Introduction to Algebraic Topology Lecture #9 Products

Greg Arone

March 19, 2020 Cohomology groups

Let X be a topological space and A an abelian group. Define the n-th group of cochains on X with values in A to be the following group n ∼ (X ∆n ) C (X ; A) = hom(Cn(X ), A) = A . The groups of cochains fit in a cochain complex

1 n n+1 C 0(X ; A) −→δ C 1(X ; A) → · · · −→δ C n(X ; A) −−→δ C n+1(X ; A) → · · ·

The cohomology groups of X with coefficients in A are defined to be the cohomology groups of this cochain complex:

Hn(X ; A) = ker(δn+1)/im(δn).

I will often omit the coefficients group A from the notation Basic properties of cohomology groups Cohomology groups are contravariant functors from Spaces to abelian groups. This means a map of spaces f : X → Y , induces a homomorphism on cohomology going in the opposite direction: f ∗ : H∗(Y ) → H∗(X ) with (f ◦ g)∗ = g ∗ ◦ f ∗. Cohomology satisfies the following properties: I Homotopy invariance: If f is homotopic to g then f ∗ = g ∗. I Dimension axiom: Hn(∗) = 0 for n 6= 0. I Multiplicativity: there is a natural isomorphism ∗ a Y ∗ H ( Xα) → H (Xα). α α

I Excision: If X = U1 ∪U0 U2, there is a long exact sequence in cohomology n n n n n−1 · · · ← H (U0) ← H (U1)⊕H (U2) ← H (X ) ← H (U0) ← · · · Everything we did with homology, has an analogue for cohomology

I Simplicial and cellular cohomology. ∗ I Relative cohomology of a pair H (X , X0), fitting in a long exact sequence

n n n n−1 · · · ← H (X0; A) ← H (X ; A) ← H (X , X0; A) ← H (X0; A) ← · · ·

∗ Note that H (X , X0) are the cohomology groups of ∗ ∗ ker(C (X ) → C (X0)) – cochains that vanish on C∗(X0). ∗ ' ∗ I Excision implies an isomorphism He (X /X0) −→ H (X , X0) for nice pairs. I Universal coefficients theorem makes cohomology effectively computable in terms of homology. The cup product The real importance of cohomology comes from the fact that it has an extra structure, a natural product. Let φ ∈ C m(X ) and ψ ∈ C n(X ). We define a new cochain φ ∪ ψ ∈ C m+n(X ) by the following formula. Suppose σ : ∆m+n → X is a singular chain

φ ∪ ψ(σ) = φ(σ|[v0,..., vm]) · ψ(σ|[vm,..., vm+n]).

Obviously, the cup product is linear in each variable, and thus induces a homomorphism

C m(X ) ⊗ C n(X ) → C m+n(X ) The product rule

Lemma

δm+n(φ ∪ ψ) = δm(φ) ∪ ψ + (−1)mφ ∪ δn(ψ)

The proof is by a routine calculation. Corollary 1. cocycle ∪ cocycle = cocycle 2. coboundary ∪ cocycle = coboundary 3. cocycle ∪ coboundary = coboundary

It follows that the cup product induces a product on cohomology

Hm(X ; R) ⊗ Hn(X ; R) → Hm+n(X ; R).

Here R can be any commutative ring. Properties of cup product

The cup product is associative (on chain level): suppose θ ∈ C l (X ), φ ∈ C m(X ), ψ ∈ C n(X ), and σ ∈ X ∆l+m+n . Then

(θ ∪ φ) ∪ ψ(σ) = θ ∪ (φ ∪ ψ)(σ) =

= θ(σ|[v0,..., vl ]) · φ(σ|[vl ,..., vl+m]) · ψ(σ|[vl+m,..., vl+m+n])

One the other hand note that for σ ∈ X ∆m+n

φ ∪ ψ(σ) = φ(σ|[v0,..., vm]) · ψ(σ|[vm,..., vm+n])

ψ ∪ φ(σ) = ψ(σ|[v0,..., vn]) · φ(σ|[vn,..., vm+n]) 6= φ ∪ ψ(σ). The cup product is not commutative on chain level. However... Homotopy commutativity Theorem Suppose [φ] ∈ Hm(X ) and [ψ] ∈ Hn(X ). Then

[φ ∪ ψ] = (−1)mn[ψ ∪ φ].

One says that the cup product is graded commutative on cohomology. Sketch of proof. n For a singular simplex σ : ∆ → X letσ ¯ = σ|[vn,..., v0]. If φ is a cochain define φ¯(σ) = φ(¯σ). It is easy to check that φ¯ ∪ ψ¯ = ψ ∪ φ. However the assignent φ 7→ φ¯ is not chain homotopic to the identity. Instead, define a homomorphism ρ: C∗(X ) → C∗(X ) by n ρ(σ) = (−1)(2)σ¯. One can show that ρ is a chain homomorphism, chain homotopic to the identity. On the other hand, a routine calculation shows that

ρ∗(φ) ∪ ρ∗(ψ) = (−1)mnρ∗(ψ ∪ φ).

Remark: There is a more conceptual proof of the commutativity of the cup product, and other similar results, using the method of acyclic models. Relative cup product

Theorem Suppose U, V ⊂ X are open subsets. Then the cup product on X restricts to a bilinear homomorphism

Hm(X , U) ⊗ Hn(X , V ) → Hm+n(X , U ∪ V ).

The proof follows from the following diagram

C ∗(X , U) ⊗ C ∗(X , V ) → C ∗(X , U + V ) ←−' C ∗(X , U ∪ V ) ↓ ↓ C ∗(X ) ⊗ C ∗(X ) → C ∗(X )

Here C ∗(X , U + V ) denotes the complex of cochains on X that vanish on all chains in U and all chains in V . The homomorphism C ∗(X , U ∪ V ) → C ∗(X , U + V ) is a quasi-isomorphism by excision. Cup length

Theorem Suppose that X = U ∪ V , where U, V are open and contractible. Then the cup product between any cohomology classes of positive dimension is zero. Proof. Consider the diagram

Hm(X , U) ⊗ Hn(X , V ) → Hm+n(X , U ∪ V ) = 0 =∼↓ ↓ Hm(X ) ⊗ Hn(X ) → Hm+n(X )

More generally, if X = U1 ∪ ... ∪ Un where Ui are open and contractible, then the product of any n elements of positive degree in cohomology is zero. IOW, the cup length of X is smaller than n. Products of spaces and tensor products of cohomology groups The cup product turns H∗(X ) = ⊕ Hn(X ) into a graded n commutative ring. Let R = ⊕Ri , S = ⊕Sj be graded commutative rings. Then R ⊗ S is again a graded commutative ring with (R ⊗ S)n = ⊕i+j=nRi ⊗ Sj . The multiplication is defined by the formula

0 0 ji 0 0 0 (ri ⊗ sj ) · (ri 0 ⊗ sj0 ) = (−1) ri ri 0 ⊗ sj sj0

There is a natural homomorphism of rings

H∗(X ) ⊗ H∗(Y ) → H∗(X × Y )

∗ ∗ φ ⊗ ψ 7→ PX (φ) ∪ PY (ψ) Kunneth formula Theorem The natural homomorphism

∗ ∗ ∗ H (X ; R) ⊗R H (Y ; R) → H (X × Y ; R)

is an isomorphism if H∗(Y ; R) is a finitely generated, free R-module. This is a special case of the Kunneth formula. If R is a field, then every module is free, so the conclusion holds if either X or Y is a finite CW complex. Idea of proof.

I If H∗(Y ) is finitely generated and free, then the functor X 7→ H∗(X ) ⊗ H∗(Y ) satisfies the axiom of cohomology theories. So does the functor X 7→ H∗(X × Y ). I So we have a natural transformation of cohomology theories. Clearly it is an isomorphism when X = ∗. It follows that it is an isomorphism when X is a CW-complex, by induction on cells. A general space X can be approximated by a CW complex. Example: product of spheres

∗ n The cohomology of a sphere: H (S ) = Λ[xn] is the exterior algebra on one generator of degree n. The cohomology of a product of spheres: ∗ m n ∼ ∼ H (S × S ) = Λ[xm] ⊗ Λ[xn] = Λ[xm, xn] is the exterior algebra on two generators. As a graded abelian group

∗ m n ∼ H (S × S ) = Z[1] ⊕ Z[xm] ⊕ Z[xn] ⊕ Z[xm ∪ xn],

2 2 with the multiplication satisfying xm = xn = 0 and mn xm ∪ xn = (−1) xn ∪ xm As an easy illustration of the extra information provided by cup product, we have the following lemma: Lemma Assuming m, n > 0, any map f : Sm+n → Sm × Sn is zero on cohomology (and therefore also on homology). Lemma Assuming m, n > 0, any map f : Sm+n → Sm × Sn is zero on cohomology (and therefore also on homology). Proof. Since Hm(Sm+n) =∼ Hn(Sm+n) =∼ {0}, it follows that ∗ ∗ f (xm) = f (xn) = 0. But then ∗ ∗ ∗ f (xm ∪ xn) = f xm ∪ f (xn) = 0.

Further applications: ∗ n 1. H (RP ; Z/2) has far reaching consequences, from Borsuk-Ulam theorem to the (non)-existence of an interesting n k product structure on R for n 6= 2 . 2. Poincare duality - deep structure on the cohomology of manifolds. 3. Other forms of duality, one application is the theorem of n invariance of domain: if a subset of R is homeomorphic to an open subset, then it actually is open... Stay tuned in Thank you for listening Stay safe