Lecture 34: Cap Product and “Cech” Cohomology

34. CAP PRODUCT AND “CECH” COHOMOLOGY 93 ∩ : H p (X) ⊗ Hn (X) → Hn−p (X) . Notice how the dimensions work. Long ago a bad choice was made: If cohomology were graded with negative integers, the way the gradations work here would look better. There are also two slant products. Maybe I won’t talk about them. In the next lecture, I’ll check a few things about cap products, and then get into the machinery of Poincaré duality. 34 Cap product and “Cech” cohomology We have a few more things to say about the cap product, and will then use it to give a statement of Poincaré duality. Proposition 34.1. The cap product enjoys the following properties. (1) (a ∪ b) ∩ x = a ∩ (b ∩ x) and 1 ∩ x = x: H∗ (X) is a module for H ∗ (X). (2) Given a map f : X → Y , b ∈ H p (Y ), and x ∈ Hn (X), f∗ (f ∗ (b) ∩ x) = b ∩ f∗ (x) . (3) Let  : H∗ (X) → R be the augmentation. Then ε(b ∩ x) = hb, xi . (4) Cap and cup are adjoint: ha ∩ b, xi = ha, b ∩ xi . Proof. (1) Easy. (2) Let β be a cocycle representing b, and σ an n-simplex in X. Then f∗ (f ∗ (β) ∩ σ) = f∗ ((f ∗ (β)(σ ◦ αp )) · (σ ◦ ωq )) = f∗ (β(f ◦ σ ◦ αp ) · (σ ◦ ω)) = β(f ◦ σ ◦ αp ) · f∗ (σ ◦ ωq ) = β(f ◦ σ ◦ αp ) · (f ◦ σ ◦ ωq ) = β ∩ f∗ (σ) 94 CHAPTER 3. COHOMOLOGY AND DUALITY This formula goes by many names: the “projection formula,” or “Frobenius reciprocity.” (3) We get zero unless p = n. Again let σ ∈ Sinn (X), and compute: ε(β ∩ σ) = ε(β(σ) · c0σ(n) ) = β(σ)ε(c0σ(n) ) = β(σ) = hβ, σi . Here now is a statement of Poincaré duality. It deals with the homological structure of compact topological manifolds. We recall the notion of an orientation, and Theorem 31.9 asserting the existence of a fundamental class [M ] ∈ Hn (M ; R) in a compact R-oriented n-manifold. Theorem 34.2 (Poincaré duality). Let M be a topological n-manifold that is compact and oriented with respect to a PID R. Then there is a unique class [M ] ∈ Hn (M ; R) that restricts to the orientation class in Hn (M, M − a; R) for every a ∈ M . It has the property that − ∩ [M ] : H p (M ; R) → Hq (M ; R) , p + q = n, is an isomorphism for all p. You might want to go back to Lecture 25 and verify that RP3 × RP3 satisfies this theorem. Our proof of Poincaré duality will be by induction. In order to make the induction go we will prove a substantially more general theorem, one that involves relative homology and cohomology. So we begin by understanding how the cap product behaves in relative homology. Suppose A ⊆ X is a subspace. We have: 0 0  S p (X) ⊗ Sn (A) i∗ ⊗1 / S p (A) ⊗ Sn (A) ∩ 1⊗i∗  S p (X) ⊗ Sn (X) ∩  S p (X) ⊗ Sn (X, A)  0  / Sq (A) i∗  / Sq (X)  / Sq (X, A)  0 The left sequence is exact because 0 → Sn (A) → Sn (X) → Sn (X, A) → 0 splits and tensoring with S p (X) (which is not free!) therefore leaves it exact. The solid arrow diagram commutes precisely by the chain-level projection formula. There is therefore a uniquely defined map on cokernels. This chain map yields the relative cap product ∩ : H p (X) ⊗ Hn (X, A) → Hq (X, A) It renders H∗ (X, A) a module for the graded algebra H ∗ (X). I want to come back to an old question, about the significance of relative homology. Suppose that K ⊆ X is a subspace, and consider the relative homology H∗ (X, X −K). Since the complement of X − K in X is K, these groups should be regarded as giving information about K. If I enlarge 34. CAP PRODUCT AND “CECH” COHOMOLOGY 95 K, I make X − K smaller: K ⊆ L induces H∗ (X, X − L) → H∗ (X − K); the relative homology is contravariant in the variable K (regarded as an object of the poset of subspaces of X). Excision gives insight into how H∗ (X, X − K) depends on K. Suppose K ⊆ U ⊆ X with K ⊆ Int(U ). To simplify things, let’s just suppose that K is closed and U is open. Then X − U is closed, X − K is open, and X − U ⊆ X − K, so excision asserts that the inclusion map H∗ (U, U − K) → H∗ (X, X − K) is an isomorphism. The cap product puts some structure on H∗ (X, X − K): it’s a module over H ∗ (X). But we can do better! We just decided that H∗ (X, X − K) = H∗ (U, U − K), so the H ∗ (X) action factors through an action by H ∗ (U ), for any open set U containing K. How does this refined action change when I decrease U ? Lemma 34.3. Let K ⊆ V ⊆ U ⊆ X, with K closed and U, V open. Then: H p (U ) ⊗ Hn (X, X − K) ∩ * Hq (X, X − K) i∗ ⊗1 ∩ 4  H p (V ) ⊗ Hn (X, X − K) commutes. Proof. This is just the projection formula again! Let UK be the set of open neighborhoods of K in X. It is partially ordered by reverse inclusion. This poset is directed, since the intersection of two opens is open. By the lemma, H p : UK → Ab is a directed system. Definition 34.4. The Čech cohomology of K is Ȟ p (K) = lim H p (U ) . −→ U ∈UK I apologize for this bad notation; its possible dependence on the way K is sitting in X is not recorded. The maps in this directed systen are all maps of graded algebras, so the direct limit is naturally a commutative graded algebra. Since tensor product commutes with direct limits, we now get a cap product pairing ∩ : Ȟ p (K) ⊗ Hn (X, X − K) → Hq (X, X − K) satifying the expected properties. This is the best you can do. It’s the natural structure that this relative homology has: H∗ (X, X − K) is a module over Ȟ ∗ (K). There are compatible restriction maps H p (U ) → H p (K), so there is a natural map Ȟ ∗ (K) → H ∗ (K) . This map is often an isomorphism. Suppose K ⊆ X satisfies the following “regular neighborhood” condition: For every open U ⊇ K, there exists an open V with U ⊇ V ⊇ K such that K ,→ V is a homotopy equivalence (or actually just a homology isomorphism). 96 CHAPTER 3. COHOMOLOGY AND DUALITY Lemma 34.5. Under these conditions, Ȟ ∗ (K) → H ∗ (K) is an isomorphism. Proof. We will check that the map to H p (K) satisfies the conditions we established in Lecture 23 to be a direct limit. So let x ∈ H p (K). Let U be a neighborood of K in X such that H p (U ) → H p (K) is an isomorphism. Then indeed x is in the image of H p (U ). Then let U be a neighborhood of K and let x ∈ H p (U ) restrict to 0 in H p (K). Let V be a sub-neighborood such that H p (V ) → H p (K) is an isomorphism. Then x restricts to 0 in H p (V ). On the other hand, here’s an example that distinguishes Ȟ ∗ from H ∗ . This is a famous example. The “topologist’s sine curve” is the subspace of R2 defined as follows. It is union of three subsets, A, B, and C. A is the graph of sin(π/x) where 0 < x < 1. B is the interval 0 × [−1, 1]. C is a continuous curve from (0, −1) to (1, 0) and meeting A ∪ B only at its endpoints. This is a counterexample for a lot of things; you’ve probably seen it in 18.901. What is the singular homology of the topologist’s sine curve? Use Mayer-Vietoris! I can choose V to be some connected portion of the continuous curve from (0, −1) to (1, 0), and U to contain the rest of the space in a way that intersects V in two open intervals. Then V is contractible, and U is made up of two contractible connected components. (This space is not locally path connected, and one of these path components is not closed.) The Mayer-Vietoris sequence looks like ∂ 0 → H1 (X) − → H0 (U ∩ V ) → H0 (U ) ⊕ H0 (V ) → H0 (X) → 0 . The two path components of U ∩ V do not become connected in U , so ∂ = 0 and we find that ∼ = ε : H∗ (X) − → H∗ (∗) and hence H ∗ (X) ∼ = H ∗ (∗). ∗ How about Ȟ ? Let X ⊂ U be an open neighborhood. The interval 0 × [−1, 1] has an neighborhood, for some small , that’s contained in U . This implies that there exists a neighborhood X ⊆ V ⊆ U such that V ' S 1 . This implies that lim H ∗ (U ) ∼ = H ∗ (S 1 ) −→ U ∈UX by a cofinality argument that we will detail later. So Ȟ ∗ (X) = 6 H ∗ (X). Nevertheless, under quite general conditions the Čech cohomology of a compact Hausdorff space is a topological invariant. The Čech construction forms a limit over open covers of the cohomology of the nerve of the cover. It is a topological invariant by construction. Theorem 34.6. Let X be a compact subset of some Euclidean space. If there is an open neighborhood of which it is a retract, then Ȟ ∗ (X; R) is canonically isomorphic to the cohomology defined using the Čech construction, and is therefore independent of the embedding into Euclidean space. See Dold’s beautiful book [3] for this and other topics discussed in this chapter. Bibliography [1] M. G. Barratt and J. Milnor, An example of anomalous singular homology, Proc. Amer. Math. Soc. 13 (1962) 293–297. [2] G. Bredon, Topology and Geometry, Springer-Verlag, 1993. [3] A. Dold, Lectures on Algebraic Topology, Springer-Verlag, 1980. [4] S. Eilenberg and J. C. Moore, Homology and fibrations, I: Coalgebras, cotensor product and its derived functors, Comment. Math. Helv. 40 (1965) 199–236. [5] S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, 1952. [6] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. [7] D. Kan, Adjoint funtors, Trans. Amer. Math. Soc. 87 (1958) 294–329. [8] J. Milnor, On axiomatic homology theory, Pacific J. Math 12 (1962) 337–341. [9] J. C. Moore, On the homotopy groups of spaces with a single non-vanishing homology group, Ann. Math. 59 (1954) 549–557. [10] C. T. C Wall, Finiteness conditions for CW complexes, Ann. Math. 81 (1965) 56–69. 109