## Highlights - The diﬀerential ∂ : Cq \(X; R\) → Cq−1 \(X; R\) can be deﬁned in two ways\. The ﬁrst is purely algebraic, the second is geometric and involves the notion of the degree of a map f : S n → S n \. If you don’t know what the degree of such a map is, look it up\. If you know the deﬁnition of degree, then look up the diﬀerential-topological deﬁnition of degree for a smooth map f : S n → Sn\. (James Davis, Paul Kirk 19) - Deﬁnition 1\.4\. A cellular map f : X → Y is a continuous function between CW-complexes so that f \(Xq \) ⊂ Yq for all q\. A cellular map induces a chain map f∗ : C∗ \(X; R\) → C∗ \(Y ; R\), since f restricts to a map of pairs f : \(Xq , Xq−1 \) → \(Yq , Yq−1 \)\. Thus for every q, cellular homology is a functor (James Davis, Paul Kirk 20) - So for example, the circle S 1 has a cell structure with one 0-cell and one 1-cell\. The boundary map is trivial, so H1 \(S 1 \) ∼ = ∼ = Z\. A generator [S 1 ] ∈ H1 \(S \) is speciﬁed by taking the 1-cell which parameterizes the circle 1 in a counterclockwise fashion\. We can use this to deﬁne the Hurewicz map ρ : π1 \(X, x0 \) → H1 \(X; Z\) (James Davis, Paul Kirk 20) - Theorem 1\.6\. Suppose that X is path–connected\. Then the Hurewicz map ρ : π1 \(X, x0 \) → H1 \(X; Z\) is a surjection with kernel the commutator subgroup of π1 \(X, x0 \)\. Hence H1 \(X; Z\) is isomorphic to the abelianization of π1 \(X, x0 \)\. (James Davis, Paul Kirk 21) - 1\.1\.4\. Construction of the simplicial chain complex of a simplicial complex\. Deﬁnition 1\.7\. An \(abstract\) simplicial complex K is a pair \(V, S\) where V is a set and S is a collection of non-empty ﬁnite subsets of V satisfying: 1\. If v ∈ V then {v} ∈ S\. 2\. If τ ⊂ σ ∈ S and τ is non-empty, then τ ∈ S\. Elements of V are called vertices\. Elements of S are called simplices\. A q-simplex is an element of S with q + 1 vertices\. If σ ∈ S is a q-simplex we say dim \(σ\) = q\. (James Davis, Paul Kirk 21) - Deﬁnition 1\.8\. The geometric realization of a simplicial complex K is the quotient space |K| = σ∈S ∆ dim \(σ\) ∼ \. In other words, we take a geometric q–simplex for each abstract q–simplex of K, and glue them together\. (James Davis, Paul Kirk 22) - The identiﬁcations are given as follows (James Davis, Paul Kirk 22) - A triangulation of a topological space X is a homeomorphism from the geometric realization of a simplicial complex to X\. (James Davis, Paul Kirk 22) - It is not hard, using the acyclic models theorem, to show that the simplicial and cubical singular homology functors are naturally isomorphic (James Davis, Paul Kirk 22) - Deﬁnition 1\.9\. The tensor product of A and B is the R-module A ⊗R B deﬁned as the quotient F \(A × B\) R\(A × B\) where F \(A × B\) is the free R-module with basis A × B and R\(A × B\) the submodule generated by 1\. \(a1 + a2 , b\) − \(a1 , b\) − \(a2 , b\) 2\. \(a, b1 + b2 \) − \(a, b1 \) − \(a, b1 \) 3\. r\(a, b\) − \(ra, b\) 4\. r\(a, b\) − \(a, rb\)\. (James Davis, Paul Kirk 23) - One denotes the image of a basis element \(a, b\) in A ⊗R B by a ⊗ b\. Note that one has the relations 1\. \(a1 + a2 \) ⊗ b = a1 ⊗ b + a2 ⊗ b, 2\. a ⊗ \(b1 + b2 \) = a ⊗ b1 + a ⊗ b2 , 3\. \(ra ⊗ b\) = r\(a ⊗ b\) = \(a ⊗ rb\)\. (James Davis, Paul Kirk 23) - Any element of A ⊗ B can be expressed as a ﬁnite sum uct ai ⊗ bi , but it may not be possible to take n = 1, nor is the representation as a sum unique\. (James Davis, Paul Kirk 23) - The universal property of the tensor product is that this map is initial in the category of bilinear maps with domain A × B\. (James Davis, Paul Kirk 23) - Proposition 1\.10\. Given a R-bilinear map φ : A×B → M , there is unique R-module map φ̄ φ̄ : A ⊗R B → M so that φ̄ ◦ π = φ\. (James Davis, Paul Kirk 23) - Proposition 1\.10 is useful for deﬁning maps out of tensor products, and the following exercise indicates that this is the deﬁning property of tensor products\. (James Davis, Paul Kirk 24) - \. The basic properties of the tensor product are given by the next theorem\. Theorem 1\.11\. 1\. A ⊗ B ∼ = ∼ =B⊗A 2\. R ⊗ B ∼ = ∼ =B 3\. \(A ⊗ B\) ⊗ C ∼ = ∼ = A ⊗ \(B ⊗ C\) 4\. \(⊕α Aα \) ⊗ B ∼ = ∼ = ⊕ α \(Aα ⊗ B\) 5\. Given R-module maps f : A → C and g : B → D, there is an Rmodule map f ⊗ g : A ⊗ B → C ⊗ D so that a ⊗ b → f \(a\) ⊗ g\(b\)\. 6\. The functor − ⊗ M is right exact\. That is, given an R-module M , and an exact sequence A f f − →B g g − → C → 0, the sequence (James Davis, Paul Kirk 24) - the sequence A ⊗ M f ⊗Id f ⊗Id −−−→ B ⊗ M g⊗Id g⊗Id −−−→ C ⊗ M → 0 is exact\. (James Davis, Paul Kirk 24) - Example 1\.12\. Let M be an abelian group\. Applying properties 5 and 2 of Theorem 1\.11 we see that if we tensor the short exact sequence 0 → Z ×n ×n −−→ Z → Z/n → 0 by M we obtain the exact sequence M ×n ×n −−→ M → Z/n ⊗Z M → 0\. Notice that Z/n ⊗Z M ∼ = ∼ = M/nM and that the sequence is not short exact if M has torsion whose order is not relatively prime to n\. Thus − ⊗ M is not left exact\. (James Davis, Paul Kirk 25) - 1\.2\.2\. Adjoint functors\. Note that a R-bilinear map β : A × B → C is the same as an element of HomR \(A, HomR \(B, C\)\) (James Davis, Paul Kirk 25) - Proposition 1\.14 \(Adjoint Property of Tensor Products\)\. There is an isomorphism of R-modules HomR \(A ⊗R B, C\) ∼ = ∼ = HomR \(A, HomR \(B, C\)\), natural in A, B, C given by φ ↔ \(a → \(b → φ\(a ⊗ b\)\)\)\. (James Davis, Paul Kirk 26) - This is more elegant than the universal property for three reasons: It is a statement in terms of the category of R-modules, it gives a reason for the duality between tensor product and Hom, and it leads us to the notion of adjoint functor\. Deﬁnition 1\.15\. \(Covariant\) functors F : C → D and G : D → C form an adjoint pair if there is a 1-1 correspondence MorD \(F c, d\) ←→ MorC \(c, Gd\), for all c ∈ Ob C and d ∈ Ob D, natural in c and d\. The functor F is said to be the left adjoint of G and G is the right adjoint of F \. (James Davis, Paul Kirk 26) - The adjoint property says that for any R-module B, the functors − ⊗ R B : R-MOD → R-MOD and HomR \(B, −\) : R-MOD → R-MOD form an adjoint pair\. Here R-MOD is the category whose objects are Rmodules and whose morphisms are R-maps\. (James Davis, Paul Kirk 26) - random functor may not have a left \(or right\) adjoint, but if it does, the adjoint is unique up to natural isomorphism (James Davis, Paul Kirk 26) - Hence taking duals deﬁnes a contravariant functor from the category of R-modules to itself\. (James Davis, Paul Kirk 26) - The following computational facts may help with Exercise 8\. 1\. HomR \(R, M \) ∼ = ∼ = M\. 2\. HomR \(⊕α Aα , M \) ∼ = ∼ = α HomR \(Aα , M \)\. 3\. HomR \(A, α Mα \) ∼ = ∼ = α HomR \(A, Mα \)\. (James Davis, Paul Kirk 27) - The starting observation is that the singular \(or cellular\) homology functor is a composite of two functors, the singular complex functor S∗ : { spaces, cts\. maps } → { chain complexes, chain maps } and the homology functor H∗ :{chain complexes, chain maps}→{graded R-modules, homomorphisms}\. The strategy is to place interesting algebraic constructions between S∗ and H∗ ; i\.e\. to use functors {Chain Complexes} → {Chain Complexes} to construct new homology invariants of spaces (James Davis, Paul Kirk 27) - Deﬁnition 1\.16\. Taking the homology of C∗ ⊗ M yields the homology of C∗ with coeﬃcients in M : H∗ \(C∗ ; M \) = ker ∂ : C∗ ⊗ M → C∗ ⊗ M Im ∂ : C∗ ⊗ M → C∗ ⊗ M \. Applying this to the singular complex of a space leads to the following deﬁnition\. Deﬁnition 1\.17\. The homology of S∗ \(X; R\) ⊗ M is called the singular homology of X with coeﬃcients in the R-module M and is denoted by H∗ \(X; M \)\. (James Davis, Paul Kirk 28) - Recall that for a chain complex \(C∗ , ∂\), a cycle is an element of ker ∂ and a boundary is an element of Im ∂\. The terminology for cochain complexes is obtained by using the “co” preﬁx: Deﬁnition 1\.20\. A cocycle is an element in the kernel of δ and a coboundary is an element in the image of δ\. (James Davis, Paul Kirk 29) - The Kronecker pairing on the homology and cohomology of a space should be thought of as an analogue \(in fact it is a generalization\) of integrating a diﬀerential n-form along an n-dimensional submanifold\. (James Davis, Paul Kirk 30) - The map H n \(C∗ ; R\) → HomR \(Hn \(C∗ ; R\), R\) need not be injective nor surjective\. (James Davis, Paul Kirk 30) - Suppose X is a smooth manifold\. Let Ω q \(X\) be the vector space of diﬀerential q-forms on a manifold\. Let d : Ω q \(X\) → Ωq+1 \(X\) be the exterior derivative\. Then \(Ω ∗ \(X\), d\) is an R-cochain complex, whose cohomology is denoted by H ∗ DR \(Ω ∗ \(X\), d\) and is called the DeRham cohomology of X\. This gives geometric analogues: q-form and q-cochain, d and δ, closed form and cocycle, exact form and coboundary\. (James Davis, Paul Kirk 31) - DeRham’s theorem states that the DeRham cohomology of a manifold X is isomorphic to the singular cohomology H ∗ \(X; R\)\. More precisely, let Sqsmooth qsm \(X; R\) be the free R-module generated by smooth singular simplices σ : ∆ q → X\. There is the chain map S ∗smooth \(X; R\) → S∗ \(X; R\) given by inclusion and the cochain map Ω \(X\) → S∗smooth \(X; R\) given by integrating a q-form along a q-chain\. DeRham’s theorem follows from the fact that both maps are chain homotopy equivalences (James Davis, Paul Kirk 31) - 1\.4\.1\. Relative cohomology\. Recall that the relative singular chain complex of a pair \(X, A\) is deﬁned by taking the chain groups Sq \(X, A\) = Sq \(X\)/Sq \(A\)\. (James Davis, Paul Kirk 31) - Lemma 1\.24\. Given a short exact sequence of R-modules 0 −→ A α α −→ B β β −→ C −→ 0, show that α splits if and only if β splits\. \(If either of these possibilities occur, we say the short exact sequence splits\.\) Show that in this case B ∼ = ∼ = A ⊕ C\. (James Davis, Paul Kirk 32) - Deﬁnition 1\.29\. An \(ordinary\) cohomology theory is a contravariant functor H ∗ : { \(space, subspace\) pairs, continuous maps of pairs } → { graded R-modules, homomorphisms }\. (James Davis, Paul Kirk 34) - for each pair \(X, A\) so that the sequence of Corollary 1\.26 is exact\. \(Long exact sequence of a pair\) 2\. If f, g : \(X, A\) → \(Y, B\) are homotopic maps, then g ∗ = f ∗ : H q \(Y, B\) → H q \(X, A\)\. \(Homotopy invariance\) 3\. If U ⊂ X, U ⊂ Int A, then H q \(X, A\) → H q \(X − U, A − U \) is an isomorphism\. \(Excision\) 4\. If pt is a point, H q \(pt\) = 0 when q = 0\. \(Dimension Axiom\) (James Davis, Paul Kirk 35) - 1\. There exist natural connecting homomorphisms δ : H q \(A\) → H q+1 \(X, A\) (James Davis, Paul Kirk 35) - Čech cohomology theory is another theory that satisﬁes the axioms \(at least for the subcategory of pairs of compact spaces\), but the Č Čech cohomology of the topologist’s sine curve is not isomorphic to the singular cohomology (James Davis, Paul Kirk 35) - \. Thus the axioms do not determine the cohomology of space (James Davis, Paul Kirk 35) - cohomology\. Thus the axioms do not determine the cohomology of space\. However they do for ﬁnite CW-complexes\. An informal way of saying this is that the proof that cellular cohomology equals singular cohomology uses only the axioms (James Davis, Paul Kirk 35) - Theorem 1\.31 \(Uniqueness\)\. Let H ∗ and ̂ Ĥ ∗ be contravariant functors from the category {pairs of ﬁnite CW-complexes, cellular maps} to {graded R-modules, homomorphisms} satisfying the Eilenberg-Steenrod Axioms\. Let pt be a point\. 1\. If H 0 \(pt\) ∼ = ∼ = ̂ Ĥ 0 \(pt\), then there is a natural isomorphism of functors H ∗ → Ĥ ∗ \. 2\. Any natural transformation H ∗ → Ĥ ∗ inducing an isomorphism for a point, is an isomorphism for all CW-complexes\. (James Davis, Paul Kirk 36) - There are also many functors from spaces to R-modules for which the dimension axiom of Eilenberg and Steenrod does not hold\. These are called generalized \(co\)homology theories\. (James Davis, Paul Kirk 36) - 1\.6\.1\. Cellular approximation theorem\. Recall that a cellular map f : X → Y is a map between CW-complexes which satisﬁes f \(X n \) ⊂ Y n for all n\. The cellular approximation theorem says that any map between CWcomplexes is homotopic to a cellular map (James Davis, Paul Kirk 36) - We have already seen that the functors − ⊗R M , HomR \(M, −\), and HomR \(−, M \) are not exact in general\. For example taking R = Z, M = Z/2, and the short exact sequence 0 → Z ×2 ×2 −→ Z → Z/2 → 0, (James Davis, Paul Kirk 38) - Exercise 20\. If F is a free module, show that − ⊗R F and HomR \(F, −\) are exact functors\. Show by example that HomR \(−, F \) need not be exact\. The idea of homological algebra is to ﬁnd natural functors which measure the failure of a functor to preserve short exact sequences (James Davis, Paul Kirk 39) - \(A ﬁrst stab at this for − ⊗R M might be to take the kernel of A ⊗ M → B ⊗ M as the value of this functor\. Unfortunately, this does not behave nicely with respect to morphisms\.\) (James Davis, Paul Kirk 39) - Before we embark on the proof of this theorem, we prove that these axioms characterize the functors Tor and Ext\. Theorem 2\.3 \(Uniqueness\)\. Any two functors satisfying T1\), T2\), and T3\) are naturally isomorphic\. Any two functors satisfying E1\), E2\), and E3\) are naturally isomorphic\. (James Davis, Paul Kirk 40) - The technique of the above proof is called dimension shifting, and it can be useful for computations (James Davis, Paul Kirk 41) - Proposition 2\.4\. Let R be a commutative ring and a ∈ R a non-zerodivisor \(i\.e\. ab = 0 implies b = 0\)\. Let M be an R-module\. Let M/a = M/aM and a M = {m ∈ M |am = 0}\. Then 1\. R/a ⊗ M ∼ = ∼ M/a, = 2\. Tor1 \(R/a, M \) ∼ = ∼ = aM , 3\. Hom\(R/a, M \) ∼ = ∼ = aM , 4\. Ext 1 \(R/a, M \) ∼ = ∼ = M/a\. (James Davis, Paul Kirk 41) - Proposition 2\.5\. 1\. If R is a ﬁeld, then Tor n R n \(−, −\) and ExtR n \(−, −\) are zero for n > 0\. 2\. If R is a P\.I\.D\., then Tor n R n \(−, −\) and ExtR n \(−, −\) are zero for n > 1\. Proof\. (James Davis, Paul Kirk 42) - Corollary 2\.7\. The functors Tor R n \(A, B\) and Tor R n \(B, A\) are naturally isomorphic\. (James Davis, Paul Kirk 43) - Tor and Ext are higher derived versions of ⊗R and Hom, so they have analogous properties\. For example we oﬀer without proof: 1\. Tor n R n \(⊕α Aα , B\) ∼ = ∼ = ⊕ α Tor n R n \(Aα , B\), 2\. Extn R \(⊕α Aα , B\) ∼ = ∼ = α Extn R R \(Aα , B\), and 3\. Ext n R \(A, α Bα \) ∼ = ∼ = α Ext n R \(A, Bα \)\. (James Davis, Paul Kirk 43) - Much of what we say can be done in the more general setting of abelian categories, these are categories where the concept of exact sequence makes sense \(for example the category of sheaves or the category of representations of a Lie algebra\) provided there are “enough projectives” or “enough injectives” in the category\. (James Davis, Paul Kirk 43) - Recall, if F is a free module over R, A, B are R-modules, and F A ✲ B ✲ 0 α ❄ β ✲ α ✲ is a diagram with α onto, then there exists a γ : F → A so that F A ✲ B ✲ 0 α ❄ β ✠ γ ✲ α ✲ commutes\. We say “the problem F A α✲ B ✲ 0 ❄ β p p p p p p✠p γ ✲ ✲ has a solution (James Davis, Paul Kirk 44) - Deﬁnition 2\.8\. An R-module P is called projective if for any A, B, α, β with α onto, the problem P A α✲ B ✲ 0 ❄ β p p p p p ✠ p✠p γ ✲ ✲ has a solution γ\. Lemma 2\.9\. An R-module P is projective if and only if there exists an R-module Q so that P ⊕ Q is a free R-module (James Davis, Paul Kirk 44) - Thus projective modules generalize free modules by isolating one of the main properties of free modules\. Furthermore the deﬁnition of a projective module is purely in terms of arrows in R-MOD, and hence is more elegant than the deﬁnition of a free module\. On the other hand they are less familiar\. Exercise 22\. Let P be a projective module\. 1\. Any short exact sequence 0 → A → B → P → 0 is split\. 2\. If P is ﬁnitely generated there is a ﬁnitely generated Q so that P ⊕ Q is free\. (James Davis, Paul Kirk 45) - Proposition 2\.10\. 1\. Any module over a ﬁeld is projective\. 2\. Any projective module over a P\.I\.D\. is free (James Davis, Paul Kirk 45) - Proof\. All modules over a ﬁeld are free, hence projective\. A projective module P is a submodule of the free module P ⊕ Q, and for P\.I\.D\.’s submodules of free modules are free\. (James Davis, Paul Kirk 46) - Note that R must be complicated, i\.e\. not a ﬁeld nor a P\.I\.D\. For example, if R = Z/6, then P = Z/2 is a projective module\. \(To see this, use the Chinese remainder theorem Z/6 = Z/2 × Z/3\)\. Here is a more interesting example, related to K-theory\. Let R be the ring of continuous functions on the circle, R = C 0 \(S 1 , R\)\. Let E → S 1 be the twisted real line bundle over S 1 \(so E = open Möbius band\) Then as vector bundles E = ∼ = S ∼ 1 × R, but E ⊕ E ∼ = ∼ = S 1 × R2 \. So, if M = C 0 \(E\) \(continuous sections of E\), M is not free \(why?\), but M ⊕ M ∼ = ∼ = C 0 \(S 1 , R\) ⊕ C 0 \(S 1 , R\) = R ⊕ R\. Thus M is projective\. (James Davis, Paul Kirk 46) - One of the quantities measured by the functor K0 of algebraic K-theory is the diﬀerence between projective and free modules over a ring\. See Chapter 11 for another aspect of algebraic K-theory, namely the geometric meaning of the functor K1 \. (James Davis, Paul Kirk 46) - Corollary 2\.11\. For a projective module P , for n > 0, and for any module M , both Tor R n \(P, M \) and Ext n R \(P, M \) vanish\. (James Davis, Paul Kirk 46) - Deﬁnition 2\.12\. An R-module M is called injective if M A ✛β B ✛ 0 pp p pp p p p p pp ✒ p pp ✒ ✛ ✻ α ✛ has a solution for all A, B, α, β \(with β injective\) (James Davis, Paul Kirk 47) - Theorem 2\.13\. An abelian group A is injective if and only if is divisible, \(i\.e\. the equation nx = a has a solution x ∈ A for each n ∈ Z, a ∈ A\.\) (James Davis, Paul Kirk 47) - Thus some examples of injective abelian groups are Q and Q/Z\. \(Note that a quotient of a divisible group is divisible, hence injective\.\) (James Davis, Paul Kirk 47) - Lemma 2\.18 \(Splicing lemma\)\. If the sequences A → B α α − → C → 0 and 0→C β β − → D → E are exact, then A → B β◦α β◦α −−→ D → E is exact\. (James Davis, Paul Kirk 49) - Comment about Commutative Algebra\. A Dedekind Domain is a commutative domain \(no zero divisors\) in which every module has a projective resolution of length 1\. Equivalently submodules of projective modules are projective\. A P\.I\.D\. is a Dedekind domain (James Davis, Paul Kirk 49) - submodules of projective modules are projective\. A P\.I\.D\. is a Dedekind domain\. From the point of view of category theory, they are perhaps more natural than P\.I\.D\.’s (James Davis, Paul Kirk 49) - If ζn = e 2πi/n is a primitive n-th root of unity, then Z[ζn ] is a Dedekind domain\. Projective modules \(in fact ideals\) which are not free ﬁrst arise at n = 23\. Non-free ideals are what makes Fermat’s Last Theorem so hard to prove (James Davis, Paul Kirk 49) - A commutative Noetherian ring R has height equal to n \(ht\(R\) = n\) if the longest chain of non-trivial prime ideals in R has length n: 0 ⊂ P1 ⊂ · · · ⊂ Pn ⊂ R\. (James Davis, Paul Kirk 50) - The homological dimension of R, hdim\(R\), is the least upper bound on the length of projective resolutions for all ﬁnitely generated modules over R\. The homological dimension of a ﬁeld is 0 and a Dedekind domain is 1\. If a ring has homological dimension n, then any module M has a projective resolution with Pk = 0 for k > n\. The numbers ht\(R\) and hdim\(R\) are related\. For a large class of rings \(regular rings\) they are equal\. (James Davis, Paul Kirk 50) - Exercise 27\. An R-module F is called ﬂat if − ⊗R F is exact\. (James Davis, Paul Kirk 51) - free module is ﬂat and clearly a summand of a ﬂat module is ﬂat, so projectives are ﬂat\. There are modules which are ﬂat, but not projective; show that Q is a ﬂat but not projective Z-module\. In fact over a P\.I\.D\. a module is ﬂat if and only if it is torsion free (James Davis, Paul Kirk 51) - Tor can be computed using a ﬂat resolution rather than a projective one (James Davis, Paul Kirk 51) - Deﬁnition 2\.21\. A projective chain complex P∗ = {· · · → P2 → P1 → P0 } is a chain complex where all the modules Pi are projective\. An acyclic chain complex C∗ = {· · · → C2 → C1 → C0 } is a chain complex where Hi \(C∗ \) = 0 for all i > 0\. (James Davis, Paul Kirk 51) - Theorem 2\.22 \(Fundamental lemma of homological algebra\.\)\. Let P∗ be a projective chain complex and C∗ be an acyclic chain complex over a ring R\. Then given a homomorphism ϕ : H0 \(P∗ \) → H0 \(C∗ \), there is a chain map f∗ : P∗ → C∗ inducing ϕ on H0 \. Furthermore, any two such chain maps are chain homotopic\. (James Davis, Paul Kirk 51) - Corollary 2\.23\. Any two deleted projective resolutions of M are chain homotopy equivalent\. (James Davis, Paul Kirk 51) - Lemma 2\.26 \(Horseshoe lemma\)\. Let 0 → A → B → C → 0 be a short exact sequence of R-modules\. Let PA and PC be deleted projective resolutions of A and C\. Then there exists a deleted projective resolution PB of B, ﬁtting into a short exact sequence of chain complexes 0 → PA → PB → PC → 0 which induces the original sequence on H0 \. (James Davis, Paul Kirk 54) - Tor\(A, B\) ∼ = ∼ = torsion\(A\) ⊗Z torsion\(B\) (James Davis, Paul Kirk 57) - Ext\(A, B\) ∼ = ∼ = torsion\(A\) ⊗Z B\. (James Davis, Paul Kirk 58) - Theorem 2\.29 \(universal coeﬃcient theorem for cohomology\)\. Let R be a principal ideal domain\. Suppose that M is a module over R, and \(C∗ , ∂\) is a free chain complex over R \(i\.e\. each Cq is a free R-module\)\. Then the sequence 0 → ExtR \(Hq−1 \(C∗ \), M \) → H q \(C∗ ; M \) → Hom\(Hq \(C∗ \), M \) → 0 (James Davis, Paul Kirk 59) - Proof of Theorem 2\.29\. There is a short exact sequence of graded, free R-modules 0 → Z∗ i i − → C∗ ∂ ∂ − → B∗ → 0 (James Davis, Paul Kirk 59) - where Zq denotes the q-cycles and Bq denotes the q-boundaries\. (James Davis, Paul Kirk 60) - Applying the universal coeﬃcient theorem to the singular or cellular complexes of a space or a pair of spaces one obtains the following\. Corollary 2\.32\. If \(X, A\) is a pair of spaces A ⊂ X, R a P\.I\.D\., M a module over R, then for each q the sequence 0 → ExtR \(Hq−1 \(X, A; R\), M \) → H q \(X, A; M \) → Hom\(Hq \(X, A; R\),M \) → 0 is short exact, natural, and splits \(though the splitting is not natural\)\. (James Davis, Paul Kirk 61) - Corollary 2\.31\. If R is a ﬁeld, M is a vector space over R, and C∗ is a chain complex over R, then H q \(C∗ ; M \) ∼ ∼ = Hom\(Hq \(C∗ \), M \)\. Moreover the Kronecker pairing is non-degenerate (James Davis, Paul Kirk 61) - Exercise 31\. Let f : RP 2 → S 2 be the map pinching the 1-skeleton to a point\. Compute the induced map on Z and Z/2 cohomology to show the splitting is not natural (James Davis, Paul Kirk 61) - Then for space X whose homology is ﬁnitely generated in every dimension \(e\.g\. a ﬁnite CW -complex\), the universal coeﬃcient theorem shows that H q \(X\) ∼ = ∼ = free\(Hq \(X\)\) ⊕ torsion\(Hq−1 \(X\)\)\. (James Davis, Paul Kirk 61) - ∗ = Hom\(A, Z\) and the torsion dual A ∼ = ∼ = Hom\(A, Q/Z\), then the universal coeﬃcient theorem says that H q \(X\) ∼ = ∼ = Hq \(X\) ∗ ⊕ \(torsion\(Hq−1 \(X\)\)\)\. The right hand side is then a contravariant functor in X but the isomorphism is still not natural\. (James Davis, Paul Kirk 62) - Theorem 2\.33\. If R is a P\.I\.D\., M is a ﬁnitely generated R-module, and C∗ is a free chain complex over R then there is a split short exact sequence 0 → H q \(C∗ \) ⊗ M → H q \(C∗ ; M \) → TorR 1 R 1 \(H q+1 \(C∗ \), M \) → 0\. Notice the extra hypothesis that M be ﬁnitely generated in this statement\. (James Davis, Paul Kirk 62) - The following universal coeﬃcient theorem measures the difference between ﬁrst tensoring a complex with a module M and then passing to homology versus ﬁrst passing to homology and then tensoring with M \. (James Davis, Paul Kirk 62) - Theorem 2\.34 \(universal coeﬃcient theorem for homology\)\. Suppose that R is a P\.I\.D\., C∗ a free chain complex over R, and M a module over R\. Then there is a natural short exact sequence\. 0 → Hq \(C∗ \) ⊗ M → Hq \(C∗ ⊗ M \) → TorR 1 \(Hq−1 \(C∗ \), M \) → 0 which splits, but not naturally\. (James Davis, Paul Kirk 62) - There is another universal coeﬃcient theorem for homology \(see [36, pg\. 248] for the proof\)\. It addresses the question of how a diﬀerent version of the Kronecker pairing fails to pass to a perfect pairing on \(co\)homology\. (James Davis, Paul Kirk 63) - Theorem 2\.36\. Let R be a P\.I\.D\., C∗ a free chain complex over R such that Hq \(C∗ \) is ﬁnitely generated for each q, and let M be an R-module\. Then the sequence 0 → Ext1R 1R \(H q+1 \(C∗ \), M \) → Hq \(C∗ ; M \) α α → Hom\(H q \(C∗ \), M \) → 0 is short exact, natural, and splits\. (James Davis, Paul Kirk 63) - In particular if X is a compact smooth manifold, by the above corollary and DeRham cohomology we see the q-th Betti number is the dimension of the real vector space of closed q-forms modulo exact q-forms\. (James Davis, Paul Kirk 64) - 1\. A graded R-module A∗ can be thought ofeither a collection of Rmodules {Ak }k∈Z , or as a module A = fe k Ak with a direct sum decomposition\. 2\. A homomorphism of graded R-modules is a element of k Hom\(Ak , Bk \)\. 3\. The tensor product of graded R-modules A∗ and B∗ is the graded R-module \(A∗ ⊗ B∗ \)n = p+q=n \(Ap ⊗ Bq \)\. (James Davis, Paul Kirk 66) - The cellular chain complex C∗ \(X × Y \) can be identiﬁed with \(i\.e\. is isomorphic to\) the tensor product C∗ \(X\) ⊗ C∗ \(Y \)\. (James Davis, Paul Kirk 68) - Theorem 3\.3 \(Künneth exact sequence\)\. Suppose C∗ , D∗ are chain complexes over a P\.I\.D\. R, and suppose Cq is a free R-module for each q\. Then there is a natural exact sequence 0 → ⊕ p+q=n Hp \(C∗ \)⊗Hq \(D∗ \) × alg −−−→Hn \(C∗ ⊗D∗ \)→ ⊕ p+q=n Tor R \(Hp \(C∗ \), Hq−1 \(D∗ \)\)→0 which splits \(non-naturally\)\. (James Davis, Paul Kirk 68) - The natural transformations A and B determine chain homotopy equivalences A : S∗ \(X × Y \) → S∗ \(X\) ⊗ S∗ \(Y \) and B : S∗ \(X\) ⊗ S∗ \(Y \) → S∗ \(X × Y \) for any pair of spaces X and Y \. We will call these maps the Eilenberg-Zilber maps\. (James Davis, Paul Kirk 70) - The confusing, abstract, but important point is that A and B are not canonical, but only natural\. That is, they are obtained by the method of acyclic models, and so constructed step by step by making certain arbitrary choices\. However, these choices are made consistently for all spaces (James Davis, Paul Kirk 70) - It is harder to imagine a candidate for the reverse map A : S∗ \(X × Y \) → S∗ \(X\) ⊗ S∗ \(Y \), but one can do this explicitly using projections to X and Y – the “Alexander-Whitney diagonal approximation”\. (James Davis, Paul Kirk 70) - However, even if one constructs the maps A and B explicitly, they will only be chain homotopy equivalences, not isomorphisms; S∗ \(X × Y \) is simply bigger than S∗ \(X\) ⊗ S∗ \(Y \) (James Davis, Paul Kirk 70) - 3\.3\.1\. The homology cross product and the Künneth formula\. Exercise 32 implies that the natural map × alg : Hp X ⊗ Hq Y → Hp+q \(S∗ X ⊗ S∗ Y \) given on the chain level by [a] ⊗ [b] → [a ⊗ b] is well-deﬁned\. Denote by B∗ the isomorphism induced by the Eilenberg-Zilber map on homology, so B∗ : H∗ \(S∗ \(X\) ⊗ S∗ \(Y \)\) → H∗ \(S∗ \(X × Y \)\) = H∗ \(X × Y \)\. Composing ×alg with B∗ , we obtain × : Hp X ⊗ Hq Y → Hp+q \(X × Y \)\. Deﬁnition 3\.5\. If α ∈ Hp X, β ∈ Hq Y , the image of α ⊗ β under this map is called the homology cross product of α and β, and is denoted by α × β\. (James Davis, Paul Kirk 71) - The Eilenberg-Zilber theorem has the following important consequence\. Theorem 3\.6 \(Künneth formula\)\. If R is a P\.I\.D\., there exists a split exact sequence 0 → n ⊕ p=0 Hp X ⊗ Hn−p Y → Hn \(X × Y \) → n−1 ⊕ p=0 Tor\(Hp X, Hn−1−p Y \) → 0\. The ﬁrst map is given by cross products\. (James Davis, Paul Kirk 71) - Corollary 3\.7\. If R is a ﬁeld, then H∗ \(X × Y \) = H∗ \(X\) ⊗ H∗ \(Y \)\. (James Davis, Paul Kirk 71) - The Künneth formula implies that if R is a P\.I\.D\., α × β = 0 if α = 0 and β = 0\. (James Davis, Paul Kirk 71) - Deﬁnition 3\.8\. If a ∈ H p X, b ∈ H q Y , the image of a ⊗ b under the composite H p X ⊗ H q Y ×alg ×alg A −−−→ H p+q \(\(S∗ X ⊗ S∗ Y \)∗ \) − ∗ A∗ −→ H ∗ \(X × Y \) map is called the cohomology cross product of a and b, and is denoted by a × b\. 3\.3\.3\. The cup product\. Combining the the diagonal map ∆ : X → X × X, x → \(x, x\) with the cross product leads to the important cup product: Deﬁnition 3\.9\. If If a ∈ H p X, b ∈ H q X, then the cup product of a and b is deﬁned by a ∪ b = ∆∗ \(a × b\) ∈ H p+q X\. (James Davis, Paul Kirk 72) - Deﬁnition 3\.11\. A diagonal approximation τ is a chain map τ : S∗ X → S ∗ X ⊗ S ∗ X for every space X, so that 1\. τ \(σ\) = σ ⊗ σ for every 0-simplex σ\. 2\. τ is natural with respect to continuous maps of a spaces\. (James Davis, Paul Kirk 74) - If A is an Eilenberg-Zilber map and ∆ : X → X × X is the diagonal map, then τ = A ◦ ∆∗ is a diagonal approximation\. Thus we can rephrase the deﬁnition of the cup product a ∪ b = τ ∗ \(a ×alg b\)\. (James Davis, Paul Kirk 74) - 3\.3\.4\. The cap product\. Recall that the Kronecker pairing is a natural bilinear evaluation map \(sometimes called “integration” by analogy with the deRham map\) , : S ∗ X × S∗ X → R deﬁned for a ∈ S q X, z ∈ Sp X by a, z = a\(z\) if p = q, 0 otherwise\. (James Davis, Paul Kirk 77) - This pairing can be extended to a “partial evaluation” or “partial integration” map E : S ∗ X ⊗ S∗ X ⊗ S ∗ X → S ∗ X by “evaluating the ﬁrst factor on the last factor”, i\.e\. E\(a ⊗ z ⊗ w\) = a\(w\) ⊗ z\. (James Davis, Paul Kirk 77) - Exercise 37\. Let a, b ∈ H ∗ X and z ∈ H∗ X\. Show that 1\. a, b ∩ z = a ∪ b, z\. 2\. a ∩ \(b ∩ z\) = \(a ∪ b\) ∩ z\. (James Davis, Paul Kirk 78) - Thus the cap product makes the homology H∗ \(X\) a module over the ring H ∗ \(X\)\. (James Davis, Paul Kirk 78) - Corollary 3\.16\. The cap product descends to a well deﬁned product ∩ : H q X × Hp+q X → Hp X \([α], [z]\) → [α ∩ z] after passing to \(co\)homology\. (James Davis, Paul Kirk 78) - 3\.3\.5\. The slant product\. We next introduce the slant product which bears the same relation to the cross product as the cap product does to the cup product \(this could be on an SAT test\)\. (James Davis, Paul Kirk 78) - Often one distinguishes between internal products which are deﬁned in terms of one space X \(such as the cup and cap products\) and external products which involve the product of two spaces X × Y \. (James Davis, Paul Kirk 79) - It is nevertheless often useful to work on the chain level, since there is subtle homotopy-theoretic information contained in the singular complex which leads to extra structure such as such as Steenrod operations and Massey products\. (James Davis, Paul Kirk 79) - Deﬁnition 3\.19\. The Alexander-Whitney map A : S∗ \(X × Y \) → S∗ \(X\) ⊗ S∗ \(Y \) is the natural transformation given by the formula A\(σ\) = p+q=n p \(pX ◦ σ\) ⊗ \(pY ◦ σ\)q \. (James Davis, Paul Kirk 80) - Exercise 40\. Using the Alexander-Whitney diagonal approximation, 1\. prove that S ∗ \(X; R\) is an associative ring with unit 1 represented by the cochain c ∈ S 0 \(X; R\) = Hom\(S0 \(X\); R\) which takes the value 1 on every singular 0-simplex in X\. (James Davis, Paul Kirk 80) - We have already seen that cohomology is an associative and graded commutative ring with unit in Theorem 3\.13\. However, the methods used there cannot be used to show that S ∗ \(X\) is an associative ring; in fact it is not for a random choice of Eilenberg-Zilber map A\. (James Davis, Paul Kirk 81) - The Alexander-Whitney map is a particularly nice choice of EilenbergZilber map because it does give an associative ring structure on S ∗ \(X\)\. This ring structure, alas, is not \(graded\) commutative \(Steenrod squares give obstructions to its being commutative\) (James Davis, Paul Kirk 81) - Notice that the deRham cochain complex of diﬀerential forms on a smooth manifold is graded commutative, since diﬀerential forms satisfy a ∧ b = ±b ∧ a\. It is possible to give a natural construction of a commutative chain complex over the rationals which gives the rational homology of a space; this was done using rational diﬀerential forms on a simplicial complex by Sullivan\. This fact is exploited in the subject of rational homotopy theory [14]\. On the other hand it is impossible to construct a functor from spaces to commutative, associative chain complexes over Z which gives the integral homology of a space\. (James Davis, Paul Kirk 81) - See Vick’s book [41] for a nice example of computing the cohomology ring of the torus directly using the Alexander-Whitney diagonal approximation\. (James Davis, Paul Kirk 81) - The cup product of an q-dimensional cocycle with a q-dimensional cocycle generalizes the Kronecker pairing in the following sense\. Proposition 3\.21\. For α ∈ H q X and z ∈ Hq X, α, z = <∗ \(α ∩ z\)\. (James Davis, Paul Kirk 82) - disappears when passing to cohomology\. We now give a formal argument\. We begin with some algebraic observations\. Suppose \(X, A\) and \(Y, B\) are two pairs of spaces\. Then S∗ X S∗ A ⊗ S∗ Y S∗ B ∼ = ∼ = S∗ X ⊗ S ∗ Y S∗ X ⊗ S ∗ B + S∗ A ⊗ S∗ Y \. (James Davis, Paul Kirk 83) - Since a chain map between free chain complexes inducing an isomorphism on homology is a chain homotopy equivalence, this is equivalent to requiring that the inclusion map induces an isomorphism on homology\. (James Davis, Paul Kirk 83) - Let V = H k \(M ,Z\)/T \. So \(V ,· \) is an inner product space over Z\. This inner product space can have two kinds of symmetry\. (James Davis, Paul Kirk 87) - If dim M = 2k, then H k \(M ;Z\)/T × H k \(M ;Z\)/T → Z called the intersection form of M \. It is well-deﬁned and unimodular over Z, i\.e\. has determinant equal to ±1\. Let V = H k \(M ,Z\)/T \. So \(V (James Davis, Paul Kirk 87) - Case 1\. k is odd\. Thus dim M = 4F + 2\. Then v · w = −w · v for v, w ∈ V , so \(V ,· \) is a skew-symmetric and unimodular inner product space over Z\. (James Davis, Paul Kirk 87) - \(V, · \) has matrix 0 1 −1 0 0 1 −1 0 \. \. \. \(all other entries zero\) in this basis\. Such a basis is called a “symplectic basis”\. The closed surface of genus r is an example; describe a symplectic basis geometrically (James Davis, Paul Kirk 87) - basis geometrically\. Hence unimodular skew-symmetric pairings over Z are classiﬁed by their rank\. In other words, the integer intersection form of a 4k − 2-dimensional manifold M contains no more information than the dimension of H 2k+1 \(M \)\. (James Davis, Paul Kirk 87) - There are 3 invariants of such unimodular symmetric forms: 1\. The rank of \(V, · \) is the rank of V as a free abelian group\. 2\. The signature of \(V, · \) is the diﬀerence of the number of positive eigenvalues and the number of negative eigenvalues in any matrix (James Davis, Paul Kirk 87) - representation of \(V, ·\)\. \(The eigenvalues of a symmetric real matrix are all real\.\) (James Davis, Paul Kirk 88) - Exercise 50\. Show that although the eigenvalues of Q are not well deﬁned, their signs are well-deﬁned, so that the signature is well-deﬁned\. \(This is often called Sylvester’s Theorem of Inertia\) (James Davis, Paul Kirk 88) - The type \(odd or even\) of \(V, ·\) is deﬁned to be even if and only if v · v is even for all v ∈ V \. Otherwise the type is said to be odd\. (James Davis, Paul Kirk 88) - The form \(V, · \) is called deﬁnite if the absolute value of its signature equals its rank; i\.e\. the eigenvalues of Q are either all positive or all negative\. The main result about unimodular integral forms is the following\. For a proof see e\.g\. [29]\. Theorem 3\.27\. If \(V, · \), \(W, · \) are two unimodular, symmetric, indeﬁnite forms over Z, then V and W are isometric \(i\.e\. there exists an isomorphism V → W preserving the inner product\) if and only if they have the same rank, signature, and type\. (James Davis, Paul Kirk 88) - In fact, any odd indeﬁnite form is equivalent to ⊕ \(1\)⊕ m \(−1\), and any even indeﬁnite form is equivalent to (James Davis, Paul Kirk 88) - ⊕ 0 1 1 0 ⊕ m E8 where E8 = (James Davis, Paul Kirk 88) - The classiﬁcation of deﬁnite forms is not known\. It is known that: 1\. for each rank, there are ﬁnitely many isomorphism types\. 2\. If \(V ,· \) is deﬁnite and even, then sign\(V ,· \) ≡ 0 Mod 8\. (James Davis, Paul Kirk 88) - 3\. There are 1 even, positive deﬁnite rank 8 forms\. 2 ” rank 16 ” 24 ” rank 24 ” 710 51 ” rank 40 ” (James Davis, Paul Kirk 89) - Deﬁnition 3\.28\. The signature, sign M , of a compact, oriented 4k-manifold without boundary M is the signature of its intersection form H 2k \(M ; Z\)/T × H 2k \(M ; Z\)/T → Z The following sequence of exercises introduce the important technique of bordism in geometric topology\. (James Davis, Paul Kirk 89) - Exercise 52\. 1\. Let M be a closed, odd–dimensional manifold\. Show that the Euler characteristic χ\(M \) = 0\. Prove it for non-orientable manifolds, too\. 2\. Let M be a closed, orientable manifold of dimension 4k + 2\. Show that χ\(M \) is even\. 3\. Let M be a closed, oriented manifold of dimension 4k\. Show that the signature sign M is congruent to χ\(M \) mod 2\. 4\. Let M be the boundary of a compact manifold W \. Show χ\(M \) is even\. 5\. Let M be the boundary of an compact, oriented manifold W and suppose the dimension of M is 4k\. Show sign M = 0\. 6\. Give examples of manifolds which are and are not boundaries\. (James Davis, Paul Kirk 89) - We have seen that even–dimensional manifolds admit intersection forms on the free part of their middle dimensional cohomology\. For odd-dimensional manifolds one can construct the linking form on the torsion part of the “middle dimensional” cohomology as well (James Davis, Paul Kirk 89) - Exercise 53\. If M is a compact, closed, oriented manifold of dimension n, show that the torsion subgroups of H p \(M \) and H n−p+1 \(M \) are isomorphic (James Davis, Paul Kirk 89) - The zig-zag lemma gives a long exact sequence in cohomology δ i · · · → H q−1 \(X; Q/Z\) − → H q \(X; Z\) − → H q \(X; Q\) → · · · \. Exercise 54\. Prove that the map δ : H q−1 \(X; Q/Z\) → H q \(X; Z\) maps onto the torsion subgroup T of H q \(X; Z\)\. (James Davis, Paul Kirk 90) - \(The map δ is a Bockstein homomorphism; see Section 10\.4\) The bilinear map Q/Z × Z → Q/Z, \(a, b\) → ab is non-degenerate, in fact induces an isomorphism Q/Z ⊗ Z → Q/Z\. (James Davis, Paul Kirk 90) - Exercise 55\. Prove that the linking pairing of M T × T → Q/Z deﬁned by \(a, b\) → δ −1 \(a\) ∪ b, [M ] is well-deﬁned\. Here δ −1 \(a\) means any element z in H k−1 \(M ; Q/Z\) with δ\(z\) = a\. Show that this pairing is skew symmetric (James Davis, Paul Kirk 90) - metric if dim\(M \) = 4k − 1\. It is a little bit harder to show that this pairing is non-singular \( (James Davis, Paul Kirk 90) - symmetric if dim\(M \) = 4k + 1 and sym- (James Davis, Paul Kirk 90) - Deﬁnition 4\.1\. A topological group G acts on a space X if there is a group homomorphism G → Homeo\(X\) such that the “adjoint” G×X →X \(g, x\) → g\(x\) is continuous\. We will usually write g · x instead of g\(x\)\. (James Davis, Paul Kirk 92) - ∈ X is the set Gx = {g · x|g ∈ G}\. The orbit space or quotient space X/G is the quotient space X/ , with the equivalence relation x g · x\. The ﬁxed set is X G = {x ∈ X|g · x = x for all g ∈ G}\. (James Davis, Paul Kirk 92) - The orbit of a point x ∈ X is (James Davis, Paul Kirk 92) - An action is called free if g\(x\) = x for all x ∈ X and for all g = e\. (James Davis, Paul Kirk 92) - An action is called eﬀective if the homomorphism G → Homeo\(X\) is injective\. (James Davis, Paul Kirk 93) - A variant of this deﬁnition requires the homomorphism G → Homeo\(X\) to be continuous with respect to the compact-open topology on Homeo\(X\), or some other topology, depending on what X is \(for example, one could ∞ take the C ∞ topology on Diﬀ\(X\) if X is a smooth manifold\)\. (James Davis, Paul Kirk 93) - Deﬁnition 4\.2\. Let G be a topological group acting eﬀectively on a space F \. A ﬁber bundle E over B with ﬁber F and structure group G is a map p : E → B together with a collection of homeomorphisms {ϕ : U × F → p −1 \(U \)} for open sets U in B \(ϕ is called a chart over U \) such that (James Davis, Paul Kirk 93) - In a ﬁber bundle, the map ψϕ,ϕ must have a very special form, namely 1\. The homeomorphism ψϕ,ϕ \(u, −\) : F → F is not arbitrary, but is given by the action of an element of G, i\.e\. ψϕ,ϕ \(u, f \) = g · f for some g ∈ G independent of f \. The element g is denoted by θϕ,ϕ \(u\)\. 2\. The topology of G is integrated into the structure by requiring that θ ϕ,ϕ : U → G be continuous\. (James Davis, Paul Kirk 94) - The requirement that G act eﬀectively on F implies that the functions θ ϕ,ϕ : U → G are unique\. Although we have included the requirement that G acts eﬀectively of F in the deﬁnition of a ﬁber bundle, there are certain circumstances when we will want to relax this condition, particularly when studying liftings of the structure group, for example, when studying local coeﬃcients\. (James Davis, Paul Kirk 94) - It is not hard to see that a locally trivial bundle is the same thing as a ﬁber bundle with structure group Homeo\(F \)\. (James Davis, Paul Kirk 95) - One subtlety about the topology is that the requirement that G be a topological group acting eﬀectively on F says only that the homomorphism G → Homeo\(F \) is injective, but the inclusion G → Homeo\(F \) need not be an embedding, nor even continuous\. (James Davis, Paul Kirk 95) - This is a useful method of understanding bundles since it relates them to \(Čech\) cohomology\. Cohomologous cochains deﬁne isomorphic bundles, and so equivalence classes of bundles over B with structure group G can be (James Davis, Paul Kirk 95) - identiﬁed with H 1 \(B; G\) \(this is one starting point for the theory of characteristic classes; (James Davis, Paul Kirk 96) - G need not be abelian \(and so what does H 1 \(B; G\) mean? (James Davis, Paul Kirk 96) - f F has the discrete topology, any locally trivial bundle over B with ﬁber F is a covering space; (James Davis, Paul Kirk 96) - If F h (James Davis, Paul Kirk 96) - 4\.3\. Examples of ﬁber bundles (James Davis, Paul Kirk 96) - 4\.3\.2\. Bundles over S 2 \. For every integer n ≥ 0, we can construct an S 1 bundle over S 2 with structure group SO\(2\); n is called the Euler number of the bundle\. (James Davis, Paul Kirk 97) - or n ≥ 1, deﬁne a 3-dimensional lens space L3n n = S 3 /Zn , where the action is given by letting the generator of Zn on act on S 3 ⊂ C2 by \(z1 , z2 \) → \(ζ n z 1 , ζn z2 \) (James Davis, Paul Kirk 97) - Deﬁne the S 1 -bundle with Euler number n ≥ 1 by p : L3n 3n → S 2 = C ∪ ∞ by [z1 , z2 ] → z1 /z2 \. (James Davis, Paul Kirk 97) - When n = 1 we obtain the famous Hopf bundle S 1 → S 3 → S 2 \. For n > 1 the Hopf map S 3 → S 2 factors through the quotient map S 3 → L3n 3n , and the ﬁbers of the bundle with Euler number n are S 1 /Zn which is again homeomorphic to S 1 \. (James Davis, Paul Kirk 97) - For n = 2, the lens space is just real projective space RP 3 \. D (James Davis, Paul Kirk 97) - For n = 0, we have the product bundle p : S 2 × S 1 → S 2 \. For n ≥ 1, deﬁne a 3-dimensional lens space L3n n = S 3 /Zn (James Davis, Paul Kirk 97) - Show that the spaces S\(T S 2 \), SO\(3\), and RP 3 are all homeomorphic (James Davis, Paul Kirk 97) - 4\.3\.3\. Clutching\. Suppose a topological group G acts on a space F \. Let X be a space and let ΣX be the unreduced suspension of X, (James Davis, Paul Kirk 97) - Then given a map β : X → G, deﬁne (James Davis, Paul Kirk 98) - This bundle is called the bundle over ΣX with clutching function β : X → G ⊂Aut\(F \)\. (James Davis, Paul Kirk 98) - Clutching provides a good way to describe ﬁber bundles over spheres\. For X a CW-complex, all bundles over ΣX arise by this clutching construction\. This follows from the fact that any ﬁber bundle over a contractible CW-complex is trivial \(this can be proven using obstruction theory\) (James Davis, Paul Kirk 98) - 4\.3\.4\. Local coeﬃcients and other structures\. An important type of ﬁber bundle is the following\. Let A be a group and G a subgroup of the automorphism group Aut\(A\)\. Then any ﬁber bundle E over B with ﬁber A and structure group G has the property that each ﬁber p−1 {b} has a group structure\. This group is isomorphic to A, but the isomorphism is not canonical in general\. We have already run across an important case of this, namely vector bundles, where A = Rn and G = GLn \(R\)\. (James Davis, Paul Kirk 98) - In particular, if A is a abelian group with the discrete topology, then p : E → B is a covering space and is called a system of local coeﬃcients on B\. T (James Davis, Paul Kirk 98) - The basic principle at play here is if the structure group preserves a certain structure on F , then every ﬁber p−1 {b} has this structure\. (James Davis, Paul Kirk 98) - Principal bundles are special cases of ﬁber bundles, but nevertheless can be used to construct any ﬁber bundle\. Conversely any ﬁber bundle determines a principal bundle\. A principal bundle is technically simpler, since the ﬁber is just F = G with a canonical action\. (James Davis, Paul Kirk 99) - Let G be a topological group\. It acts on itself by left translation\. G → Homeo\(G\), g → \(x → gx\)\. Deﬁnition 4\.3\. A principal G-bundle over B is a ﬁber bundle p : P → B with ﬁber F = G and structure group G acting by left translations\. (James Davis, Paul Kirk 99) - Proposition 4\.4\. If p : P → B is a principal G-bundle, then G acts freely on P on the right with orbit space B\. (James Davis, Paul Kirk 99) - As a familiar example, any regular covering space p : E → B is a principal G-bundle with G = π1 B/p∗ π1 E\. Here G is given the discrete topology\. In particular, the universal covering B̃ → B of a space is a principal π1 Bbundle\. A non-regular covering space is not a principal G-bundle\. (James Davis, Paul Kirk 100) - Exercise 63\. Any free \(right\) action of a ﬁnite group G on a \(Hausdorﬀ\) space E gives a regular cover and hence a principal G-bundle E → E/G\. (James Davis, Paul Kirk 100) - Theorem 4\.5\. Suppose that X is a compact Hausdorﬀ space, and G is a compact Lie group acting freely on X\. Then the orbit map X → X/G is a principal G-bundle\. (James Davis, Paul Kirk 100) - As an application note that if a topological group G acts on spaces F and F , and if p : E → B is a ﬁber bundle with ﬁber F and structure group G, then one can use the transition functions from p to deﬁne a ﬁber bundle p : E → B with ﬁber F and structure group G with exactly the same transition functions\. This is called changing the ﬁber from F to F (James Davis, Paul Kirk 100) - This can be useful because the topology of E and E may change (James Davis, Paul Kirk 100) - With the second incarnation of the bundle the twisting becomes revealed in the homotopy type, because the total space of the ﬁrst bundle has the homotopy type of S 2 , while the total space of the second has the homotopy type of the sphere bundle S\(T S 2 \) and hence of RP 3 according to Exercise 59\. (James Davis, Paul Kirk 101) - We call this principal G-bundle the principal G-bundle underlying the ﬁber bundle p : E → B with structure group G\. (James Davis, Paul Kirk 101) - A fundamental case of changing ﬁbers occurs when one lets the ﬁber F be the group G itself, with the left translation action\. (James Davis, Paul Kirk 101) - Deﬁnition 4\.6\. Let p : P → B be a principal G-bundle\. Suppose G acts on the left on a space F , i\.e\. an action G × F → F is given\. Deﬁne the Borel construction P ×G F to be the quotient space P × F/ ∼ where \(x, f \) ∼ \(xg, g −1 f \)\. (James Davis, Paul Kirk 101) - Thus principal bundles are more basic that ﬁber bundles, in the sense that the ﬁber and its G-action are explicit, namely G acting on itself by left translation\. Moreover, any ﬁber bundle with structure group G is associated to a principal G-bundle by specifying an action of G on a space F \. Many properties of bundles become more visible when stated in the context of principal bundles\. (James Davis, Paul Kirk 102) - An important set of examples comes from this construction by starting with the tangent bundle of a smooth manifold M \. The principal bundl (James Davis, Paul Kirk 102) - Deﬁne a space F \(E\) to be the space of frames in E, so that a point in F \(E\) is a pair \(b, f \) where b ∈ B and f = \(f1 , · · · , fn \) is a basis for the vector space p−1 \(b\) (James Davis, Paul Kirk 102) - Prove that q : F \(E\) → B is a principal GL\(n, R\)-bundle, and that E = F \(E\) ×GL\(n,R\) Rn (James Davis, Paul Kirk 102) - We say q : E ×G F → B is the ﬁber bundle associated to the principal bundle p : E → B via the action of G on F \. (James Davis, Paul Kirk 102) - F \(T M \) is called the frame bundle of M \. (James Davis, Paul Kirk 103) - Any representation of GL\(n, R\) on a vector space V gives a vector bundle with ﬁber isomorphic to V \. Important p representations include the alternating representations GL\(n, R\) → po p \(Hom\(R n , R\)\) (James Davis, Paul Kirk 103) - from which (James Davis, Paul Kirk 103) - called the frame bundle of M \. Any representation of GL\(n, R\) on a vector space V gives a vector bundle with ﬁber isomorphic to V \. Important p representations include the alternating representations GL\(n, R\) → po p \(Hom\(R n , R\)\) from which one obtains the vector bundles of diﬀerential p-forms over M (James Davis, Paul Kirk 103) - Recall that a local coeﬃcient system is a ﬁber bundle over B with ﬁber A and structure group G where A is a \(discrete\) abelian group and G acts via a homomorphism G → Aut\(A\) (James Davis, Paul Kirk 103) - Lemma 4\.7\. Every local coeﬃcient system over a path-connected \(and semilocally simply connected\) space B is of the form A ✲ B̃ ×π1 B A q q ❄ B i\.e\., is associated to the principal π1 B-bundle given by the universal cover B̃ of B where the action is given by a homomorphism π1 B → Aut\(A\)\. (James Davis, Paul Kirk 103) - In other words the group G ⊂ Aut\(A\) can be replaced by the discrete group π1 B\. (James Davis, Paul Kirk 103) - In some circumstances, given a subgroup H of G and a ﬁber bundle p : E → B with structure group G, one can view the bundle as a ﬁber bundle with structure group H\. When this is possible, we say the structure group can be reduced to H\. (James Davis, Paul Kirk 104) - Proposition 4\.8\. Let H be a topological subgroup of the topological group G\. Let H act on G by left translation\. Let q : Q → B be a principal H-bundle\. Then G ✲ Q ×H G q ❄ B is a principal G-bundle (James Davis, Paul Kirk 104) - Deﬁnition 4\.9\. Given a principal G-bundle p : E → B we say the structure group G can be reduced to H for some subgroup H ⊂ G if there exists a principal H-bundle Q → B and a commutative diagram Q ×H G r ✲ E ◗ ✑ ✑ ◗ ✑ s ◗ ✑ ✰ B so that the map r is G-equivariant\. For a ﬁber bundle, we say the structure group reduces if the structure group of the underlying principal bundle reduces\. (James Davis, Paul Kirk 104) - In this more general context, for example, Lemma 4\.7 states that any ﬁber bundle over B with discrete ﬁber can have its structure group reduced to π1 B\. (James Davis, Paul Kirk 104) - Deﬁnition 4\.10\. A real vector bundle is called orientable if its structure group can be reduced to the subgroup GL+ \(n, R\) of matrices with positive determinan (James Davis, Paul Kirk 105) - For example, a smooth manifold is orientable if and only if its tangent bundle is orientable (James Davis, Paul Kirk 105) - Deﬁnition 4\.11\. A morphism of ﬁber bundles with structure group G and ﬁber F from E → B to E → B is a pair of continuous maps f˜ : E → E and f : B → B so that the diagram f˜ f˜ E ✲ E E ❄ ❄ B ✲ B f f commutes and so that for each chart (James Davis, Paul Kirk 105) - One important type of ﬁber bundle map is a gauge transformation\. This is a bundle map from a bundle to itself which covers the identity map of the base, i\.e\. the following diagram commutes\. (James Davis, Paul Kirk 106) - Deﬁnition 4\.12\. Suppose that a ﬁber bundle p : E → B with ﬁber F and structure group G is given, and that f : B → B is some continuous function\. Deﬁne the pullback of p : E → B by f to be the space f ∗ \(E\) = {\(b , e\) ∈ B × E | p\(e\) = f \(b \)}\. (James Davis, Paul Kirk 106) - The following exercise shows that any map of ﬁber bundles is given by a pullback\. (James Davis, Paul Kirk 106) - 4\.7\.1\. Fiber bundles over paracompact bases are ﬁbrations\. State and prove the theorem of Hurewicz \(Theorem 6\.8\) which says that a map f : E → B with B paracompact is a ﬁbration \(see Deﬁnition 6\.7\) provided that B has an open cover {Ui } so that f : f −1 \(Ui \) → Ui is a ﬁbration for each i\. In particular, any locally trivial bundle over a paracompact space is a ﬁbration\. (James Davis, Paul Kirk 107) - 4\.7\.2\. Classifying spaces\. For any topological group G there is a space BG and a principal G-bundle EG → BG so that given any paracompact space B, the pullback construction induces a bijection between the set [B, BG] of homotopy classes of maps from B to BG and isomorphism classes of principal G-bundles over B (James Davis, Paul Kirk 107) - Show that given any action of G on F , any ﬁber bundle E → B with structure group G and ﬁber F is isomorphic to the pullback f ∗ \(EG ×G F \) where f : B → BG classiﬁes the principal G-bundle underlying E → B\. Use this theorem to deﬁne characteristic classes for principal bundles (James Davis, Paul Kirk 108) - When studying the homotopy theory of non-simply connected spaces, one is often led to consider an action of the fundamental group on some abelian group\. Local coeﬃcient systems are a tool to organize this information\. (James Davis, Paul Kirk 110) - There are two approaches to constructing the complexes giving the homology and cohomology of a space with local coeﬃcients\. The ﬁrst is more algebraic, and takes the point of view that the fundamental chain complex associated to a space X is the singular \(or cellular\) complex of the universal cover X̃, viewed as a chain complex over the group ring Z[π1 X]\. From this point of view local coeﬃcients are nothing more than modules over the group ring Z[π1 X]\. (James Davis, Paul Kirk 110) - coeﬃcient of a (James Davis, Paul Kirk 110) - The second approach is more topological; one takes a local coeﬃcient system over X \(i\.e\. a ﬁber bundle over X whose ﬁbers are abelian groups and whose transition functions take values in the automorphisms of the group\) and deﬁne a chain complex by taking the chains to be formal sums of singular simplices \(or cells\) such that the coeﬃcient of a simplex is an element in the ﬁber over that simplex \(hence the terminology local coeﬃcients\)\. (James Davis, Paul Kirk 110) - Deﬁnition 5\.1\. The group ring Zπ is a ring associated to a group π\. Additively it is the free abelian group on π, i\.e\., elements are \(ﬁnite\) linear combinations of the group elements m 1 g 1 + · · · + mk gk mi ∈ Z, gi ∈ π\. (James Davis, Paul Kirk 111) - wo examples of group rings \(with their standard notation\) are Z[Z] = Z[t, t−1 ] = {a−j t−j + · · · + a0 + · · · + ak tk | an ∈ Z} \(this ring is called the ring of Laurent polynomials\) and Z[Z/2] = Z[t]/\(t2 − 1\) = {a + bt | a, b ∈ Z}\. (James Davis, Paul Kirk 111) - We will work with modules over Zπ\. (James Davis, Paul Kirk 111) - Let A be an abelian group and ρ : π → AutZ \(A\) be a homomorphism\. \(The standard terminology is to call either ρ or A a representation of π\.\) The representation ρ endows A with the structure of a left Zπ-module by taking the action (James Davis, Paul Kirk 111) - Thus a representa (James Davis, Paul Kirk 112) - sentation of a group π on an abelian group is the same thing as a Zπ-module\. (James Davis, Paul Kirk 112) - Deﬁnition 5\.2\. If R is a ring \(possibly non-commutative\), M is a right R-module, and N is a left R-module \(sometimes one writes MR and R N \), then the tensor product M ⊗R N is an abelian group satisfying the adjoint property Hom Z \(M ⊗R N, A\) ∼ = ∼ = HomR \(M, HomZ \(N, A\)\) for any abelian group A\. The corresponding universal property is that there is a Z-bilinear map φ : (James Davis, Paul Kirk 112) - The tensor product is constructed by taking the free abelian group on M × N and modding out by the expected relations\. Elements of M ⊗R N are denoted by m i ⊗ n i \. The relation mr ⊗ n = m ⊗ rn holds\. (James Davis, Paul Kirk 112) - Exercise 72\. Compute the abelian group Z+ ⊗Z[Z/2] Z− \(see Exercise 71\)\. (James Davis, Paul Kirk 112) - The starting point in the algebraic construction of homology with local coeﬃcients is the observation that the singular chain complex of the universal cover of a space is a right Zπ-module\. (James Davis, Paul Kirk 112) - → X be the universal cover of X, with its usual right π1 XLet X action obtained by identifying π with the group of covering transformations (James Davis, Paul Kirk 112) - Deﬁnition 5\.3\. Given a Zπ-module A, form the tensor product S ∗ \(X; A\) = S∗ \(X\)⊗ Zπ A\. X\)⊗ This is a chain complex whose homology is called the homology of X with local coeﬃcients in A and is denoted by H∗ \(X; A\)\. (James Davis, Paul Kirk 113) - Notice that (James Davis, Paul Kirk 113) - t since the ring Zπ is non-commutative \(except if π is abelian\), the tensored chain complex only has the structure of a chain complex over Z, not Zπ\. Thus the homology group H∗ \(X; A\) is only a Z-module\. (James Davis, Paul Kirk 113) - If the Zπ-module is speciﬁed by a representation ρ : π1 X →Aut\(A\) for some abelian group A, and we wish to emphasize the representation, we will sometimes embellish A with the subscript ρ and write H∗ \(X; Aρ \) for the homology with coeﬃcients in A\. It is also common to call H∗ \(X; Aρ \) the homology of X twisted by ρ : π1 X →Aut\(A\)\. (James Davis, Paul Kirk 113) - into a left Zπ-module by the \(standard\) proceThus transform S∗ \(X\) dure: g · z := z · g −1 \. (James Davis, Paul Kirk 113) - 5\.2\. Examples and basic properties The \(ordinary\) homology and cohomology groups are just special cases of the homology and cohomology with local coeﬃcients corresponding to twisting by the trivial representations ρ as we now show\. (James Davis, Paul Kirk 113) - Deﬁnition 5\.4\. Given a left Zπ-module A form the cochain complex S ∗ \(X; A\) = HomZπ \(S∗ \( X A\)\. X\), \(This means the set of group homomorphisms f : S∗ \( X → A which satisfy X\) f \(rz\) = rf \(z\) for all r ∈ Zπ and z ∈ S∗ \( X X\)\.\) The cohomology of this complex is called the cohomology of X with local coeﬃcients in A, and is denoted by H ∗ \(X; A\)\. (James Davis, Paul Kirk 113) - If the module A is deﬁned by a representation ρ : π1 X →Aut\(A\) for an abelian group A, the cohomology with local coeﬃcients may be denoted by H ∗ \(X; Aρ \) and is often called the cohomology of X twisted by ρ\. (James Davis, Paul Kirk 113) - In other words, the homology with local coeﬃcients given by the tautological representation ρ : π → Aut\(Zπ\) ρ\(g\) = \( mh h → mh gh\) with \(untwisted\) Z coeﬃcients\. equals the homology of X (James Davis, Paul Kirk 114) - These examples and the two exercises show that the \(untwisted\) homology of any cover of X with any coeﬃcients can be obtained as a special case of the homology of X with appropriate local coeﬃcients\. (James Davis, Paul Kirk 115) - One might ask whether the same facts hold for cohomology\. They do not without some modiﬁcation\. If A = Zπ, then the cochain complex HomZπ \(S∗ X, A\) is not in general isomorphic to X, HomZ \(S∗ X, Z\) and Z\)\. It turns out that if X is compact so H \(X; Zπ\) is not equal to H k \(X; k H k \(X; Zπ\) ∼ Z\), the compactly supported cohomology of X\. = Hck \(X; (James Davis, Paul Kirk 115) - → X is the If p : X universal cover, then X inherits a CW-structure from X – the cells of X are the path components of the inverse images of cells of X (James Davis, Paul Kirk 115) - The following exercises are important in gaining insight into what information homology with local coeﬃcients captures\. Exercise 76\. Compute the cellular chain complex C∗ \(S 1 \) as a Z[t, t−1 ]S module\. Compute Hk \(S 1 ; Aρ \) and H k 1 \(S ; Aρ \) for any abelian group A and any homomorphism ρ : π1 S 1 = Z → Aut\(A\)\. (James Davis, Paul Kirk 115) - = Exercise 77\. Let ρ : π1 \(RP n \) − → Z/2 = Aut\(Z\)\. Compute Hk \(RP n ; Zρ \) k n and H \(RP ; Zρ \) and compare to the untwisted homology and cohomology\. Exercise 78\. Let p and q be a relatively prime pair of integers and denote by L\(p,q\) the 3-dimensional Lens space L\(p,q\) = S 3 /\(Z/p\), where Z/p = t acts on S 3 ⊂ C2 via q t\(Z,W \) = \(ζZ,ζ q W \) \(ζ = e2πi/p \)\. Let ρ : Z/p → Aut\(Z/n\) = Z/\(n − 1\) for n prime\. Compute H k \(L\(p,q\); \(Z/n\)ρ \) and H k \(L\(p,q\); \(Z/n\)ρ \)\. (James Davis, Paul Kirk 116) - Exercise 79\. Let K be the Klein bottle\. Compute Hn \(K; Zρ \) for all twistings ρ of Z \(i\.e\. all ρ : π1 K → Z/2 = Aut\(Z\)\)\. (James Davis, Paul Kirk 116) - An important application of local coeﬃcients is its use in studying the algebraic topology of non-orientable manifolds\. (James Davis, Paul Kirk 116) - For example, consider RP n for n even\. The orientation double cover is S n ; the deck transformation reverses orientation\. For RP n for n odd, the orientation double cover is a disjoint union of two copies of RP n , oriented with the opposite orientations\. If M is a connected manifold, deﬁne the orientation character or the ﬁrst Stiefel–Whitney class w : π1 M → {±1} (James Davis, Paul Kirk 116) - by w[γ] = 1 if γ lifts to a loop in the orientation double cover and w[γ] = −1 if γ lifts to a path which is not a loop\. Intuitively, w[γ] = −1 if going around the loop γ reverses the orientation\. M is orientable if and only if w is trivial\. (James Davis, Paul Kirk 116) - Corollary 5\.6\. Any manifold with H 1 \(M ; Z/2\) = 0 is orientable\. (James Davis, Paul Kirk 117) - Notice that Aut\(Z\) = {±1} and so the orientation character deﬁnes a representation w : π1 X → Aut\(Z\)\. The corresponding homology and cohomology Hk \(X; Zw \), H k \(X; Zw \) is called the homology and cohomology of X twisted by the orientation character w, or with local coeﬃcients in the orientation sheaf\. (James Davis, Paul Kirk 117) - The Poincaré duality theorem \(Theorem 3\.26\) has an extension to the non-orientable situation\. (James Davis, Paul Kirk 117) - The cap products in Theorem 5\.7 are induced by the bilinear maps on coeﬃcients Z × Zw → Zw and Zw × Zw → Z (James Davis, Paul Kirk 117) - the local coeﬃcient system itself, i\.e\. the ﬁber bundle with discrete abelian group ﬁbers (James Davis, Paul Kirk 118) - The homology of the chain complex \(Sk \(X; E\), ∂\) is called the homology with local coeﬃcients in E\. (James Davis, Paul Kirk 119) - Here is the example involving orientability of manifolds, presented in terms of local coeﬃcients instead of the orientation representation\. Let M be an n-dimensional manifold\. Deﬁne a local coeﬃcient system E → M , by (James Davis, Paul Kirk 119) - setting E= x∈M Hn \(M, M − {x}\)\. A basis for the topology of E is given by (James Davis, Paul Kirk 120) - Then E → X is a local coeﬃcient system with ﬁbers Hn \(M, M − {x}\) ∼ = ∼ = Z, called the orientation sheaf of M \. \(Note the orientation double cover MO is the subset of E corresponding to the subset ±1 ∈ Z\) (James Davis, Paul Kirk 120) - In particular, there is a Mayer-Vietoris sequence for homology with local coeﬃcients which gives a method for computing\. Some care must be taken in using this theorem because local coeﬃcients do not always extend (James Davis, Paul Kirk 122) - Given a homomorphism ρ : π1 \(X − U \) → Aut\(A\) and an inclusion of pairs \(X − U, B − U \) → \(X, B\) excision holds \(i\.e\. the inclusion of pairs induces isomorphisms in homology with local coeﬃcients\) only if ρ extends over π1 \(X\)\. In particular the morphism of local coeﬃcients must \(exist and\) be isomorphisms on ﬁbers\. (James Davis, Paul Kirk 122) - Proposition 5\.14\. Let B be a path connected CW-complex, π = π1 \(B\) and V a Zπ-module\. then 1\. Let Vπ denote the quotient of V by the subgroup generated by the elements {v − γ · v | v ∈ V, γ ∈ π1 B } \(the group Vπ is called the group of coinvariants\)\. Then H0 \(B; V \) ∼ = ∼ = Vπ \. (James Davis, Paul Kirk 122) - 2\. Let V π π denote the subgroup of V consisting of elements ﬁxed by π, i\.e\. V π π = {v ∈ V | γ · v = v for all γ ∈ π} \(the group V π is called the group of invariants\)\. Then H 0 \(B; V \) ∼ = ∼ = V π π \. (James Davis, Paul Kirk 123) - 5\.5\.1\. The Hopf degree theorem\. This theorem states that the degree of a map f : S n → S n determines its homotopy class\. See Theorems 6\.67 and 8\.5\. Prove the theorem using the simplicial approximation theorem (James Davis, Paul Kirk 123) - Colimits and limits are important categorical constructions in algebra and topology\. Special cases include the notions of a cartesian product, a disjoint union, a pullback, a pushout, a quotient space X/A, and the topology of a CW-complex (James Davis, Paul Kirk 124) - If all the Xi are subsets of a set A and if all the fi ’s are inclusions of subspaces show that the colimit can be taken to X = ∪Xi \. The topology is given by saying U ⊂ X is open if and only if U ∩ Xi is open for all i\. Thus such a colimit can be thought of as some sort of generalization of a union (James Davis, Paul Kirk 124) - Consider the categories {· → · → · → · → · · · }, {· · · ← · ← · ← · ←}, {· → · ← ·}, {· ← · → ·}, {· ·}, and discuss how colimits and limits over these categories give the above colimit, the above limit, the pullback, the pushout, the cartesian product and the disjoint union\. (James Davis, Paul Kirk 124) - Now let I be a category and let T be the category of topological spaces\. Let X : I → T , i → Xi be a functor, so you are given a topological space for every object i, and the morphisms of I give oodles of maps between the Xi satisfying the same composition laws as the morphisms in I do\. Deﬁne colim I Xi and lim I Xi (James Davis, Paul Kirk 124) - Finally suppose that Y is a CW-complex and Y0 ⊂ Y1 ⊂ Y2 ⊂ Y3 ⊂ · · · is an increasing union of subcomplexes whose union is Y \. Show that Hn Y = colim i→∞ Hn \(Y i i \)\. Deﬁne Milnor’s lim 1 \(see [25] or [42]\) and show that there is an exact sequence 0 → lim ← 1 H n−1 \(Yi \) → H n Y → lim ← H n \(Yi \) → 0\. (James Davis, Paul Kirk 125) - 6\.1\. Compactly generated spaces Given a map f : X×Y → Z, we would like to topologize the set of continuous functions C\(Y, Z\) in such way that f is continuous if and only if the adjoint ˜ f˜ : X → C\(Y, Z\), f˜\(x\)\(y\) = f \(x, y\) is continuous (James Davis, Paul Kirk 126) - continuous\. Here are three examples: 1\. We would like an action of a topological group G × Z → Z to correspond to a continuous function G → Homeo\(Z\), where Homeo\(Z\) is given the subspace topology inherited from C\(Z, Z\)\. 2\. We would like a homotopy f : I × Y → Z to correspond to a path ˜ f˜ : I → C\(Y, Z\) of functions\. 3\. The evaluation map C\(Y, Z\) × Y → Z, \(f, y\) → f \(y\) should be continuous\. \(Is the evaluation map an adjoint?\) (James Davis, Paul Kirk 126) - Unfortunately, such a topology on C\(Y, Z\) is not possible, even for Hausdorﬀ topological spaces, unless you bend your point of view (James Davis, Paul Kirk 126) - The category of compactly generated spaces is a framework which permits one to make such constructions without worrying about these technical issues (James Davis, Paul Kirk 127) - Deﬁnition 6\.1\. A topological space X is said to be compactly generated if X is Hausdorﬀ and if a subset A ⊂ X is closed if and only if A ∩ C is closed for every compact C ⊂ X\. (James Davis, Paul Kirk 127) - 3\. CW-complexes with ﬁnitely many cells in each dimension\. (James Davis, Paul Kirk 127) - Examples of compactly generated spaces include: 1\. locally compact Hausdorﬀ spaces \(e\.g\. manifolds\), 2\. metric spaces, and (James Davis, Paul Kirk 127) - We will use the notation K for the category of compactly generated spaces\. \(This is taken as a full subcategory of the category of all topological spaces, i\.e\. every continuous function between compactly generated spaces is a morphism in K\.\) (James Davis, Paul Kirk 127) - Hausdorﬀ space can be turned into a compactly generated space by the following trick\. Deﬁnition 6\.2\. If X is Hausdorﬀ, let k\(X\) be the set X with the new topology deﬁned by declaring a subset A ⊂ X to be closed in k\(X\) if and only if A ∩ C is closed in X for all C ⊂ X compact\. (James Davis, Paul Kirk 127) - 6\.1\.1\. Basic facts about compactly generated spaces\. 1\. If X ∈ K, then k\(X\) = X\. 2\. If f : X → Y is a function, then k\(f \) : k\(X\) → k\(Y \) is continuous if and only if f |C : C → Y is continuous for each compact C ⊂ X\. 3\. Let C\(X, Y \) denote the set of continuous functions from X to Y \. Then k∗ : C\(X, k\(Y \)\) → C\(X, Y \) is a bijection if X is in K\. (James Davis, Paul Kirk 127) - 4\. The singular chain complexes of a Hausdorﬀ space Y and the space k\(Y \) are the same\. (James Davis, Paul Kirk 128) - 6\.1\.2\. Products in K\. Unfortunately, the product of compactly generated spaces need not be compactly generated\. However, this causes little concern, as we now see\. Deﬁnition 6\.3\. Let X, Y be compactly generated spaces\. The categorical product of X and Y is the space k\(X × Y \)\. (James Davis, Paul Kirk 128) - The following useful facts hold about the categorical product\. 1\. k\(X × Y \) is in fact a product in the category K\. 2\. If X is locally compact and Y is compactly generated, then X × Y = k\(X × Y \)\. In particular, I × Y = k\(I × Y \)\. Thus the notion of homotopy is unchanged\. (James Davis, Paul Kirk 128) - 6\.1\.3\. Function spaces\. The standard way to topologize the set of functions C\(X, Y \) is to use the compact-open topology\. Deﬁnition 6\.4\. If X and Y are compactly generated spaces, let C\(X, Y \) denote the set of continuous functions from X to Y , topologized with the compact-open topology\. This topology has as a subbasis sets of the form U \(K, W \) = {f ∈ C\(X, Y \)|f \(K\) ⊂ W } where K is a compact set in X and W an open set in Y \. (James Davis, Paul Kirk 128) - If Y is a metric space, this is the notion, familiar from complex analysis, of uniform convergence on compact sets (James Davis, Paul Kirk 128) - Unfortunately, even for compactly generated spaces X and Y , C\(X, Y \) need not be compactly generated\. We know how to handle this problem: deﬁne Map\(X, Y \) = k\(C\(X, Y \)\)\. As a set, Map\(X, Y \) is the set of continuous maps from X to Y , but its topology is slightly diﬀerent from the compact open topology (James Davis, Paul Kirk 128) - Theorem 6\.5 \(adjoint theorem\)\. For X, Y , and Z compactly generated, f \(x, y\) → f˜\(x\)\(y\) gives a homeomorphism Map\(\(X × Y \), Z\) → Map\(X, Map\(Y, Z\)\) Thus − × Y and Map\(Y, −\) are adjoint functors from K to K\. (James Davis, Paul Kirk 129) - 6\.1\.4\. Quotient maps\. We discuss yet another convenient property of compactly generated spaces\. For topological spaces, one can give an example of quotient maps p : W → Y and q : X → Z so that p × q : W × X → Y × Z is not a quotient map\. However, one can show the following\. (James Davis, Paul Kirk 129) - The following useful properties of Map\(X, Y \) hold\. 1\. Let e : Map\(X, Y \) × X → Y be the evaluation e\(f, x\) = f \(x\)\. Then if X, Y ∈ K, e is continuous\. 2\. If X, Y, Z ∈ K, then: \(a\) Map\(X, Y × Z\) is homeomorphic to Map\(X, Y \) × Map\(X, Z\), \(b\) Composition deﬁnes a continuous map Map\(X, Y \) × Map\(Y, Z\) → Map\(X, Z\)\. (James Davis, Paul Kirk 129) - Theorem 6\.6\. 1\. If p : W → Y and q : X → Z are quotient maps, and X and Z are locally compact Hausdorﬀ, then p × q is a quotient map\. 2\. If p : W → Y and q : X → Z are quotient maps and all space are compactly generated, then p × q is a quotient map, provided we use the categorical product\. (James Davis, Paul Kirk 129) - By this convention, we lose no information concerning homology and homotopy, but we gain the adjoint theorem\. (James Davis, Paul Kirk 129) - Deﬁnition 6\.7\. A continuous map p : E → B is a ﬁbration if it has the homotopy lifting property \(HLP\); i\.e\. the problem g̃ Y × {0} ✲ E Y ×I ✲ B G g̃ ✲ ❄p p ❄ p ✲ G p p p p pp p p p p pp p p p p p p p p✒ pp p pp ✒ ̃ G̃ has a solution for every space Y (James Davis, Paul Kirk 130) - In other words, given the continuous maps p, G, g̃, and the inclusion Y × {0} → Y × I, the problem is to ﬁnd a continuous map G̃ making the diagram commute\. (James Davis, Paul Kirk 130) - A covering map is a ﬁbration\. In studying covering space theory this fact is called the covering homotopy theorem\. For covering maps the lifting is unique, but this is not true for an arbitrary ﬁbration\. (James Davis, Paul Kirk 130) - The following theorem of Hurewicz says that if a map is locally a ﬁbration, then it is so globally\. Theorem 6\.8\. Let p : E → B be a continuous map\. Suppose that B is paracompact and suppose that there exists an open cover {Uα } of B so that p : p −1 \(Uα \) → Uα is a ﬁbration for each Uα \. Then p : E → B is a ﬁbration\. (James Davis, Paul Kirk 130) - Corollary 6\.9\. If p : E → B is a ﬁber bundle over a paracompact space B, then p is a ﬁbration\. (James Davis, Paul Kirk 130) - Deﬁnition 6\.11\. If p : E → B is a ﬁbration, and f : X → B a continuous map, deﬁne the pullback of p : E → B by f to be the map f ∗ \(E\) → X where f ∗ \(E\) = {\(x,e\) ∈ X × E|f \(x\) = p\(e\)} ⊂ X × E and the map f ∗ \(E\) → B is the restriction of the projection X × E → X\. (James Davis, Paul Kirk 131) - A ﬁbration need not be a ﬁber bundle\. Indeed, the deﬁnition of a ﬁbration is less rigid than that of a ﬁber bundle and it is not hard to alter a ﬁber bundle slightly to get a ﬁbration which is not locally trivial\. Nevertheless, a ﬁbration has a well deﬁned ﬁber up to homotopy\. The following theorem asserts this, and also states that a ﬁbration has a substitute for the structure group of a ﬁber bundle, namely the group of homotopy classes of self-homotopy equivalences of the ﬁber (James Davis, Paul Kirk 131) - Theorem 6\.12\. Let p : E → B be a ﬁbration\. Assume B is path connected\. Then all ﬁbers Eb = p −1 \(b\) are homotopy equivalent\. Moreover every path α : I → B deﬁnes a homotopy class α∗ of homotopy equivalences Eα\(0\) → Eα\(1\) which depends only on the homotopy class of α rel endpoints, in such a way that multiplication of paths corresponds to composition of homotopy equivalences\. In particular, there exists a well-deﬁned group homomorphism [α] → \(α −1 \)∗ π1 \(B, b0 \) → Homotopy classes of self-homotopy equivalences of Eb0 \. (James Davis, Paul Kirk 132) - Since homotopy equivalences induce isomorphisms in homology or cohomology, a ﬁbration with ﬁber F gives rise to local coeﬃcients systems whose ﬁber is the homology or cohomology of F , as the next corollary asserts\. (James Davis, Paul Kirk 135) - Corollary 6\.13\. Let p : E → B be a ﬁbration and let F = p−1 \(b0 \)\. Then p gives rise to local coeﬃcient systems over B with ﬁber Hn \(F ; M \) or H n \(F ; M \) for any n and any coeﬃcient group M \. These local coeﬃcients are obtained from the representations via the composite homomorphism π1 \(B,b0 \) → Homotopy classes of self-homotopy equivalences F → F → Aut\(A\) where A = Hn \(F ; M \) or A = H n \(F ; M \) (James Davis, Paul Kirk 135) - We see that a ﬁbration gives rise to many local coeﬃcient systems, by taking homology or cohomology of the ﬁber\. More generally one obtains a local coeﬃcient system given any homotopy functor from spaces to abelian groups \(or R-modules\), such as the generalized homology theories which we introduce in Chapter 8\. (James Davis, Paul Kirk 135) - Deﬁnition 6\.14\. Let \(Y, y0 \) be a based space\. The path space Py0 Y is the space of paths in Y starting at y0 , i\.e\. Py0 Y = Map\(I,0; Y ,y0 \) ⊂ Map\(I,Y \), (James Davis, Paul Kirk 135) - The loop space Ωy0 Y is the space of all loops in Y based at y0 , i\.e\. Ωy0 Y = Map\(I,{0,1}; Y ,{y0 }\)\. (James Davis, Paul Kirk 136) - Often the subscript y0 is omitted in the above notation\. Let Y I I = Map\(I, Y \)\. This is called the free path space\. Let p : Y I I → Y be the evaluation at the end point of a path: p\(α\) = α\(1\)\. (James Davis, Paul Kirk 136) - 1\. The map p : Y I I → Y , where p\(α\) = α\(1\), is a ﬁbration\. Its ﬁber over y0 is the space of paths which end at y0 , a space homeomorphic to Py0 Y \. 2\. The map p : Py0 Y → Y is a ﬁbration\. Its ﬁber over y0 is the loop space Ωy0 Y \. 3\. The free path space Y I I is homotopy equivalent to Y \. The projection p : Y I I → Y is a homotopy equivalence\. 4\. The space of paths in Y starting at y0 , Py0 Y , is contractible\. (James Davis, Paul Kirk 136) - Deﬁnition 6\.16\. A ﬁber homotopy between two morphisms \( ˜i f˜i , fi \) i = 0, 1 of ﬁbrations is a commutative diagram E×I H✲ E B×I ✲ B H ✲ H H ❄ p×Id ❄ p ✲ H with H0 = f0 , H1 = f1 , H 0 = f H ˜0 f˜0 , and H H 1 = f ˜1 f˜1 \. (James Davis, Paul Kirk 138) - Given two ﬁbrations over B, p : E → B and p : E → B, we say they have the same ﬁber homotopy type if there exists a map ˜ f˜ from E to E covering the identity map of B, and a map g̃ from E to E covering the identity map of B, such that the composites E ✲ E E E B B ❅ ❘ ❅ g̃◦f˜ ✲ ✠ ❅ ❘ ❅ ✲ ˜ f˜◦g̃ ✠ are each ﬁber homotopic to the identity via a homotopy which is the identity on (James Davis, Paul Kirk 138) - Notice that a ﬁber homotopy equivalence f ˜ f˜ : E → E induces a homotopy equivalence Eb0 → Eb 0 on ﬁbers\. (James Davis, Paul Kirk 138) - Let f : X → Y be a continuous map\. We will replace X by a homotopy equivalent space Pf and obtain a map Pf → Y which is a ﬁbration (James Davis, Paul Kirk 138) - Let q : Y I I → Y be the path space ﬁbration, with q\(α\) = α\(0\); evaluation at the starting point (James Davis, Paul Kirk 138) - Deﬁnition 6\.17\. The pullback Pf = f ∗ \(Y I I \) of the path space ﬁbration along f is called the mapping path space\. Pf = f ∗ \(Y I I \) ✲ YI X ✲ Y f ✲ ❄ ❄ q ✲ f \(6\.1\) An element of Pf is a pair \(x, α\) where α is a path in Y and x is a point in X which maps via f to the starting point of α\. (James Davis, Paul Kirk 139) - The mapping path ﬁbration p : Pf → Y is obtaining by evaluating at the end point p\(x, α\) = α\(1\) (James Davis, Paul Kirk 139) - Given a map f : X → Y , it is common to be sloppy and say “F is the ﬁber of f ”, or “F → X → Y is a ﬁbration” to mean that after replacing X by the homotopy equivalent space Pf and the map f by the ﬁbration Pf → Y , the ﬁber is a space of the homotopy type of F \. (James Davis, Paul Kirk 142) - Deﬁnition 6\.19\. A map i : A → X is called a coﬁbration, or satisﬁes the homotopy extension property \(HEP\), if the following diagram has a solution (James Davis, Paul Kirk 142) - Coﬁbration is a “dual” notion to ﬁbration, using the adjointness of the functors − × I and −I , and reversing the arrows\. To see this, note that since a map A × I → B is the same as a map A → B I , the diagram deﬁning a ﬁbration f : X → Y can be written X ✛ eval\. at 0 XI Z ✲ Y I\. ❄ f ✛ eval\. at 0 ✻ ✲ p p pp p p p p p pp p p p pp p pp The diagram deﬁning a coﬁbration f : Y → X can be written as (James Davis, Paul Kirk 143) - For “reasonable” spaces, any coﬁbration i : A → X can be shown to be an embedding whose image is closed in X\. (James Davis, Paul Kirk 143) - Deﬁnition 6\.20\. Let X be compactly generated, A ⊂ X a subspace\. Then \(X,A\) is called an NDR–pair \(NDR stands for “neighborhood deformation retract”\) if there exist (James Davis, Paul Kirk 143) - In particular the neighborhood U = {x ∈ X|u\(x\) < 1} of A deformation retracts to A\. (James Davis, Paul Kirk 144) - Theorem 6\.22 \(Steenrod\)\. Equivalent are: 1\. \(X,A\) is an NDR pair\. 2\. \(X × I, X × 0 ∪ A × I\) is a DR pair\. 3\. X × 0 ∪ A × I is a retract of X × I\. 4\. i : A → X is a coﬁbration\. (James Davis, Paul Kirk 144) - paracompact spaces are ﬁbrations\. Theorem 6\.23\. If X is a CW-complex, and A ⊂ X a subcomplex, then \(X,A\) is a NDR pair (James Davis, Paul Kirk 145) - The next result should remind you of the result that ﬁber bundles over (James Davis, Paul Kirk 145) - Exercise 91\. If \(X, A\) and \(Y, B\) are coﬁbrations, so is their product \(X, A\) × \(Y, B\) = \(X × Y, X × B ∪ A × Y \)\. We next establish that a pushout of a coﬁbration is a coﬁbration; this is dual to the fact that pullback of a ﬁbration is a ﬁbration (James Davis, Paul Kirk 145) - Deﬁnition 6\.24\. A pushout of maps f : A → B and g : A → C is a commutative diagram A ✲ B C ✲ D f ✲ ❄ g ❄ ✲ which is initial among all such commutative diagrams, i\.e\. any (James Davis, Paul Kirk 145) - problem of the form A ✲ B C ✲ D E f ✲ ❄ g ❆ ❆ ❆ ❆ ❆ ❆ p ❆ ❆ ❄ ✲ ❍ ❍ ❍ ❍ ❍ ❘p ❍ ❥ ❍ pp p p p p ❆ ❆ p❆ p ❘p ❆ ❍ ❥ ❍ has a unique solution\. (James Davis, Paul Kirk 145) - Pushouts always exist\. They are constructed as follows\. When A is empty the pushout is the disjoint union B C\. A concrete realization is given by choosing base points b0 ∈ B and c0 ∈ C and setting BC = {\(b, c0 , 0\) ∈ B × C × I | b ∈ B} ∪ {\(b0 , c, 1\) ∈ B × C × I | c ∈ C}\. In general, a concrete realization for the pushout of f : A → B and g : A → C is BC f \(a\) ∼ g\(a\) (James Davis, Paul Kirk 146) - Let f : A → X be a continuous map\. We will replace X by a homotopy equivalent space Mf and obtain a map A → Mf which is a coﬁbration\. In short, every map is equivalent to a coﬁbration (James Davis, Paul Kirk 146) - Deﬁnition 6\.26\. The mapping cylinder of a map f : A → X is the space Mf = \(A × I\) X \(a, 1\) ∼ f \(a\) (James Davis, Paul Kirk 146) - If f : A → X is a coﬁbration, then h is a homotopy equivalence rel A, in particular h induces a homotopy equivalence of the coﬁbers Cf → X/f \(A\)\. (James Davis, Paul Kirk 148) - Deﬁnition 6\.28\. A sequence of functions A f f − →B g g − →C of sets \(not spaces or groups\) with base points is called exact at B if f \(A\) = g −1 \(c0 \) where c0 is the base point of C\. (James Davis, Paul Kirk 150) - The following two theorems form the cornerstone of constructions of exact sequences in algebraic topology\. Theorem 6\.29 \(basic property of ﬁbrations\)\. Let p : E → B be a ﬁbration, with ﬁber F = p −1 \(b0 \) and B path-connected\. Let Y be any space\. Then the sequence of sets [Y ,F ] i∗ i∗ −→ [Y ,E] p∗ p∗ −→ [Y ,B] is exact\. (James Davis, Paul Kirk 150) - Theorem 6\.30 \(basic property of coﬁbrations\)\. Let i : A → X be a coﬁbration, with coﬁber X/A\. Let q : X → X/A denote the quotient map\. Let Y be any path-connected space\. Then the sequence of sets [X/A,Y ] q ∗ −→ [X,Y ] i ∗ i∗ −→ [A,Y ] is exact\. (James Davis, Paul Kirk 150) - ﬁnition 6\.31\. Deﬁne K∗ to be the category of compactly generated spaces with a non-degenerate base point, i\.e\. \(X,x0 \) is an object of K∗ if the inclusion {x0 } ⊂ X is a coﬁbration\. The morphisms in K∗ are the base point preserving continuous maps\. (James Davis, Paul Kirk 151) - Deﬁnition 6\.32\. The smash product of based spaces is X ∧Y = X ×Y X ∨Y = X ×Y X × {y0 } ∪ {x0 } ∪ Y (James Davis, Paul Kirk 152) - Note that the smash product X∧Y is a based space\. Contrary to popular belief, the smash product is not the product in the category K∗ , although the wedge product X ∨ Y = \(X × {y0 }\) ∪ \({x0 } × Y \) ⊂ X × Y is the sum in K∗ \. The smash product is the adjoint of the based mapping space (James Davis, Paul Kirk 152) - Theorem 6\.33 \(adjoint theorem\)\. There is a \(natural\) homeomorphism Map\(X ∧ Y, Z\)0 ∼ = ∼ = Map\(X, Map\(Y, Z\)0 \)0 (James Davis, Paul Kirk 152) - Deﬁnition 6\.34\. The \(reduced\) suspension of a based space \(X, x0 \) is SX = S 1 ∧ X\. The \(reduced\) cone is CX = I ∧ X\. (James Davis, Paul Kirk 152) - In other words, if ΣX is the unreduced suspension and cone\(X\) is the unreduced cone \(= ΣX/X × {0}\), then there are quotient maps ΣX → SX cone\(X\) → CX given by identifying {x0 } × I shaded in the following ﬁgure\. (James Davis, Paul Kirk 152) - Exercise 94\. If X ∈ K∗ \(i\.e\. the inclusion {x0 } → X is a coﬁbration\), then the quotient maps ΣX → SX and cone\(X\) → CX are homotopy equivalences (James Davis, Paul Kirk 153) - Proposition 6\.35\. The reduced suspension SS n is homeomorphic to S n+1 and the reduced cone CS n is homeomorphic to D n+1 \. (James Davis, Paul Kirk 153) - Theorem 6\.37 \(loops and suspension are adjoints\)\. The spaces Map\(SX, Y \)0 and Map\(X, ΩY \)0 are naturally homeomorphic (James Davis, Paul Kirk 153) - We will see eventually that the homotopy type of a ﬁber of a ﬁbration measures how far the ﬁbration is from being a homotopy equivalence\. \(For example, if the ﬁber is contractible then the ﬁbration is a homotopy equivalence\. (James Davis, Paul Kirk 154) - \) More generally given a map f : X → Y , one can turn it into a ﬁbration Pf → Y as above; the ﬁber of this ﬁbration measures how far f is from a homotopy equivalence\. (James Davis, Paul Kirk 154) - After turning f : X → Y into a ﬁbration Pf → Y one then has an inclusion of the ﬁber F ⊂ Pf \. Why not turn this into a ﬁbration and see what happens? Now take the ﬁber of the resulting ﬁbration and continue the process \. \. \. (James Davis, Paul Kirk 154) - Deﬁnition 6\.38\. If f : X → Y is a map, the homotopy ﬁber of f is the ﬁber of the ﬁbration obtained by turning f into a ﬁbrations (James Davis, Paul Kirk 154) - Similarly, the homotopy coﬁber of f : X → Y is the mapping cone Cf , the coﬁber of X → Mf \. (James Davis, Paul Kirk 154) - \. Let F → E → B be a ﬁbration\. Let Z be the homotopy ﬁber of F → E, so Z → F → E is a ﬁbration \(up to homotopy\)\. Then Z is homotopy equivalent to the loop space ΩB\. (James Davis, Paul Kirk 154) - \. Let A → X → X/A be a coﬁbration sequence\. Let W be the homotopy coﬁber of X → X/A, so that X → X/A → W is a coﬁbration \(up to homotopy\)\. Then W is homotopy equivalent to the \(unreduced\) suspension ΣA\. (James Davis, Paul Kirk 154) - 1 \. Let A → X be a base point preserving coﬁbration\. Then any two consecutive maps in the sequence A → X → X/A → SA → SX → · · · → S n A → S n X → S n \(X/A\) → · · · have the homotopy type of a coﬁbration followed by projection onto the coﬁber\. 2\. Let E → B be a ﬁbration with ﬁber F \. Then any two consecutive maps in the sequence · · · → Ωn F → Ωn E → Ωn B → · · · → ΩF → ΩE → ΩB → F → E → B have the homotopy type of a ﬁbration preceded by the inclusion of its ﬁber\. To prove 1 \., one must use reduced mapping cylinders and reduced cones\. 6\.12\. Puppe sequences Lemma 6\.41\. Let X and Y be spaces in K∗ \. 1\. [X,ΩY ]0 = [SX,Y ]0 is a group\. 2\. [X,Ω\(ΩY \)]0 = [SX, ΩY ]0 = [S 2 X, Y ]0 is an abelian group\. Sketch of proof\. The equalities follow from Theorem 6\.37, the adjointness of loops and suspension\. The multiplication can be looked at in two ways: ﬁrst on [SX, Y ]0 as coming from the map ν : SX → SX ∨ SX (James Davis, Paul Kirk 157) - Theorem 6\.40\. 1\. Let A → X be a coﬁbration\. Then any two consecutive maps in the sequence A → X → X/A → ΣA → ΣX → · · · → Σn A → Σn X → Σn \(X/A\) → · · · have the homotopy type of a coﬁbration followed by projection onto the coﬁber\. 1 \. Let A → X be a base point preserving coﬁbration\. Then any two consecutive maps in the sequence A → X → X/A → SA → SX → · · · → S n A → S n X → S n \(X/A\) → · · · have the homotopy type of a coﬁbration followed by projection onto the coﬁber\. 2\. Let E → B be a ﬁbration with ﬁber F \. (James Davis, Paul Kirk 157) - given by collapsing out the “equator” X × 1/2\. (James Davis, Paul Kirk 157) - A loop space is a example of an H-group and a suspension is an example of a co-H-group (James Davis, Paul Kirk 158) - , but here is the basic idea\. An H-group Z is a based space with a “multiplication” map µ : Z × Z → Z and an “inversion” map ϕ : X → X which satisfy the axioms of a group up to homotopy \(e\.g\. is associative up to homotopy\)\. For a topological group G and any space X, Map\(X, G\) is a group, similarly for an H-group Z, [X, Z]0 is a group\. To deﬁne a co-H-group, one reverses all the arrows in the deﬁnition of H- (James Davis, Paul Kirk 158) - group, so there is a co-multiplication ν : W → W ∨ W and a co-inversion (James Davis, Paul Kirk 158) - ψ : W → W \. Then [W, X]0 is a group\. Finally, there is a formal, but occasionally very useful result\. If W is a co-H-group and Z is an H-group, then the two multiplications on [W, Z]0 agree and are abelian\. Nifty, huh? One consequence of this is that π1 \(X, x0 \) of an H-group \(e\.g\. a topological group\) is abelian (James Davis, Paul Kirk 159) - Theorem 6\.42 \(Puppe sequences\)\. Let Y ∈ K∗ \. 1\. If F → E → B is a ﬁbration, the following sequence is a long exact sequence of sets \(i ≥ 0\), groups \(i ≥ 1\), and abelian groups \(i ≥ 2\)\. · · · → [Y ,Ωi F ]0 → [Y ,Ωi E]0 → [Y ,Ωi B]0 → · · · → [Y ,ΩB]0 → [Y ,F ]0 → [Y ,E]0 → [Y ,B]0 where Ω i Z denotes the iterated loop space Ω\(Ω\(· · · \(ΩZ\) · · · \)\. (James Davis, Paul Kirk 159) - 2\. If \(X,A\) is an coﬁbration, the following sequence is a long exact sequence of sets \(i ≥ 0\), groups \(i ≥ 1\), and abelian groups \(i ≥ 2\)\. · · · → [S i \(X/A\), Y ]0 → [S i X, Y ]0 → [S i A, Y ]0 → · · · → [SA, Y ]0 → [X/A, Y ]0 → [X, Y ]0 → [A, Y ]0 (James Davis, Paul Kirk 159) - In particular, πn X = π1 \(Ω n−1 X\)\. (James Davis, Paul Kirk 159) - For example, to get a hold of the group structure for writing down a proof, use πn \(X, x0 \) = [\(I n , ∂I n \), \(X, x0 \)]\. For the proof of the exact sequence of a pair \(coming later\) use πn \(X, x0 \) = [\(D n , S n−1 \), \(X, x0 \)]\. For ﬁnding a geometric interpretation of the boundary map in the homotopy long exact sequence of a ﬁbration given below, use πn \(X, x0 \) = [\(S n−1 × I, \(S n−1 × ∂I\) ∪ \(∗ × I\)\), \(X, x0 \)]\. (James Davis, Paul Kirk 160) - Finally, the homotopy type of a CWcomplex X is determined by the homotopy groups of X together with a cohomological recipe \(the k-invariants\) for assembling these groups\. (James Davis, Paul Kirk 160) - Exercise 98\. Show that πn \(X × Y \) = πn \(X\) ⊕ πn \(Y \)\. (James Davis, Paul Kirk 160) - Corollary 6\.44 \(long exact sequence of a ﬁbration\)\. Let F → E → B be a ﬁbration\. Then the sequence · · · → πn F → πn E → πn B → πn−1 F → πn−1 E → · · · → π1 F → π1 E → π1 B → π0 F → π0 E → π0 B is exact (James Davis, Paul Kirk 160) - Theorem 6\.45\. Let ̃ X̃ → X be a connected covering space of a connected space X\. Then the induced map πn \(X ̃ X̃\) → πn \(X\) is injective if n = 1, and an isomorphism if n > 1\. (James Davis, Paul Kirk 161) - This is a special case of Theorem 4\.5\. Theorem 6\.46 \(Gleason\)\. Let G be a compact Lie group acting freely on a compact manifold X\. Then X → X/G is a principal ﬁber bundle with ﬁber G\. (James Davis, Paul Kirk 161) - We will use the following theorem from equivariant topology to conclude that certain maps are ﬁbrations\. (James Davis, Paul Kirk 161) - Let K = R, C, H, or O \(the real numbers, complex numbers, quaternions, and octonions\)\. Each of these has a norm N : K → R+ (James Davis, Paul Kirk 161) - There are four Hopf ﬁbrations \(these are ﬁber bundles\): S 0 → S 1 → S 1 S 1 → S 3 → S 2 S 3 → S 7 → S 4 and S 7 → S 15 → S 8 \. These are constructed by looking at the various division algebras over R\. (James Davis, Paul Kirk 161) - Let EK = {\(x, y\) ∈ K ⊕ K|N \(x\)2 + N \(y\)2 = 1}\. Let GK = {x ∈ K|N \(x\) = 1}\. (James Davis, Paul Kirk 162) - Exercise 100\. GK is a compact Lie group homeomorphic to S r for r = 0, 1, 3\. For K = O, GK is homeomorphic to S 7 , but it is not a group; associativity fails (James Davis, Paul Kirk 162) - Let GK act on EK by g · \(x, y\) = \(gx, gy\) \(Note N \(gx\)2 + N \(gy\)2 = N \(x\) 2 + N \(y\) 2 if N \(g\) = 1\.\) This action is free\. This is easy to show for K = R, C, or H, since K is associative (James Davis, Paul Kirk 162) - Returning to the other exact sequences, it follows from the cellular approximation theorem that πn S 4 = πn−1 S 3 for n ≤ 6 \(since πn \(S 7 \) = 0 for n ≤ 6\), and that πn S 8 = πn−1 S 7 for n ≤ 14\. We will eventually be able to say more (James Davis, Paul Kirk 163) - Let S 1 act on S 2n−1 = {\(z1 , \. \. \. , zn \) ∈ Cn | Σ|zi |2 = 1} by t\(z1 , · · · , zn \) = \(tz1 , · · · , tzn \) if t ∈ S1 = {z ∈ C | |z| = 1}\. Exercise 102\. Prove that S 1 acts freely\. The orbit space is denoted by CP n−1 and called complex projective space\. (James Davis, Paul Kirk 163) - Exercise 103\. Using the long exact sequence for a ﬁbration, show that CP ∞ is an Eilenberg–MacLane space of type K\(Z, 2\), i\.e\. a CW-complex with π2 the only non-zero homotopy group and π2 ∼ = ∼ = Z\. (James Davis, Paul Kirk 164) - \. Calculate the cellular chain complexes for CP k and HP k \. 2\. Compute the ring structure of H ∗ \(CP k ; Z\) and H ∗ \(HP k ; Z\) using Poincaré duality\. 3\. Examine whether OP k can be deﬁned this way, for k > 1\. 4\. Show these reduce to Hopf ﬁbrations for k = 1\. (James Davis, Paul Kirk 164) - The Stiefel manifold Vk \(R n \) is the space of orthonormal k-frames in R n : Vk \(R n \) = {\(v1 , v2 , \. \. \. , vk \) ∈ \(Rn \)k | vi · vj = δij } given the topology as a subspace of \(R n \) k = R nk \. (James Davis, Paul Kirk 164) - The Grassmann manifold or grassmannian Gk \(R n \) is the space of kdimensional subspaces \(a\.k\.a\. k-planes\) in R n \. It is given the quotient topology using the surjection Vk \(R n \) → Gk \(Rn \) taking a k-frame to the k-plane it spans\. (James Davis, Paul Kirk 164) - Let G be a compact Lie group\. Let H ⊂ G be a closed subgroup \(and hence a Lie group itself\)\. The quotient G/H is called a homogeneous space\. The \(group\) quotient map G → G/H is a principal H-bundle since H acts freely on G by right translation\. If H has a closed subgroup K, then H acts on the homogeneous space H/K\. Changing the ﬁber of the above bundle results in a ﬁber bundle G/K → G/H with ﬁber H/K\. (James Davis, Paul Kirk 164) - Exercise 105\. Identify G/H with the grassmannian and G/K with the Stiefel manifold\. Conclude that the map taking a frame to the plane it spans deﬁnes a principal O\(k\) bundle Vk \(R n \) → Gk \(Rn \)\. (James Davis, Paul Kirk 165) - Let γk \(R n \) = {\(p, V \) ∈ Rn × Gk \(Rn \) | p is a point in the k-plane V }\. There is a natural map γk \(R n \) → Gk \(Rn \) given by projection on the second coordinate\. The ﬁber bundle so deﬁned is a vector bundle with ﬁber R k \(a k-plane bundle\) R k → γk \(Rn \) → Gk \(Rn \)\. It is called the canonical \(or tautological\) vector bundle over the grassmannian\. (James Davis, Paul Kirk 165) - Exercise 107\. Show there are ﬁbrations O\(n − k\) → O\(n\) → Vk \(Rn \) O\(n − 1\) → O\(n\) → S n−1 taking a matrix to its last k columns (James Davis, Paul Kirk 165) - πi \(O\(n − 1\)\) ∼ = \(6\.3\) πi \(O\(n − 1\)\) ∼ = πi \(O\(n\)\) for i < n − 2, and πi \(Vk \(R n \)\) = 0 for i < n − k − 1\. (James Davis, Paul Kirk 165) - The isomorphism of Equation \(6\.3\) is an example of “stability” in algebraic topology\. (James Davis, Paul Kirk 165) - A famous theorem of Bott says: Theorem 6\.49 \(Bott periodicity\)\. πk O ∼ = ∼ = πk+8 O for k ∈ Z+ \. (James Davis, Paul Kirk 165) - Moreover the homotopy groups of O are computed to be (James Davis, Paul Kirk 165) - clutching \(see Section 4\.3\.3\) corresponds to a bundle over S k+1 with structure group O\(n\)\. \(Alternatively, one may use that πk+1 \(BO\(n\)\) ∼ = ∼ = πk \(O\(n\)\) using the long exact sequence of homotopy groups of the ﬁbration O\(n\) → EO\(n\) → BO\(n\)\)\. (James Davis, Paul Kirk 166) - The generators of the ﬁrst eight homotopy groups of O are given by Hopf bundles\. (James Davis, Paul Kirk 166) - is a principal G-bundle EG → BG where EG is contractible\. This bundle classiﬁes principal G-bundles in the sense that given a principal G-bundle p : G → E → B over a CW-complex B \(or more generally a paracompact space\), there is a map of principal G-bundles E ✲ EG B ✲ BG f ❄ p ✲ ˜ f˜ ❄ ✲ f and that the homotopy class [f ] ∈ [B, BG] is uniquely determined\. It follows that the \(weak\) homotopy type of BG is uniquely determined (James Davis, Paul Kirk 166) - A project for Chapter 4 was to show that for every topological group G, there (James Davis, Paul Kirk 166) - Corollary 6\.50\. The inﬁnite grassmannian Gk \(R ∞ \) is a model for BO\(k\)\. The principal O\(k\) bundle O\(k\) → Vk \(R ∞ \) → Gk \(R ∞ \) is universal and classiﬁes principal O\(k\)-bundles\. The canonical bundle R k → γk \(R ∞ \) → Gk \(R ∞ \) classiﬁes R k -vector bundles with structure group O\(k\) \(i\.e\. R k -vector bundles equipped with metric on each ﬁber which varies continuously from ﬁber to ﬁber\)\. (James Davis, Paul Kirk 166) - The fact that the grassmannian classiﬁes orthogonal vector bundles makes sense from a geometric point of view\. (James Davis, Paul Kirk 166) - Moreover, Gk \(R ∞ \) is also a model for BGLk \(R\) and hence is a classifying space for k-plane bundles over CW-complexes (James Davis, Paul Kirk 167) - This follows either by redoing the above discussion, replacing k-frames by sets of k-linearly independent vectors, or by using the fact that O\(k\) → GLk \(R\) is a homotopy equivalence, with the homotopy inverse map being given by the Gram-Schmidt process\. (James Davis, Paul Kirk 167) - Similar statements apply in the complex setting to unitary groups U \(n\)\. Let Gk \(C n \) = complex k-planes in C n Gk \(C n \) = U \(n\)/\(U \(k\) × U \(n − k\)\), the complex grassmanian Vk \(C n \) = U \(n\)/U \(n − k\), the unitary Stiefel manifold\. There are principal ﬁber bundles U \(n − k\) → U \(n\) → Vk \(Cn \) (James Davis, Paul Kirk 167) - and U \(k\) → Vk \(Cn \) → Gk \(Cn \)\. (James Davis, Paul Kirk 167) - Moreover, V1 \(C n \) ∼ = ∼ = S 2n−1 , therefore πk \(U \(n\)\) ∼ = ∼ = πk \(U \(n − 1\)\) if k < 2n − 2 and so letting U = lim n→∞ U \(n\), we conclude that πk U = πk \(U \(n\)\) for n > 1 + k 2 \. (James Davis, Paul Kirk 167) - Theorem 6\.51 \(Bott periodicity\)\. πk U ∼ = ∼ = πk+2 U for k ∈ Z+ \. Moreover, πk U = Z if k is odd, and 0 if k is even\. Exercise 108\. Prove that π1 U = Z and π2 U = 0\. (James Davis, Paul Kirk 168) - Taking determinants give ﬁbrations SO\(n\) → O\(n\) det det −−→ {±1} and SU \(n\) → U \(n\) det det −−→ S 1 \. In particular, SO\(n\) is the identity path–component of O\(n\), so πk \(SO\(n\)\) = πk \(O\(n\)\) for k ≥ 1\. (James Davis, Paul Kirk 168) - Exercise 109\. Prove that SO\(2\) = U \(1\) = S 1 , SO\(3\) ∼ = ∼ = RP 3 , SU \(2\) ∼ = ∼ = S 3 , and that the map p : S × S → SO\(4\) given by \(a, b\) → \(v → av b̄\) 3 3 where a, b ∈ S 3 ⊂ H and v ∈ H ∼ = ∼ = R 4 is a 2-fold covering map (James Davis, Paul Kirk 168) - Exercise 110\. Using Exercise 109 and the facts: 1\. πn S n = Z \(Hopf degree Theorem\)\. 2\. πk S n = 0 for k < n \(Hurewicz theorem\)\. 3\. πk S ∼ n = ∼ πk+1 S = n+1 for k < 2n − 1 \(Freudenthal suspension theorem\)\. 4\. There is a covering Z → R → S 1 \. 5\. πn S n−1 = Z/2 for n > 3 \(this theorem is due to V\. Rohlin and G\. Whitehead; see Corollary 9\.27\)\. Compute as many homotopy groups of S n ’s, O\(n\), Grassmann manifolds, Stiefel manifolds, etc\. as you can\. (James Davis, Paul Kirk 168) - Deﬁnition 6\.52\. The relative homotopy group \(set if n = 1\) of the pair \(X, A\) is πn \(X, A, x0 \) = [D n , S n−1 , p; X, A, x0 ], the set of based homotopy classes of base point preserving maps from the pair \(D n , S n−1 \) to \(X, A\)\. This is a functor from pairs of spaces to sets \(n = 1\), groups \(n = 2\), and abelian groups \(n > 2\)\. (James Davis, Paul Kirk 168) - Theorem 6\.53 \(long exact sequence in homotopy of a pair\)\. The homotopy set πn \(X, A\) is a group for n ≥ 2, and is abelian for n ≥ 3\. Moreover, there is a long exact sequence · · · → πn A → πn X → πn \(X, A\) → πn−1 A → · · · → π1 \(X, A\) → π0 A → π0 X\. (James Davis, Paul Kirk 169) - Exercise 111\. Concoct an argument from this picture and use it to ﬁgure out why π1 \(X, A\) is not a group\. Also use it to prove that the long exact sequence is exact\. (James Davis, Paul Kirk 169) - Lemma 6\.54\. Let f : E → B be a ﬁbration with ﬁber F \. Let A ⊂ B be a subspace, and let G = f −1 \(A\), so that F → G f f − → A is a ﬁbration\. Then f induces isomorphims f∗ : πk \(E, G\) → πk \(B, A\) for all k\. In particular, taking A = {b0 } one obtains the commuting ladder (James Davis, Paul Kirk 169) - with all vertical maps isomorphisms, taking the long exact sequence of the pair \(E, F \) to the long exact sequence in homotopy for the ﬁbration F → E → B\. (James Davis, Paul Kirk 170) - Notice that the commutativity of this diagram and the fact that f∗ is an isomorphism gives an alternative deﬁnition of the connecting homomorphism πk \(B\) → πk−1 \(F \) in the long exact sequence of the ﬁbration F → E → B\. (James Davis, Paul Kirk 170) - An alternative and useful perspective on Theorem 6\.53 is obtained by replacing a pair by a ﬁbration as follows\. Turn A → X into a ﬁbration, with A replacing A and L\(X, A\) the ﬁber\. (James Davis, Paul Kirk 170) - the composite πk−1 \(L\(X, A\)\) e ◦∂ −1 ∗ e ◦∂ −1 ∗ −− −−→ πk \(X, A\) (James Davis, Paul Kirk 170) - is an isomorphism (James Davis, Paul Kirk 171) - where the top sequence is the long exact sequence for the ﬁbration L\(X, A\) → A → X and the bottom sequence is the long exact sequence of the pair \(X, A\)\. (James Davis, Paul Kirk 171) - Homotopy groups are harder to compute and deal with than homology groups, essentially because excision fails for relative homotopy groups\. In Chapter 8 we will discuss stable homotopy and generalized homology theories, in which \(properly interpreted\) excision does hold\. Stabilization is a procedure which looks at a space X only in terms of what homotopy information remains in S n X as n gets large\. The ﬁber L\(X, A\) and coﬁber X/A are stably homotopy equivalent\. (James Davis, Paul Kirk 171) - Theorem 6\.57\. This deﬁnes an action of π1 \(Y, y0 \) on the based set [X, Y ]0 , and [X, Y ] is the quotient set of [X, Y ]0 by this action if Y is path connected\. (James Davis, Paul Kirk 173) - Now π1 \(Y, y0 \) can be identiﬁed with group of covering transformations of ̃ Ỹ \. Thus, π1 \(Y, y0 \) acts on [X, Y ̃ Ỹ ] by post composition i\.e\. α : ̃ Ỹ → Ỹ acts on f : X → Ỹ by α ◦ f \. (James Davis, Paul Kirk 174) - Corollary 6\.60\. For any space Y , π1 \(Y, y0 \) acts on πn \(Y, y0 \) for all n with quotient [S n , Y ], the set of free homotopy classes\. (James Davis, Paul Kirk 174) - One could restrict to simply connected spaces Y and never worry about the distinction between based and unbased homotopy classes of maps into Y \. This is not practical in general, and so instead one can make a dimensionby-dimension deﬁnition\. Deﬁnition 6\.61\. We say Y is n-simple if π1 Y acts trivially on πn Y \. We say Y is simple if Y is n-simple for all n\. (James Davis, Paul Kirk 174) - Proposition 6\.62\. If F is n-simple, then the ﬁbration F → E → B deﬁnes a local coeﬃcient system over B with ﬁber πn F \. (James Davis, Paul Kirk 174) - \(A good example to think about is the Klein bottle mapping onto the circle\.\) (James Davis, Paul Kirk 174) - Thus, we obtain a homomorphism ρ : π1 B → Aut\(πn \(F \)\), i\.e\. a local coeﬃcient system over B\. (James Davis, Paul Kirk 175) - Theorem 6\.63\. The group π1 A acts on πn \(X, A\), πn X, and πn A for all n\. Moreover, the long exact sequence of the pair · · · → πn A → πn X → πn \(X, A\) → πn−1 A → · · · is π1 A-equivariant\. (James Davis, Paul Kirk 175) - Deﬁnition 6\.65\. The Hurewicz map ρ : πn X → Hn X is deﬁned by ρ\([f ]\) = f∗ \([S n ]\), where f : S n → X represents an element of πn X, [S n ] ∈ Hn S n ∼ = ∼ = Z is the generator \(given by the natural orientation of S n \) and f∗ : Hn S n → Hn X the induced map\. (James Davis, Paul Kirk 176) - The following theorem is the subject of one of the projects for this chapter\. It says that for simply connected spaces, the ﬁrst non-vanishing homotopy and homology groups coincide\. The Hurewicz theorem is the most important result in algebraic topology\. (James Davis, Paul Kirk 177) - Theorem 6\.66 \(Hurewicz theorem\)\. 1\. Let n > 0\. Suppose that X is path-connected\. If πk \(X, x0 \) = 0 for all k < n, then Hk \(X\) = 0 for all 0 < k < n, and the Hurewicz map ρ : πn X → H n X is an isomorphism if n > 1, and a surjection with kernel the commutator subgroup of π1 X if n = 1\. (James Davis, Paul Kirk 177) - 2\. Let n > 1\. Suppose X and A are path-connected\. If πk \(X, A\) = 0 for all k < n then Hk \(X, A\) = 0 for all k < n, and ρ : π + n+ \(X, A\) → Hn \(X, A\) is an isomorphism\. In particular ρ : πn \(X, A\) → Hn \(X, A\) is an epimorphism\. (James Davis, Paul Kirk 177) - Corollary 6\.67 \(Hopf degree theorem\)\. The Hurewicz map ρ : πn S n → Hn S n is an isomorphism\. Hence a degree zero map f : S n → S n is nullhomotopic\. (James Davis, Paul Kirk 177) - Although we have stated this as a corollary of the Hurewicz theorem, it can be proven directly using only the \(easy\) simplicial approximation theorem\. (James Davis, Paul Kirk 177) - Deﬁnition 6\.68\. 1\. A space X is called n-connected if πk X = 0 for k ≤ n\. \(Thus “simply connected” is synonymous with 1-connected\)\. 2\. A pair \(X, A\) is called n-connected if πk \(X, A\) = 0 for k ≤ n\. 3\. A map f : X → Y is called n-connected if the pair \(Mf , X\) is nconnected, where Mf = mapping cylinder of f \. (James Davis, Paul Kirk 177) - Using the long exact sequence for \(Mf , X\) and the homotopy equivalence Mf ∼ Y we see that f is n-connected if and only if f∗ : πk X → πk Y is an isomorphism for k < n and an epimorphism for k = n\. Replacing the map f : X → Y by a ﬁbration and using the long exact sequence for the (James Davis, Paul Kirk 177) - homotopy groups of a ﬁbration shows that f is n-connected if and only if the homotopy ﬁber of f is \(n − 1\)-connected\. (James Davis, Paul Kirk 178) - Corollary 6\.69 \(Whitehead theorem\)\. 1\. If f : X → Y is n-connected, then f∗ : Hq X → Hq Y is an isomorphism for all q < n and an epimorphism for q = n\. 2\. If X, Y are 1-connected, and f : X → Y is a map such that f∗ : Hq X → Hq Y is an isomorphism for all q < n and an epimorphism for q = n\. Then f is n-connected\. 3\. If X, Y are 1-connected spaces, f : X → Y a map inducing an isomorphism on Z-homology, then f induces isomorphisms f∗ : πk X ∼ = ∼ = = − → πk Y for all k\. (James Davis, Paul Kirk 178) - A map f : X → Y inducing an isomorphism of πk X → πk Y for all k is called a weak homotopy equivalence (James Davis, Paul Kirk 178) - Thus a map inducing a homology isomorphism between simply connected spaces is a weak homotopy equivalence\. Conversely a weak homotopy equivalence between two spaces gives a homology isomorphism\. (James Davis, Paul Kirk 178) - We will see later \(Theorem 7\.34\) that if X, Y are CW-complexes, then f : X → Y is a weak homotopy equivalence if and only if f is a homotopy equivalence (James Davis, Paul Kirk 178) - Corollary 6\.70\. A continuous map f : X → Y between simply connected CW-complexes inducing an isomorphism on all Z-homology groups is a homotopy equivalence\. (James Davis, Paul Kirk 178) - This corollary does not imply that if X, Y are two simply connected spaces with the same homology, then they are homotopy equivalent; one needs a map inducing the homology equivalence\. (James Davis, Paul Kirk 178) - For example, X = S 4 ∨ \(S 2 × S 2 \) and Y = CP 2 ∨ CP 2 are simply connected spaces with the same homology\. They are not homotopy equivalent because their cohomology rings are diﬀerent\. (James Davis, Paul Kirk 178) - Recall from Shapiro’s lemma \(Exercise 75\) that Hk \(X ̃ X̃; Z\) ∼ = ∼ = Hk \(X, Z[π1 X]\) for all k (James Davis, Paul Kirk 178) - The Whitehead theorem for non-simply connected spaces involves homology with local coeﬃcients: If f : X → Y is a map, let f˜ : X̃ → Ỹ be the corresponding lift to universal covers\. (James Davis, Paul Kirk 178) - and πk ̃ X̃ ∼ = ∼ = πk X for k > 1 (James Davis, Paul Kirk 179) - Theorem 6\.71\. If f : X → Y induces an isomorphism f∗ : π1 X → π1 Y , then f is n-connected if and only if it induces isomorphisms Hk \(X; Z[π]\) → Hk \(Y ; Z[π]\) for k < n and an epimorphism Hn \(X; Z[π]\) → Hn \(Y ; Z[π]\) In particular, f is a weak homotopy equivalence \(homotopy equivalence if X, Y are CW-complexes\) if only if f∗ : Hk \(X; Aρ \) → Hk \(Y ; Aρ \) is an isomorphism for all local coeﬃcient systems ρ : π → Aut\(A\)\. Thus, in the presence of a map f : X → Y , homotopy equivalences can be detected by homology\. (James Davis, Paul Kirk 179) - 6\.18\.1\. The Hurewicz theorem\. The statement is given in Theorem 6\.66\. A reference is §IV\.4-IV\.7 in [43]\. Another possibility is to give a spectral sequence proof\. Chapter 10 contains a spectral sequence proof the Hurewicz theorem\. (James Davis, Paul Kirk 179) - Obstruction theory addresses the following types of problems (James Davis, Paul Kirk 180) - Fibrations and coﬁbrations are easier to work with than arbitrary maps since they have ﬁbers and coﬁbers (James Davis, Paul Kirk 181) - Then the following exercise is an easy consequence of the homotopy lifting property, the homotopy extension property, and the method of turning maps into ﬁbrations or coﬁbrations\. Exercise 116\. Each of the four problems stated above is solvable for arbitrary continuous maps g : A → X and p : E → B between CW-complexes if and only if it is solvable for the CW-pair \(Mh , A\) and the ﬁbration Pp → E\. (James Davis, Paul Kirk 182) - Exercise 117\. \(Motivating exercise of obstruction theory\) Any map f : X → Y from an n-dimensional CW-complex to an n-connected space is null-homotopic\. (James Davis, Paul Kirk 182) - It turns out that if Y is only assumed to be \(n − 1\)-connected there is a single obstruction θ\(f \) ∈ H n \(X; πn Y \) which vanishes if and only if the map f is null-homotopic\. (James Davis, Paul Kirk 182) - In this way we obtain a cellular cochain which assigns to e n+1 ∈ X the element in πn Y \. If this cochain is the zero cochain, then the map can be extended over the \(n + 1\)-skeleton of X\. It turns out this cochain is in fact a cocycle and so represents a cohomology class in H n+1 \(X; πn Y \)\. The remarkable result is that if this cocycle represents the zero cohomology class, then by redeﬁning the map on the n-skeleton one can then extend it over the \(n + 1\)-skeleton of X \(if you take one step backwards, then you will be able to take two steps forward\)\. (James Davis, Paul Kirk 182) - This suggests one could deﬁne H n \(X; A\) to be [X, K\(A, n\)]\. This observation forms the basic link between the homological algebra approach to cohomology and homotopy theory (James Davis, Paul Kirk 183) - Corollary 7\.24\. Let K\(π, n\) and K \(π, n\) be two Eilenberg-MacLane spaces of type \(π, n\) for n > 1\. The identity map determines a canonical homotopy equivalence between them\. (James Davis, Paul Kirk 197) - Computing the cohomology of Eilenberg-MacLane spaces is very important, because of connections to cohomology operations\. Deﬁnition 7\.25\. For positive integers n and m and abelian groups π and π , a cohomology operation of type \(n, π, m, π \) is a natural transformation of functors θ : H n \(−; π\) → H m \(−; π \)\. (James Davis, Paul Kirk 198) - For example u → u∪u gives a cohomology operation of type \(n, Z, 2n, Z\)\. (James Davis, Paul Kirk 198) - It follows from our work above that for π abelian, [X, K\(π, 1\)] = H 1 \(X; π\) = Hom\(H1 X, π\) = Hom\(π1 X, π\)\. For π non-abelian we have the following theorem\. Theorem 7\.26\. For a based CW-complex X, the map on fundamental groups gives a bijection [X, K\(π, 1\)]0 → Hom\(π1 X, π\)\. (James Davis, Paul Kirk 198) - Proposition 7\.28\. Suppose that 1→L φ φ − →π γ γ − →H →1 is an exact sequence of \(not necessarily abelian\) groups\. Then the homotopy ﬁber of the map g : K\(π, 1\) → K\(H, 1\) inducing γ as in Theorem 7\.26 is K\(L, 1\) and that the inclusion of the ﬁber K\(L, 1\) → K\(π, 1\) induces the homomorphism φ\. If L, π, and H are abelian, the same assertions hold with K\(π, 1\) replaced by K\(π, n\) for any positive integer n\. (James Davis, Paul Kirk 199) - Thus short exact sequences of groups correspond exactly to ﬁbrations of Eilenberg–MacLane spaces; the sequence of groups 1→L→π→H→1 is a short exact sequence of groups if and only if the corresponding sequence of spaces and maps K\(L, 1\) → K\(π, 1\) → K\(H, 1\) is a ﬁbration sequence up to homotopy\. (James Davis, Paul Kirk 199) - Deﬁnition 7\.29\. A space is aspherical if its universal cover is contractible\. (James Davis, Paul Kirk 199) - Corollary 7\.34 below implies that a CW-complex is aspherical if and only if it is a K\(π, 1\)\. (James Davis, Paul Kirk 199) - The group Hn \(K\(π, 1\)\) is (James Davis, Paul Kirk 199) - Aspherical spaces are ubiquitous\. Compact 2-manifolds other than the sphere and projective space are K\(π, 1\)’s\. Also, K\(Z/2, 1\) = RP ∞ \. More generally K\(Z/n, 1\) = L ∞ n , where L ∞ n is the inﬁnite lens space given as S ∞ /\(Z/n\) where S ∞ ∞ ⊂ C is the inﬁnite dimensional sphere and the action is given by multiplication by a primitive n-th root of unity in every coordinate\. (James Davis, Paul Kirk 200) - Since πn \(X × Y \) = πn \(X\) ⊕ πn \(Y \), K\(Zn , 1\) = \(S 1 \)n , the n-torus\. The Cartan–Hadamard Theorem states that if M is a complete Riemannian manifold with sectional curvature everywhere ≤ 0, then for every point p ∈ M , the exponential map exp : Tp M → M is a covering map\. In particular M is aspherical (James Davis, Paul Kirk 200) - called the nth homology of the group π and is often denoted by Hn \(π\) (James Davis, Paul Kirk 200) - We also mention the still open Borel conjecture\. Compact aspherical manifolds with isomorphic fundamental groups are homeomorphic\. (James Davis, Paul Kirk 200) - The K\(π, 1\)-spaces are important for at least three reasons\. 1\. If M is a Zπ-module, then H∗ \(K\(π, 1\); M \) is an important algebraic invariant of the group and the module\. 2\. K\(π, 1\) = Bπ, and hence [X, Bπ] = Hom\(π1 X, π\)/\(φ ∼ gφg −1 \) classiﬁes regular covers with deck transformations π\. 3\. In the study of ﬂat bundles, that is, bundles whose structure group G reduces to a discrete group π, the classifying map X → BG factors through some K\(π, 1\)\. (James Davis, Paul Kirk 200) - \. A weak homotopy equivalence is a map f : X → Y which induces isomorphisms πi \(X, x\) → πi \(Y, f \(x\)\) for all i and for all base-points x in X\. (James Davis, Paul Kirk 200) - 2\. A CW-approximation of a topological space Y is a weak homotopy equivalence X→Y where X is a CW-complex\. (James Davis, Paul Kirk 201) - Theorem 7\.31\. Any space Y has a CW-approximation\. Proof\. (James Davis, Paul Kirk 201) - By the relative Hurewicz Theorem, a CW-approximation induces an isomorphism on homology\. (James Davis, Paul Kirk 201) - Theorem 7\.32 \(coﬁbrant theorem\)\. A map f : Y → Z is a weak homotopy equivalence if and only if for all CW-complexes X, f∗ : [X, Y ] → [X, Z] [g] → [f ◦ g] is a bijection (James Davis, Paul Kirk 201) - We have a weak homotopy equivalence f : Y → Z, which we may as well assume is the inclusion of a subspace by replacing Z by a mapping cylinder (James Davis, Paul Kirk 202)