Lecture 26: Coproducts, Cohomology

Chapter 3 Cohomology and duality 26 Coproducts, cohomology The next topic is cohomology. This is like homology, but it’s a contravariant rather than covariant functor of spaces. There are three reasons why you might like a contravariant functor. (1) Many geometric contructions pull back; that is, they behave contravariantly. For example, if I e → X and a map f : Y → X, I get a pullback covering space f ∗ X. e have some covering space X A better example is vector bundles (that we’ll talk about in 18.906) – they don’t push out, they pullback. So if we want to study them by means of “natural” invariants, these invariants will have to lie in a (hopefully computable) group that also behaves contravariantly. This will lead to the theory of characteristic classes. (2) The structure induced by the diagonal map from a space to its square induces stucture in contravariant functors that is more general and easier to study. (3) Cohomology turns out to be the target of the Poincaré duality map. Let’s elaborate on point (2). Every space has a diagonal map ∆ X− →X ×X. This induces a map H∗ (X; R) → H∗ (X × X; R), for any coefficient group R. Now, if R is a ring, we get a cross product map × : H∗ (X; R) ⊗R H∗ (X; R) → H∗ (X × X; R) . If R is a PID, the Künneth Theorem tells us that this map is a monomorphism. If the remaining term in the Künneth Theorem is zero, the cross product is an isomorphism. So if H∗ (X; R) is free over R (or even just flat over R), we get a “diagonal” or “coproduct” ∆ : H∗ (X; R) → H∗ (X; R) ⊗R H∗ (X; R) . If R is a field, this map is universally defined, and natural in X. This kind of structure is unfamiliar, and at first seems a bit strange. After all, the tensor product is defined by a universal property for maps out of it; maps into it just are what they are. Still, it’s often useful, and we pause to fill in some of its properties. Definition 26.1. Let R be a ring. A (graded) coalgebra over R is a (graded) R-module M equipped with a “comultiplication” ∆ : M → M ⊗R M and a “counit” map ε : M → R such that the following 69 70 CHAPTER 3. COHOMOLOGY AND DUALITY diagrams commute. M = w R ⊗R M o ε⊗1  = ∆ M ⊗R M ∆  M ⊗R M ' / M ⊗R R / M ⊗R M ∆ M 1⊗ε  1⊗∆ / ∆⊗1 M ⊗R M ⊗R M It is commutative if in addition M ∆ z M ⊗R M ∆ $ / M ⊗R M τ commutes, where τ (x ⊗ y) = (−1)|x|·|y| y ⊗ x is the twist map. Using acyclic models, you saw for homework that the the Künneth map is associative and commutative: The diagrams S∗ (X) ⊗ S∗ (Y ) ⊗ S∗ (Z)  ×⊗1 / S∗ (X × Y ) ⊗ S∗ (Z) 1⊗× × S∗ (X) ⊗ S∗ (Y × Z)  × / S∗ (X × Y × Z) and S∗ (X) ⊗ S∗ (Y )  τ × S∗ (X × Y ) T∗ / S∗ (Y ) ⊗ S∗ (X)  × / S∗ (Y × X) commute up to natural chain homotopy, where τ is as defined above on the tensor product and T : X × Y → Y × X is the swap map. Similar diagrams apply to the standard comparison map for the homology of tensor products of chain complexes, µ : H∗ (C) ⊗ H∗ (D) → H∗ (C ⊗ D) , and the result is this: Corollary 26.2. Suppose R is a PID and H∗ (X; R) is free over R. Then H∗ (X; R) has the natural structure of a commutative graded coalgebra over R. We could now just go on and talk about coalgebras. But they are less familiar, and available only if H∗ (X; R) is free over R. So instead we’re going to dualize, talk about cohomology, and get an algebra structure. Some say that cohomology is better because you have algebras, but that’s more of a sociological statement than a mathematical one. Let’s get on with it. 26. COPRODUCTS, COHOMOLOGY 71 Definition 26.3. Let N be an abelian group. A singular n-cochain on X with values in N is a function Sinn (X) → N . If N is an R-module, then I can extend linearly to get an R-module homomorphism Sn (X; R) → N. Notation 26.4. Write S n (X; N ) = Map(Sinn (X), N ) = HomR (Sn (X; R), N ) . This is going to give us something contravariant, that’s for sure. But we haven’t quite finished dualizing. The differential d : Sn+1 (X; R) → Sn (X; R) induces a “coboundary map” d : S n (X; N ) → S n+1 (X; N ) defined by (df )(σ) = (−1)n+1 f (dσ) . The sign is a little strange, and we’ll see an explanation in a minute. Anyway, we get a “cochain complex,” with a differential that increases degree by 1. We still have d2 = 0, since (d2 f )(σ) = ±d(f (dσ)) = ±f (d2 σ) = ±f (0) = 0 , so we can still take homology of this cochain complex. Definition 26.5. The nth singular cohomology group of X with coefficients in an abelian group N is ker(S n (X; N ) → S n+1 (X; N )) . H n (X; N ) = im(S n−1 (X; N ) → S n (X; N )) If N is an R-module, then H n (X; N ) is again an R-module. Let’s first compute H 0 (X; N ). A 0-cochain is a function Sin0 (X) → N ; that is, a function (not required to be continuous!) f : X → N . To compute df , take a 1-simplex σ : ∆1 → X and evaluate f on its boundary: (df )(σ) = −f (dσ) = −f (σ(e0 ) − σ(e1 )) = f (σ(e1 )) − f (σ(e0 )) . So f is a cocycle if it’s constant on path components. That is to say: Lemma 26.6. H 0 (X; N ) = Map(π0 (X), N ). Q Warning 26.7. S n (X; Z) = Map(Sinn (X); Z) = Sinn (X) Z, which is probably an uncountable product. An awkward fact is that this is never free abelian. The first thing a cohomology class does is to give a linear functional on homology, by “evaluation.” Let’s spin this out a bit. We want to tensor together cochains and chains. But to do that we should make the differential in S ∗ (X) go down, not up. Just as a notational matter, let’s write ∨ S−n (X; N ) = S n (X; N ) ∨ (X) → S ∨ n n+1 (X). Now and define a differential d : S−n −n−1 (X) to be the differential d : S (X) → S ∨ S∗ (X) is a chain complex, albeit a negatively graded one. Form the graded tensor product, with M
Sp∨ (X; N ) ⊗ Sq (X) . S∗∨ (X; N ) ⊗ S∗ (X) n = p+q=n 72 CHAPTER 3. COHOMOLOGY AND DUALITY Now evaluation is a map of graded abelian groups h−, −i : S∗∨ (X; N ) ⊗ S∗ (X) → N , where N is regarded as a chain complex concentrated in degree 0. We would like this map to be a chain map. So let f ∈ S n (X; N ) and σ ∈ Sn (X), and compute 0 = dhf, σi = hdf, σi + (−1)n hf, dσi . This forces (df )(σ) = hdf, σi = −(−1)n f (dσ) , explaining the odd sign in our definition above. Here’s the payoff: There’s a natural map
µ H−n (S∗∨ (X; N )) ⊗ Hn (S∗ (X)) − → H0 S∗∨ (X; N ) ⊗ S∗ (X) → N This gives us the Kronecker pairing h−, −i : H n (X; N ) ⊗ Hn (X) → N . We can develop the properties of cohomology in analogy with properties of homology. For example: If A ⊆ X, there is a restriction map S n (X; N ) → S n (A; N ), induced by the injection Sinn (A) ,→ Sinn (X). And as long as A is nonempty, we can split this injection, so any function Sinn (A) → N extends to Sinn (X) → N . This means that S n (X; N ) → S n (A; N ) is surjective. (This is the case if A = ∅, as well!) Definition 26.8. The relative n-cochain group with coefficients in N is S n (X, A; N ) = ker (S n (X; N ) → S n (A; N )) . This defines a sub cochain complex of S ∗ (X; N ), and we define H n (X, A; N ) = H n (S ∗ (X, A; N )) . The short exact sequence of cochain complexes 0 → S ∗ (X, A; N ) → S ∗ (X; N ) → S ∗ (A; N ) → 0 induces the long exact cohomology sequence ··· k δ H 1 (X, A; N ) k / H 1 (X; N ) / H 1 (A; N ) δ H 0 (X, A; N ) / H 0 (X; N ) / H 0 (A; N ) .