the form (Theodore W. Gamelin 307)
- Sectors. A sector can be mapped onto a half-plane with the aid of the power function z°, (Theodore W. Gamelin 308)
- D = {0 < argz < a}, (Theodore W. Gamelin 308)
- The fractional linear transformation w = (2 —4)/(z + i) maps the open upper half-plane H onto the open unit disk . (Theodore W. Gamelin 308)
- its inverse z = ¢(1 + w)/(1 — w). (Theodore W. Gamelin 308)
- 2T/ g w—‘ﬂ(z)—m, z € (Theodore W. Gamelin 308)
- The exponential function e¢* maps horizontal strips to sectors. (Theodore W. Gamelin 309)
- Since fractional linear transformations map circles to circles, the images of the two arcs lie on circles passing through 0 and oo, and so are rays from 0 to co. (Theodore W. Gamelin 310)
- fractional linear transformation { = g(z) mapping zo to 0 and 23 to co. (Theodore W. Gamelin 310)

Last Annotation: 03/21/2019

- Now a Hopf algebra, such as B in 1.3.6, is a cogroup object in the category of commutative rings R, (Douglas C. Ravenel 40)
- which is to say that Hom(B, R) = GR is a group-valued (Douglas C. Ravenel 40)
- functor (Douglas C. Ravenel 41)
- For a p-typical analog of 1.3.6 we need to replace b by cogroupoid object in the category of commutative Z(p) -algebras K. Such an object is called a Hopf algebroid (A1.1.1) and consists of a pair (A, Γ) of commutative rings with appropriate structure maps so that Hom(A, K) and Hom(Γ, K) are the sets of objects and morphisms, respectively, of a groupoid. (Douglas C. Ravenel 41)
- To get at this question we use the spectrum J, which is the fibre of a certain map bu → Σ2 bu, where bu is the spectrum representing connective complex K-theory, i.e., the spectrum obtained by spectrum representing connective complex K-theory, i.e., the spectrum obtained by delooping the space Z × BU . (Douglas C. Ravenel 51)
- spectra and the stable homotopy category as described, for example, in the first few sections of Adams [4]. (Douglas C. Ravenel 61)
- Recall that H ∗ (X) is a module over the mod (p) Steenrod algebra A, to be described explicitly in the next chapter (Douglas C. Ravenel 61)
- 2.1.10. Definition. Given a sequence of spectra and maps f 1 f2 f3 X 0 ←− X1 ←− X2 ←− X3 ←− · · · , (Douglas C. Ravenel 64)
- m Xi , is the fiber of the map ←− Y g: whose ith component is the difference Y Xi ence be Y Xi → between Y Xi en the projection pi : Xj → Xi and the composite (Douglas C. Ravenel 65)
- This lim is not a categorical inverse limit (Mac Lane [1, Section III.4] because a ←− compatible collection of maps to the Xi , does not give a unique map to lim Xi . For compatible collection of maps to the Xi , does not give a unique map to lim Xi . Fo ←− this reason some authors (e.g., Bousfield and Kan [1]) denote it instead by holim (Douglas C. Ravenel 65)
- The Adams spectral sequence of 2.2.3 is useful for computing π∗ (X), i.e., [S 0 , X]. With additional assumptions on E one can generalize to a spectral sequence for computing [W, X]. (Douglas C. Ravenel 73)
- Throughout this book, P (x) will denote a polynomial algebra (over a field which will be clear from the context) on one or more generators x, and E(x) will denote the exterior algebra on same (Douglas C. Ravenel 79)
- We start by describing the dual Steenrod algebra A∗ (Douglas C. Ravenel 79)
- In this section we will consider four spectra (M O, M U , bo, and bu) in which the change-of-rings isomorphism of A1.1.18 can be used to great advantage (Douglas C. Ravenel 80)
- H ∗ (M U ; Z) = Z[b1 , b2 , . . . ], where bi ∈ H2i . H/(p) denote the mod (p) Eilenberg–M (Douglas C. Ravenel 81)
- π ∗ (M U ) = Z[x1 , x2 , . . . ] with xi ∈ π2i (M U ). (Douglas C. Ravenel 81)
- A theorem of Milnor and Moore [3] says that every graded primitively generated Hopf algebra is isomorphic to the universal enveloping algebra of a restricted Lie algebra (Douglas C. Ravenel 88)
- The lambda algebra Λ is an associative differential bigraded algebra whose cohomology, like that of the cobar complex, is Ext (Douglas C. Ravenel 97)
- . Its greatest attraction, which will not be exploited here, is that it contains for each n > 0 a subcomplex Λ(n) whose cohomology is the E2 -term of a spectral sequence converging to the 2-component of the unstable homotopy groups of S n . In other words Λ(n) is the E 1 -term of an unstable Adams spectral sequence (Douglas C. Ravenel 97)
- 3.4.4. Theorem. (a) (Browder [1]). For p = 2 h2j is a permanent cycle iff there is a framed manifold of dimension 2j+1 − 2 with Kervaire invariant one. Such are known to exist for j ≤ 5. For more discussion see 1.5.29 and 1.5.35. (Douglas C. Ravenel 107)
- We do not know how to make this computation directly. (Douglas C. Ravenel 112)
- We do not know the image of the map in 3.4.19 (Douglas C. Ravenel 113)
- n Section 1 we made some easy Ext calculations and thereby computed the homotopy groups of such spectra as M U and bo. The latter involved the cohomology of A(1), the subalgebra of the mod (2) Steenrod algebra generated by Sq 1 and Sq 2 . (Douglas C. Ravenel 114)
- The use of the Adams spectral sequence in computing cobordism rings is becoming more popular. The spectra M O, M SO, M SU , and M Spin were originally (Douglas C. Ravenel 114)
- analyzed by other methods (see Stong [1] for references) but in theory could be analyzed with the Adams spectral sequence (Douglas C. Ravenel 115)
- The spectrum M Oh8i (the Thom spectrum associated with the 7-connected cover of BO) has been investigated by Adams spectral sequence methods in Giambalvo [2], Bahri [1], Davis [3, 6], and Bahri and Mahowald [1]. In Johnson and Wilson [5] the Adams spectral sequence is used to compute the bordism ring of manifolds with free G-action for an elementary abelian p-group G. (Douglas C. Ravenel 115)
- The most prodigious Adams spectral sequence calculation to date is that for the symplectic cobordism ring by Kochman [1, 2, 3]. (Douglas C. Ravenel 115)
- In Section 2 we described the May SS. The work of Nakamura [1] enables one to use algebraic Steenrod operations (A1.5) to compute May differentials. (Douglas C. Ravenel 115)
- The May SS is obtained from an increasing filtration of the dual Steenrod algebra A∗ . (Douglas C. Ravenel 115)
- The Adams spectral sequence was used in the proof of the Segal conjecture for Z/(2) by Lin [1] and Lin et al. [2]. (Douglas C. Ravenel 120)
- Finally, we must mention the Whitehead conjecture. The n-fold symmetric product Spn (X) of a space X is the quotient of the n-fold Cartesian product by the action of the symmetric group Σn . Dold and Thom [1] showed tha (Douglas C. Ravenel 120)
- Sp ∞ (X) = lim Spn (X) is a product of Eilenberg–Mac Lane spaces whosw homo←− topy is the homotopy of X. Symmetric products can be defined on spectra and we have Sp∞ (S 0 ) = HJ, the integer Eilenbergh–Mac Lane spectrum. After localizing at the prime p one considers S 0 → Spp (S 0 ) → Spp 2 (S 0 ) → · · · and (3.5.16) H ← S 0 ← Σ−1 Spp (S 0 )/S 0 ← Σ−2 Spp 2 (S 0 )/Spp (S 0 ) ← · · · . Whitehead conjectured that this diagram induces an long exact sequence of homotopy groups. In particular, the map Σ−1 Spp (S 0 )/S 0 → S 0 shouls induce a surjection in homotopy in positive dimensions; this is the famous theorem of Kahn surjection in homotopy in positive dimensions; this is the famous theorem of Kahn and Priddy (Douglas C. Ravenel 121)
- In Section 4 we set up the Adams–Novikov spectral sequence and use it to compute the stable homotopy groups of spheres through a middling range of dimensions, namely ≤ 24 for p = 2 and ≤ 2p3 − 2p − 1 for p > 2. (Douglas C. Ravenel 123)
- The main results are Quillen’s theorem 4.1.6, which identifies π∗ (M U ) with the Lazard ring L (A2.1.8); the Landweber–Novikov theorem 4.1.11, which describes M U∗ (M U ); the Brown– Peterson theorem 4.1.12, which gives the spectrum BP ; and the Quillen–Adams theorem 4.1.19, which describes BP∗ (BP (Douglas C. Ravenel 123)
- We begin by informally defining the spectrum M U . For more details see Milnor and Stasheff [5]. Recall that for each n ≥ 0 the group of complex unitary n × n matrices U (n) has a classifying space BU (n). It has a complex n-plane bundle γn over it which is universal in the sense that any such bundle ξ over a paracompact space X is the pullback of γn , induced by a map f : X → BU (n). Isomorphism classes of such bundles ξ are in one-to-one correspondence with homotopy classes of classes of such bundles ξ are in one-to-one correspondence with homotopy classes of maps from X to BU (n). Any Cn -bundle ξ has an associated disc bundle D(ξ) and sphere bundle S(ξ). The Thom space T (ξ) is the quotient D(ξ)/S(ξ). Alternatively, for compact X, T (ξ) is the one-point compactification of the total space of ξ. (Douglas C. Ravenel 123)
- M U (n) is T (γn ), the Thom space of the universal n-plane bundle γn over BU (n). (Douglas C. Ravenel 123)
- It follows from the celebrated theorem of Thom [1] that π∗ (M U ) is isomorphic to the complex cobordisrn ring (see Milnor [4]) which is defined as follow (Douglas C. Ravenel 124)
- A stably complex manifold is one with a complex structure on its stable normal bundle. (This notion of a complex manifold is weaker than others, e.g., algebraic, analytic, and almost complex.) All such manifolds are oriented. Two closed stably complex manifolds M1 and M2 are cobordant if there is a stably complex manifold W whose boundary is the disjoint union of M1 (with the opposite of the given orientation) and M2 . (Douglas C. Ravenel 124)
- Milnor and Novikov’s calculation of π∗ (M U ) (3.1.5) implies that two such manifolds are cobordant if they have the same Chern numbers. (Douglas C. Ravenel 124)
- On the other hand, the connection with formal group laws (A2.1.1) discovered by Quillen [2] (see 4.1.6) is essential to all that follows. This leads one to suspect that there is some unknown formal group theoretic construction of M U or its associated infinite loop space. For example, many well-known infinite loop spaces have been constructed as classifying spaces of certain types of categories (see Adams [9], section 2.6), but to our knowledge no such description exists for M U . (Douglas C. Ravenel 124)
- 4.1.1. Definition. Let E be an associative commutative ring spectrum. A complex orientation for E is a class 1 E(CP ) xE ∈ E e ≃E e 2 (S e 2 (CP ∞ ) whose restriction to E = π0 (E) is 1, where CP e E(CP 1 )≃Ee 2 (S 2 ) ∼ = π0 (E) n denotes n-dimensional complex projective space (Douglas C. Ravenel 124)
- Recall that M U is built up out of Thom spaces M U (n) of complex vector bundles over BU (n) and that the map BU (n) → M U (n) is an equivalence when n = 1. The composition is an equivalence when n = (Douglas C. Ravenel 124)
- Alternatively, xM U could be defined to be the first Conner–Floyd Chern class of the canonical complex line bundle over CP ∞ (Douglas C. Ravenel 125)
- Hence a complex orientation xE leads to a formal group law FE over E ∗ (pt.). (Douglas C. Ravenel 125)
- The spectrum BP is named after Brown and Peterson, who first constructed it via its Postnikov towe (Douglas C. Ravenel 128)
- Brown and Peterson [1] also showed that BP can be constructed from H (the integral Eilenberg–Mac Lane spectrum) by killing all of the torsion in its integral homology with Postnikov fibration (Douglas C. Ravenel 129)
- More recently, 0 Priddy [1] has shown that BP can be constructed from S(p) by adding local cells to kill off all of the torsion in its homotopy. (Douglas C. Ravenel 129)
- BP bears the same relation to p-typical formal group laws (A2.1.17) that M U bears to formal group laws as seen in 4.1.6 (Douglas C. Ravenel 129)
- Cartier’s theorem A2.1.18, which states that any formal group law over a Z(p) -algebra is canonically isomorphic to p-typical one. (Douglas C. Ravenel 129)

Last Annotation: 08/24/2016

- 1. (James Ward Brown & Ruel Vance Churchill 21)
- 2. (James Ward Brown & Ruel Vance Churchill 22)
- 4. (James Ward Brown & Ruel Vance Churchill 22)
- 5. (James Ward Brown & Ruel Vance Churchill 22)
- 11. (James Ward Brown & Ruel Vance Churchill 22)
- 1. (James Ward Brown & Ruel Vance Churchill 24)
- 4. (James Ward Brown & Ruel Vance Churchill 25)
- 8. (James Ward Brown & Ruel Vance Churchill 25)
- EXAi’. The eq ualion I:. J + 3i I = 2 rcprescms lhc ci rclc whose center (James Ward Brown & Ruel Vance Churchill 27)
- is :. 0 = ( J. -3) and whose radius is R = 2. (James Ward Brown & Ruel Vance Churchill 27)
- 3. (James Ward Brown & Ruel Vance Churchill 30)
- 4. (James Ward Brown & Ruel Vance Churchill 30)
- 5. (James Ward Brown & Ruel Vance Churchill 30)
- 8. (James Ward Brown & Ruel Vance Churchill 31)
- I.:" I = I.: 111 (James Ward Brown & Ruel Vance Churchill 31)
- 1. (James Ward Brown & Ruel Vance Churchill 33)
- 4. (James Ward Brown & Ruel Vance Churchill 33)
- <>. (James Ward Brown & Ruel Vance Churchill 33)
- 9. (James Ward Brown & Ruel Vance Churchill 33)
- 10. (James Ward Brown & Ruel Vance Churchill 33)
- 11. (James Ward Brown & Ruel Vance Churchill 33)
- 13. (James Ward Brown & Ruel Vance Churchill 34)
- (cos61 + i sin61) 11 = cosne + i sin11e ( (James Ward Brown & Ruel Vance Churchill 38)
- 1. (James Ward Brown & Ruel Vance Churchill 40)
- 3. @ (James Ward Brown & Ruel Vance Churchill 41)
- 5. (James Ward Brown & Ruel Vance Churchill 41)
- 6. (James Ward Brown & Ruel Vance Churchill 41)
- (9 9. (James Ward Brown & Ruel Vance Churchill 41)
- 10. (James Ward Brown & Ruel Vance Churchill 42)
- 2. @F (James Ward Brown & Ruel Vance Churchill 47)
- 3. (James Ward Brown & Ruel Vance Churchill 48)
- 4. (James Ward Brown & Ruel Vance Churchill 48)
- (>. (James Ward Brown & Ruel Vance Churchill 48)
- 7. (James Ward Brown & Ruel Vance Churchill 48)
- A poim :. 0 is said lo be an illterior point of a sec S whenever there is som (James Ward Brown & Ruel Vance Churchill 49)
- ncighborl10o<l of :.o thal contains only poinrs of S: il is called an exterior point of S (James Ward Brown & Ruel Vance Churchill 49)
- ncighborl10o<l of :.o thal contains only poinrs of S: il is called an exterior point of S when rhcrc exists a neighborhood of it containing no poinrs of S. If :. 0 is ncirhcr of (James Ward Brown & Ruel Vance Churchill 49)
- when rhcrc exists a neighborhood of it containing no poinrs of S. If :. 0 is ncirhcr of these. il is a boundary poillt of S. A boundary poifll is. therefore. a poi Ill all of whose (James Ward Brown & Ruel Vance Churchill 49)
- these. il is a boundary poillt of S. A boundary poifll is. therefore. a poi Ill all of wh neighborhoods contain at least one poifll in S and al least one point nor in S. (James Ward Brown & Ruel Vance Churchill 49)
- A set is open if il docs not contain any of ils boundary points. (James Ward Brown & Ruel Vance Churchill 49)
- il docs not contain any of ils boundary points. ll is left as an a set is open if and only if each of irs poinrs is an imcrior poinr. (James Ward Brown & Ruel Vance Churchill 49)
- A set is closed if it cofllains all of its boundary points. and the closure of a set Sis the (James Ward Brown & Ruel Vance Churchill 49)
- closed sec consisting of all poi Ills in S together rhe boundary of S. (James Ward Brown & Ruel Vance Churchill 49)
- . For a set S to be not open there (James Ward Brown & Ruel Vance Churchill 49)
- must be a boundary poim that is contained in the set. and for S lo be nor closed there (James Ward Brown & Ruel Vance Churchill 49)
- must be a boundary point not in it (James Ward Brown & Ruel Vance Churchill 50)
- Observe that the punctured disk 0 < l.z: I -S l is (James Ward Brown & Ruel Vance Churchill 50)
- neither open nor closed. (James Ward Brown & Ruel Vance Churchill 50)
- An open set S is co1l11ected if each pair of points z. 1 and z. 2 in it can be joined (James Ward Brown & Ruel Vance Churchill 50)
- by a polygonal lille. (James Ward Brown & Ruel Vance Churchill 50)
- A nonempty open set that (James Ward Brown & Ruel Vance Churchill 50)
- is connected is called a domai11. N (James Ward Brown & Ruel Vance Churchill 50)
- . A doma (James Ward Brown & Ruel Vance Churchill 50)
- together with some. none. or all of its boundary points is usually referred to as a region. (James Ward Brown & Ruel Vance Churchill 50)
- A poim ::o is said lo be an accumullltio11 point. or limit poinl. of a scl S if each (James Ward Brown & Ruel Vance Churchill 51)
- dclelcd neighborhood of :: 0 contains al lcasl one point of S. (James Ward Brown & Ruel Vance Churchill 51)
- . ll follows lhal if a scl Si (James Ward Brown & Ruel Vance Churchill 51)
- dclelcd neighborhood of :: 0 contains al lcasl one point of closed. then il comains each of ils accumulation poims. (James Ward Brown & Ruel Vance Churchill 51)
- Thus a scl is closed if and only if il conlains all of ils (James Ward Brown & Ruel Vance Churchill 51)
- accumulalion poill (James Ward Brown & Ruel Vance Churchill 51)
- Evidcmly. a poinl :: 0 is 1101 an accumulation poinl of a scl S whenever lhcrc cxisls (James Ward Brown & Ruel Vance Churchill 51)
- some dclclcd neighborhood of ::o lhal docs nol conlain al lcasl one poim in S. (James Ward Brown & Ruel Vance Churchill 51)
- 1. (James Ward Brown & Ruel Vance Churchill 51)
- 4. (James Ward Brown & Ruel Vance Churchill 52)
- 5. (James Ward Brown & Ruel Vance Churchill 52)
- 6. (James Ward Brown & Ruel Vance Churchill 52)
- /(:::)can be expressed in tcnns of a pair of real-vaJued functions of the real (James Ward Brown & Ruel Vance Churchill 55)
- variables x and y: (James Ward Brown & Ruel Vance Churchill 55)
- /(:::) = 11(.r. y) + iv(x. y). (James Ward Brown & Ruel Vance Churchill 55)
- If the function v in equation (I) always has value zcm. then the value of .f i (James Ward Brown & Ruel Vance Churchill 55)
- alreal. Thus .f is a real-valuedfw1clio11 of a complex variable. (James Ward Brown & Ruel Vance Churchill 55)
- . (James Ward Brown & Ruel Vance Churchill 59)
- 2. (James Ward Brown & Ruel Vance Churchill 60)
- 3. (James Ward Brown & Ruel Vance Churchill 60)
- t for each F: neighborhood Iu: 11 10 I < F of (James Ward Brown & Ruel Vance Churchill 61)
- Geomet1ically. this deli nition says that for each F: neighborhood Iu: 11 10 I < F of wo. there is a deleted ,5 neighborhood 0 < I:. :.o I < 8 of :.o such that every point (James Ward Brown & Ruel Vance Churchill 61)
- : in it has an image "’ lying in the F: ncighborhoo<l (F (James Ward Brown & Ruel Vance Churchill 61)
- scl 1::1 > I /r a 11eighhorhood of oo. (James Ward Brown & Ruel Vance Churchill 67)
- I lim : ·-· ::., . r (;::) = 00 Jim = 0 : •:,.I(:: l (James Ward Brown & Ruel Vance Churchill 67)
- (James Ward Brown & Ruel Vance Churchill 68)
- lim /(:) = x lim =0. : . "’.\... : .(I .f (I/:) (James Ward Brown & Ruel Vance Churchill 68)
- A function f is co11ti11uous at a point :o if all three of the follov,:ing conditions arc (James Ward Brown & Ruel Vance Churchill 69)
- satisfied (James Ward Brown & Ruel Vance Churchill 69)
- : lim /(:) = ;.. : . f (:o). (James Ward Brown & Ruel Vance Churchill 69)
- 2. (James Ward Brown & Ruel Vance Churchill 71)
- 4. (James Ward Brown & Ruel Vance Churchill 71)
- co ., _ _ A . /(::) f <::o) ( •. 0 )3 -= Fimal ./ol A s 11m .Lo . ;PN ) :11 :: ::o (James Ward Brown & Ruel Vance Churchill 72)
- @li 7. (James Ward Brown & Ruel Vance Churchill 72)
- 9. @S (James Ward Brown & Ruel Vance Churchill 72)
- e11. 11 (James Ward Brown & Ruel Vance Churchill 72)
- the existence <~/’the derirntire <d’aji111ctio11 at (James Ward Brown & Ruel Vance Churchill 75)
- exist there. It is. however, true that the existence <~/’the der a point implies the continuity <~(the.fi1co at that point. (James Ward Brown & Ruel Vance Churchill 75)
- O 1. (James Ward Brown & Ruel Vance Churchill 78)
- @ 2. (James Ward Brown & Ruel Vance Churchill 78)
- 6. (James Ward Brown & Ruel Vance Churchill 78)
- /(:,) = ll(X. _') +iv(.r. y) (James Ward Brown & Ruel Vance Churchill 81)
- and that/’(:.) exists at a point :.o = .ro + i yo. (James Ward Brown & Ruel Vance Churchill 81)
- llx = l’ 1.• lly = -t (James Ward Brown & Ruel Vance Churchill 81)
- the.fir.·t-orderpllrthil derirnth·es < (James Ward Brown & Ruel Vance Churchill 83)
- exist e1 (James Ward Brown & Ruel Vance Churchill 83)
- c011ti111tous l (James Ward Brown & Ruel Vance Churchill 83)
- .mti~fy the Ca11ch (James Ward Brown & Ruel Vance Churchill 83)
- Rie 11101111 eq11a ti 011 (James Ward Brown & Ruel Vance Churchill 83)
- Th ell .f’ (:.o) exis ts, (James Ward Brown & Ruel Vance Churchill 83)
- 1. (James Ward Brown & Ruel Vance Churchill 87)
- 2. (James Ward Brown & Ruel Vance Churchill 87)
- 3. f w (James Ward Brown & Ruel Vance Churchill 88)
- 8. (James Ward Brown & Ruel Vance Churchill 88)
- . J . A function function f§ o o (James Ward Brown & Ruel Vance Churchill 89)
- the complex variable:: W is. allalytic in a11au opell set§ HS if o it o has a derivative donvative everywhere LYETY WIre b § o §3 e oo N: b b 3ohye o ey s hanae T ek R iAN E ta Y ~r AR IS nd e Bt ER §§ FIEES W opengef & . N ~"§ P eity, LRICOIIENMCN TR AN ENIIN Y ViNy e~ $a8s (James Ward Brown & Ruel Vance Churchill 89)
- wi sci. that sot. (James Ward Brown & Ruel Vance Churchill 89)
- 1. (James Ward Brown & Ruel Vance Churchill 93)
- 3. (James Ward Brown & Ruel Vance Churchill 93)
- 4. (James Ward Brown & Ruel Vance Churchill 93)
- y. (fl1vo.fi111ctiom fllnd g are analytic in the swne do111ai11 D (James Ward Brown & Ruel Vance Churchill 98)
- forward lo prove. Namely. (fl1vo.fi111ctiom fllnd g are analytic in the swne do111ai11 D and~(/.:)= g(.::) 011 a subset <f D thllt lws a limit point .:: 0 in D. then/(::)= g(.::) (James Ward Brown & Ruel Vance Churchill 98)
- erery11·here in D."’ (James Ward Brown & Ruel Vance Churchill 98)
- there may exist a function .fi. which is analyt.ic in D2. such that /2(:.) = fi (:.)for each (James Ward Brown & Ruel Vance Churchill 99)
- there may exist a function .fi. which is analyt.ic in D2. such that /2(:.) = fi (:.)for each :. in the intersection D 1 n D 2 • If so. ,,.. e call Ii an a11alytic co11ti11uatio11 of .fi into the (James Ward Brown & Ruel Vance Churchill 99)
- :. in the intersection second domain D2. (James Ward Brown & Ruel Vance Churchill 99)
- that analytic continua1ion exists. ll 1s unique, (James Ward Brown & Ruel Vance Churchill 99)
- Jooe ,,= In r I-1_ i8 (James Ward Brown & Ruel Vance Churchill 110)
- . ex < R < ex + 2:r ). (James Ward Brown & Ruel Vance Churchill 110)
- is ! isymgle-vafned si11gle-l’liff 1ted§ and CORTNUOUS e B o ek v e andeconli nuous inItthe staleded domain demain »:.f”, (James Ward Brown & Ruel Vance Churchill 111)
- I~ Log;.= LAME TiR 1TRY §Y TN AY Y wsa In r& +oo i(-) S§ EN ade U food (James Ward Brown & Ruel Vance Churchill 111)
- —— PY £ e v JU =0 ¥ =0 T (James Ward Brown & Ruel Vance Churchill 111)
- is called the pri11cipal bra11ch. (James Ward Brown & Ruel Vance Churchill 111)
- s ps any poinl lhat is common lo all branch cues off is called a (James Ward Brown & Ruel Vance Churchill 111)
- =» ;‘ hra11ch T point. T (James Ward Brown & Ruel Vance Churchill 111)
- L . Pointsonthe . Poinls b on the> branch cutlor cut for F& arc are singular simgular s psd o (James Ward Brown & Ruel Vance Churchill 111)
- points ( (James Ward Brown & Ruel Vance Churchill 111)
- The origin t. The ongm and ray 0 lhe my & == ex or make up lhe make up the brnnch branch cul cat for for lhe the manch () brnnch (2) (James Ward Brown & Ruel Vance Churchill 111)
- of the logarithmic of logarttmie function. function. T (James Ward Brown & Ruel Vance Churchill 111)
- d ray 0 lhe my & == ex or make make upup lhe the brnnch branch cul cat for for lhe the manch brnnch (2)() The branch cul cut for for chc the principal branch (6) {(6) consists cass“si‘;ixof ofii“ic che (James Ward Brown & Ruel Vance Churchill 111)
- of the logarithmic of logarttmie function. function. ongin and the ray & == x. origin and the ray H = n. (James Ward Brown & Ruel Vance Churchill 111)
- t the set of values of log(i 2 ) is not the set of (James Ward Brown & Ruel Vance Churchill 112)
- values of 2 log i. (James Ward Brown & Ruel Vance Churchill 112)
- l. (James Ward Brown & Ruel Vance Churchill 116)
- 3. (James Ward Brown & Ruel Vance Churchill 116)
- :.& i= 3 0& (James Ward Brown & Ruel Vance Churchill 117)
- nt t c¢ is is any any complex complox number. number, ,, (James Ward Brown & Ruel Vance Churchill 117)
- -" = e"lo~:. o {»’S’Mg: (James Ward Brown & Ruel Vance Churchill 117)
- , :."is. . o7 s, in s general. geseral, multiple-valued multipie-valued (James Ward Brown & Ruel Vance Churchill 117)
- d _,. d:. ,, (James Ward Brown & Ruel Vance Churchill 117)
- (’ . = = ~expie ;-, exp(c log:.) ¥ sy trrrs o G 23 v -xp(iog 7). xp(log:. yic ). That yiel (James Ward Brown & Ruel Vance Churchill 117)
- ) = c expl (c I) log:.]. (James Ward Brown & Ruel Vance Churchill 117)
- d _,. (’_.. I d:: ,, .... (James Ward Brown & Ruel Vance Churchill 118)
- o define the pri11cipa/ hra11ch of the function::’· on the domain (James Ward Brown & Ruel Vance Churchill 118)
- Equal.ion (3) also serves to defi 1:.1 > 0. -;r < Arg:: < ;r. (James Ward Brown & Ruel Vance Churchill 118)
- . (.: is an entire function of::. (James Ward Brown & Ruel Vance Churchill 118)
- (James Ward Brown & Ruel Vance Churchill 118)
- d -c·· = c· Jog c. d:: (James Ward Brown & Ruel Vance Churchill 118)
- ·i I (James Ward Brown & Ruel Vance Churchill 118)
- i I ·i = e lo~ · i. (James Ward Brown & Ruel Vance Churchill 118)
- Note that the values of ;i arc all real 1111111/Jers. (James Ward Brown & Ruel Vance Churchill 118)
- sinh sinh yv tends tends to o infinity infinity as as v tends to y tends to infinity. infinity. ii it is is dear clear fro fro . P & t sand cos s are sFHEAGEE G0 UK t sin:. and cos:. arc 1101 sevictseroed ho1111ded on he complex plane, ovmy the T complex plane. (James Ward Brown & Ruel Vance Churchill 122)
- lhc mean value theorem for derivatives (James Ward Brown & Ruel Vance Churchill 135)
- in calculus docs not carry over to complex-valued functions 1n(I ). (James Ward Brown & Ruel Vance Churchill 135)
- the mean value theorem for integral.· docs nol carry over cilhcr. T (James Ward Brown & Ruel Vance Churchill 135)
- 2. (James Ward Brown & Ruel Vance Churchill 136)
- 4. (James Ward Brown & Ruel Vance Churchill 136)
- The PRC arc Ty arg CO is ey e a simple arc. or I8SNo, N ‘q‘np\s SEASEY or a sy Jordan arg, aaorqan §g -‘s.’" arc.~ ey if 11 it ATT L docs not cross GoCs nol oross it 336 TNy sci f:dih tisgit :".’: tI (James Ward Brown & Ruel Vance Churchill 137)
- 2 ) when % ) = g 8 e A ::(/J) =::(a). e ({3}SO0 AES > { . ’’’e 5% W say RN WO SV that ANS it{ C ¥ ¥Ry L is 8 % is a o simple » 3 RIS closed SENIMNSS AN OOoSY & LIy LU curve. o Ser e (James Ward Brown & Ruel Vance Churchill 137)
- ¥ ¥ or a Jordan 3 FOVQAn curve LeaTsey CUQve ANy . (James Ward Brown & Ruel Vance Churchill 137)
- ::(/J) =::(a). ({3}SO0 AES { . ’‘’e W say WO SV that it{ C L isis a o simple 3 RIS closed & LIy LU curve. or a Jordan FOVQAn c C . isposilivel)’ orie11ted when it is in the counterclockwise direction. . - (James Ward Brown & Ruel Vance Churchill 137)
- f C is lhe number L= 1"’ l::’(l)jdt. l under cercain changes i (James Ward Brown & Ruel Vance Churchill 139)
- n che unit tangent vector (James Ward Brown & Ruel Vance Churchill 140)
- ngent vector T = :::’(/) I::: I(/) I n illlerval. with a (James Ward Brown & Ruel Vance Churchill 140)
- sed interval a : : = t ::::=band nonzero throughout chc open interval a < t < b. A co11/our, or piecewise smooch arc, is an arc consisting of a finite number of (James Ward Brown & Ruel Vance Churchill 140)
- A co11/our, or piecewise sm smooth arcs joined end to end. (James Ward Brown & Ruel Vance Churchill 140)
- The points on any simple closed curve or simple closed contour C arc boundary (James Ward Brown & Ruel Vance Churchill 140)
- The points on any simple closed curve or simple closed contour C arc bound poi ms of two distinct domains. one of which is chc imcrior of C and is bounded. T (James Ward Brown & Ruel Vance Churchill 140)
- . The (James Ward Brown & Ruel Vance Churchill 140)
- poi ms of two distinct domains. one of which is ch ocher. which is chc exterior of C. is unbounded. l (James Ward Brown & Ruel Vance Churchill 140)
- e Jorda11 curve theorem. as (James Ward Brown & Ruel Vance Churchill 140)
- Dclinice integrals in calculus can be interpreted as c.u-eas. and chcy have othe (James Ward Brown & Ruel Vance Churchill 142)
- in tcrpretations as wcl I. Excepc in special cases. no corresponding helpful i ncerprccation. (James Ward Brown & Ruel Vance Churchill 142)
- geometric or physical. is available for intcgrnls in the complex plane. (James Ward Brown & Ruel Vance Churchill 142)
- he function f (:.)as being piccC\-’isc colllinuous on C. We then de or co11tour i11tegral. off along C in 1cnns of !he pammclcr I: (James Ward Brown & Ruel Vance Churchill 143)
- (James Ward Brown & Ruel Vance Churchill 143)
- 1tegral. off along C in 1cnns of !he pa j.( ..n:.)d:. = • /"" Il:.<n1:.‘u)t11. (J contour. :.’(I) is also piecewise concinu (James Ward Brown & Ruel Vance Churchill 143)
- (James Ward Brown & Ruel Vance Churchill 146)
- (James Ward Brown & Ruel Vance Churchill 146)
- the integrals of /(:) along the two paths C 1 and C 2 have t1m<:re111 (James Ward Brown & Ruel Vance Churchill 147)
- Evidently. then. the integrals of /(:) along the two paths C 1 and C 2 have t1m< 1’lt!1teJ even though those paths have the same initial and the same hnal points. (James Ward Brown & Ruel Vance Churchill 147)
- t the integral of/(:) over the simple closed contour (James Ward Brown & Ruel Vance Churchill 147)
- 01180. or C 1 C 2• has the 11011:em 'lllue (James Ward Brown & Ruel Vance Churchill 147)
- the value of a contour integral of a given f might be independent of the path taken (James Ward Brown & Ruel Vance Churchill 147)
- the value of a contour integral of a given function from one fixed poi Ill to another (James Ward Brown & Ruel Vance Churchill 147)
- ) contour integrals of a given function around every closed contour might all hav (James Ward Brown & Ruel Vance Churchill 147)
- value zero (Ex (James Ward Brown & Ruel Vance Churchill 147)
- not alwa (James Ward Brown & Ruel Vance Churchill 147)
- not always (James Ward Brown & Ruel Vance Churchill 147)
- 2. (James Ward Brown & Ruel Vance Churchill 150)
- 5. (James Ward Brown & Ruel Vance Churchill 150)
- { u• (I) is a pieccirise m11 ti1111011s co (James Ward Brown & Ruel Vance Churchill 152)
- then (James Ward Brown & Ruel Vance Churchill 152)
- Lei C denote a co11to11r <fle11g1/i L. (James Ward Brown & Ruel Vance Churchill 153)
- If(:> I :::: 1\11 (James Ward Brown & Ruel Vance Churchill 153)
- h /(:) i.· de.fined, !hen IJ.r(:)d:I:::: ML. (James Ward Brown & Ruel Vance Churchill 153)
- at since C is a contour and f is piecewise concinuous on C. a number (James Ward Brown & Ruel Vance Churchill 153)
- Note that since C is a contour and f is piecewise concinuou 1\11 such as che one appearing in inequalicy (5) v..·ill always exist. (James Ward Brown & Ruel Vance Churchill 153)
- e che (James Ward Brown & Ruel Vance Churchill 153)
- real-valued funccion 1/1:(1 lll is continuous on the closed bounded incerval a :St :S /J (James Ward Brown & Ruel Vance Churchill 153)
- wh (James Ward Brown & Ruel Vance Churchill 153)
- when f is concinuous on C: (James Ward Brown & Ruel Vance Churchill 153)
- such a function always reaches a maximum value M (James Ward Brown & Ruel Vance Churchill 153)
- on chac imerval.~ (James Ward Brown & Ruel Vance Churchill 153)
- 2. (James Ward Brown & Ruel Vance Churchill 155)
- 4. (James Ward Brown & Ruel Vance Churchill 156)
- ction ff(z) (:.) on a domain D lunulon F(::) D. or a function F() Note that an antide1ivativc antderivative is. 5 of of nogeessit necessity. (James Ward Brown & Ruel Vance Churchill 157)
- that F’(:.) F’(z) == f(:: f(2 anadviie an analytic function. (James Ward Brown & Ruel Vance Churchill 157)
- . . This is becausee e dorvalive of because the dcrivacive ¢ the difference (James Ward Brown & Ruel Vance Churchill 157)
- uiigue t’Xceptfor 1111iq11e exceptfm t111 an additire addmu’consfant. co11.·ta111. This is because e F(::) G(:.) of any cwo such amidcrivativcs is zero: a (James Ward Brown & Ruel Vance Churchill 157)
- (c) t : (James Ward Brown & Ruel Vance Churchill 158)
- (a) f (h) t .f (James Ward Brown & Ruel Vance Churchill 158)
- (James Ward Brown & Ruel Vance Churchill 158)
- (James Ward Brown & Ruel Vance Churchill 158)
- (James Ward Brown & Ruel Vance Churchill 158)
- (James Ward Brown & Ruel Vance Churchill 158)
- (James Ward Brown & Ruel Vance Churchill 158)
- (James Ward Brown & Ruel Vance Churchill 158)
- (James Ward Brown & Ruel Vance Churchill 158)
- (James Ward Brown & Ruel Vance Churchill 158)
- (James Ward Brown & Ruel Vance Churchill 158)
- f (: (James Ward Brown & Ruel Vance Churchill 158)
- f (:.) = I /:. 2• (James Ward Brown & Ruel Vance Churchill 158)
- . has an anciderivativc F(:.) = I/:. in the domain I:. I > O. (James Ward Brown & Ruel Vance Churchill 158)
- the imcgral of the function f (:.) = I/:. around the same circle nmnot (James Ward Brown & Ruel Vance Churchill 158)
- be evaluated in a similar way. F (James Ward Brown & Ruel Vance Churchill 158)
- . F(:.) is not diffcrcllliable. or even defined. along its branch cue. (James Ward Brown & Ruel Vance Churchill 158)
- 2. (James Ward Brown & Ruel Vance Churchill 164)
- 3. l (James Ward Brown & Ruel Vance Churchill 164)
- . Green’s theorem slmcs lhal (James Ward Brown & Ruel Vance Churchill 165)
- Green’s theorem slmcs lhal l. Pdx + Qdy = 1·1 (Q 1 P,.)dA. linuous on R. since il is analytic lhere. Hence (James Ward Brown & Ruel Vance Churchill 165)
- . ~(a .fi111ctio11 f is analytic nl all poi11ts i11terior to and 011 a simple (James Ward Brown & Ruel Vance Churchill 166)
- dosed co11to11r C. thc11 (James Ward Brown & Ruel Vance Churchill 166)
- jc.f(:) d: = 0. (James Ward Brown & Ruel Vance Churchill 166)
- ~fa .fi111ction f is analytic thmuglwut a simply co1111ccted domain D. (James Ward Brown & Ruel Vance Churchill 172)
- 1 c /(-:.)d-:.=0 (James Ward Brown & Ruel Vance Churchill 172)
- for e'el)’ dosed contour C lying in D. (James Ward Brown & Ruel Vance Churchill 172)
- For if C is simple and lies in D. lhc funccion (James Ward Brown & Ruel Vance Churchill 172)
- incersccls ilsclf a.finite number of limes. For if C is .f is analylic al each poinl inlerior lo and on C: a (James Ward Brown & Ruel Vance Churchill 172)
- . if C is closed bur inlerseccs ilself a hnile (James Ward Brown & Ruel Vance Churchill 172)
- ensures lhal equalion (I) holds. Furthennore. if C is closed bur inlerseccs ilself a hnile number of limes. ic consisls of a finile number of simple closed colllours. and lhe (James Ward Brown & Ruel Vance Churchill 172)
- 2. (James Ward Brown & Ruel Vance Churchill 176)
- 2. (James Ward Brown & Ruel Vance Churchill 187)
- 3. (James Ward Brown & Ruel Vance Churchill 187)
- 1. (James Ward Brown & Ruel Vance Churchill 194)
- 3. (James Ward Brown & Ruel Vance Churchill 213)
- 8. (James Ward Brown & Ruel Vance Churchill 213)
- 2. (James Ward Brown & Ruel Vance Churchill 222)
- 4. (James Ward Brown & Ruel Vance Churchill 223)
- 1. (James Ward Brown & Ruel Vance Churchill 254)
- 2. (James Ward Brown & Ruel Vance Churchill 254)
- 4. (James Ward Brown & Ruel Vance Churchill 264)
- 5. (James Ward Brown & Ruel Vance Churchill 264)
- 3 · (James Ward Brown & Ruel Vance Churchill 282)
- 4 . (James Ward Brown & Ruel Vance Churchill 282)
- 5 . (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)
- (James Ward Brown & Ruel Vance Churchill 282)

Last Annotation: 12/21/2012

- (F. William Lawvere & Stephen Hoel Schanuel 36)
- elemanın morfime dönüşmesi bu kısım önemli (F. William Lawvere & Stephen Hoel Schanuel 36)
- (F. William Lawvere & Stephen Hoel Schanuel 36)
- f=g<=> her xeleman A İÇİN f(x)=g(x) epiklik gibi önemlii (F. William Lawvere & Stephen Hoel Schanuel 40)
- (F. William Lawvere & Stephen Hoel Schanuel 72)

- say S, and (F. William Lawvere & Stephen Hoel Schanuel 36)
- If for each then point! -2-> I A, f a a = g a a, f = g. (Notice that and are points of Briefly, ‘if maps of sets agree at points f a a g a a B.) they are the same map.’ In doing the exercises you should remember that the two maps (F. William Lawvere & Stephen Hoel Schanuel 40)
- External diagrams (F. William Lawvere & Stephen Hoel Schanuel 45)
- external diagrams because they don’t show what’s going on inside. In Session 1 we met an external (F. William Lawvere & Stephen Hoel Schanuel 46)
- internal diagram (F. William Lawvere & Stephen Hoel Schanuel 47)
- Theorem (uniqueness of inverses): If f has both a retraction r and a section s then r = s. (F. William Lawvere & Stephen Hoel Schanuel 71)
- ‘reciprocal for the number 2’ meark a number satisfying ? x 2 = 1 (and therefore also 2 x ? = (F. William Lawvere & Stephen Hoel Schanuel 78)
- 66 (F. William Lawvere & Stephen Hoel Schanuel 83)
- Here is an example with finite sets. Let be the set of students in the classroom A f and the set of genders ‘female’ and `male’; and let be the obvious map that B f B A —› gives the gender. If is the set with elements and and is the map which C yes no, h answers the question ‘Did this student wear a hat today?’, then depending on who wore a hat today there are many possibilities for the map But since there are so few h. maps (F. William Lawvere & Stephen Hoel Schanuel 86)
- That is, has to be — what was the word? — an Only have a f endomap. endomaps chance to be and even then, endomaps are idempotent. Just to idempotent; most not be sure they’re not trying to trick me, I had better check: is even an p 0 q endomap? Well, its domain is — let’s see, was donefirst, so the domain of is the domain of q p o q which was And the codomain of is the codomain of which was ... yes, q, D. p o q p, (F. William Lawvere & Stephen Hoel Schanuel 130)

Last Annotation: 02/07/2019

- Setup for spectral sequences (Raoul Bott & Loring W. Tu 85)
- Spectral sequence of a filtered complex (Raoul Bott & Loring W. Tu 86)
- The Grassmanian (Raoul Bott & Loring W. Tu 154)

Last Annotation: 10/25/2019

- 1.7.5. Example: Proving the Snake Lemma. Consider the diagram (Ravi Vakil 61)

Last Annotation: 09/11/2019

Last Annotation: 06/08/2019

- 1.3 A three term recurrence. Now let’s do the Fibonacci recurrence Fn+1 =Fn+ F n _x. (n > 1; Fo = Fi = 1). (Herbert S. Wilf 11)
- mula that is to be solved by the method of generating functions. 1. Make sure that the set of values of the free variable (say n) for which the given recurrence relation is true, is clearly delineated. 2. Give a name to the generating function that you will look for, and write out that function in terms of the unknown sequence (e.g., call it A(x), and define it to be Ση>0 αηχη). 3. Multiply both sides of the recurrence by x n , and sum over all values of n for which the recurrence holds. 4. Express both sides of the resulting equation explicitly in terms of your generating function A(x). 5. Solve the resulting equation for the unknown generating function A(x). 6. If you want an exact formula for the sequence that is defined by the given recurrence relation, then attempt to get such a formula by expanding A(x) into a power series by any method you can think of. In particular, if A{x) is a rational function (quotient of two polynomials), then success will result from expanding in partial fractions and then handling each of the resulting terms separately. (Herbert S. Wilf 11)
- Given: a recurrence formula (Herbert S. Wilf 11)
- The Method Given: a recurrence formula that is to be sol (Herbert S. Wilf 11)
- It concerns the partitions of a set. By a partition of a set S we will mean a collection of nonempty, pairwise disjoint sets whose union is 5. Another name for a partition of S is an equivalence relation on S. The sets into which S is partitioned are called the classes of the partition. (Herbert S. Wilf 20)
- The problem that we will address in this example is to discover how many partitions of [n] into k classes there are. Let {£} denote this number. It is called the Stirling number of the second kind. (Herbert S. Wilf 20)
- Theorem 1.6.1. The exponential generating function of the Bell numbers is e 6 * 1 , i.e., the coefficient of xn/n in the power series expansion ofee _ 1 is the number of partitions of a set ofn elements. (Herbert S. Wilf 26)
- The next novel element of this story is the fact that we can go from generating functions to recurrence formulas, (Herbert S. Wilf 26)
- The x(d/dx) log operation (1) Take the logarithm of both sides of the equation. (2) Differentiate both sides and multiply through by x. (3) Clear the equation of fractions. (4) For each n, find the coefficients of xn on both sides of the equation and equate them. (Herbert S. Wilf 26)
- The next step, the differentiation, changes the log of the sum into a ratio of two sums, which is much nicer. (Herbert S. Wilf 27)
- Proposition. A formal power series f = Ση>ο αη%η has a reciprocal if and only if ÜQ φ 0. in that case the reciprocal is unique (Herbert S. Wilf 31)
- Proof. Let / have a reciprocal, namely 1 / / = X ] n > 0 bnxn. Then / ( I / / ) = 1 and according to (2.1.2), Co = 1 = a 0 6 0 , so ao τ^Ο. Further, in this case (2.1.2) tells us that for n > 1, cn = 0 = ^2k dkbn-k, from which we find bn = (-1/αο) Σ afc6 —k (^ > !)· (2.1.3) (Herbert S. Wilf 31)
- Thus the composition f(g(x)) of two formal power series is defined if and only if go = 0 or f is a polynomial (Herbert S. Wilf 32)
- We claim that in this case the inverse series exists if and only if the constant term is 0 and the coefficient of x is nonzero in the series / . Proposition. Let the formal power series f, g satisfy (2.1.5). Then f = fix + Î2X2 + -(ίιφ 0), and g = gxx + g2x2 + · · ■ (gi φ 0). Proof. Suppose that / = frxr - and g = gsxs H , where r, s > 0 and rs frgs Φ 0. Then f(g(x)) = x = frglx H , whence rs — 1, and r — s = 1, as claimed. ■ (Herbert S. Wilf 32)
- Therefore, to multiply the nth member of a sequence by n causes its ops generating function to be ’multiplied1 by x(d/dx), (Herbert S. Wilf 34)
- As an example, consider the recurrence (n + 1)α η+ ι = 3a n + 1 (n > 0; a0 = 1). If / is the opsgf of the sequence {an}o°> t n e n from Rule 1 and (2.2.2), 1 (Herbert S. Wilf 34)
- / = 3/ + (Herbert S. Wilf 34)
- Next suppose / ^ {αη}ο°· Then what generates the sequence {n2an}™7 Obviously we re-apply the multiply-by-n operator xD, so the answer is (xD)2f. In general, (xD)kf °JZ {nkan}n>0. (Herbert S. Wilf 34)
- Rule 2. If f ^Ζ {αη}ο°> and P is a polynomial, then P(xD)f °JZ {P(n)an}n>0. (Herbert S. Wilf 35)

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 wh