The Künneth formula

Recall the tensor product of abelian groups:

Tensor Product of Abelian Groups

Let $A$ and $B$ be abelian groups. The tensor product $A\otimes B$ is defined as the abelian group generated by the symbols $a\otimes b$ for $a\in A,b\in B$, subject to the following relations for all $a,a^{\prime }\in A$ and $b,b^{\prime }\in B$:

1. $(a+a^{\prime })\otimes b=a\otimes b+a^{\prime }\otimes b$
2. $a\otimes (b+b^{\prime })=a\otimes b+a\otimes b^{\prime }$

Consequences of the Definition

• Zero Element: The zero element of $A\otimes B$ is $0\otimes 0$. The relations imply that $a\otimes 0=0$ and $0\otimes b=0$ for any $a\in A,b\in B$.
• Inverses: The inverse of an element $a\otimes b$ is $-(a\otimes b)$, which can be shown to be equal to $(-a)\otimes b$ and $a\otimes (-b)$.
• Integer Multiplication: For any integer $n\in \mathbb{Z}$, the relations imply that $n(a\otimes b)=(na)\otimes b=a\otimes (nb)$.

Properties of Tensor Products

Let $A,B,C$, and $A_i$ be abelian groups.

1. Commutativity: $A\otimes B\cong B\otimes A$.
2. Distributivity over Direct Sums: $({⨁}_i{A}_i)\otimes B\cong {⨁}_i(A_i\otimes B)$.
3. Associativity: $(A\otimes B)\otimes C\cong A\otimes (B\otimes C)$.
4. Identity Element: $\mathbb{Z}\otimes A\cong A$.
5. Tensor Product with Cyclic Groups: ${\mathbb{Z}}_n\otimes A\cong A/nA$.
6. Induced Homomorphisms: A pair of homomorphisms $f:A\to A^{\prime }$ and $g:B\to B^{\prime }$ induces a homomorphism $f\otimes g:A\otimes B\to A^{\prime }\otimes B^{\prime }$ via the rule on generators: $(f\otimes g)(a\otimes b)=f(a)\otimes g(b)$
7. Universal Property: A bilinear map $\varphi :A\times B\to C$ induces a unique homomorphism from the tensor product into $C$, which sends $a\otimes b$ to $\varphi (a,b)$.

Cross Product

Cross Product

Let $X$ and $Y$ be topological spaces, and let $R$ be a ring. The cross product is a bilinear map $\times :H^i(X;R)\times H^j(X;R)\to H^{i+j}(X\times Y;R)$
that sends $(\alpha ,\beta )\mapsto {\mathrm{pr}}_X^{\ast }(\alpha )⌣{\mathrm{pr}}_Y^{\ast }(\beta )$, where ${\text{pr}}_X:X\times Y\to X$ and ${\text{pr}}_Y:X\times Y\to Y$ are the projection maps.

Lemma

The cross product is bilinear, so it induces $\times :H^i(X;R){\otimes }_R{H}^j(X;R)\to H^{i+j}(X\times Y;R).$

Künneth Formula for Cohomology

If $X$ and $Y$ are CW complexes and $H^k(X;R)$ or $H^k(Y;R)$ is a finitely generated free $R$-module for all $k$, then the cross product $\times :H^{\ast }(X;R){\otimes }_R{H}^{\ast }(Y;R)\to H^{\ast }(X\times Y;R)$ is an isomorphism of graded rings.

e.g.

• $H^{\ast }(S^1\times S^{1)}\cong {\Lambda }_{\mathbb{Z}}(x)\otimes {\Lambda }_{\mathbb{Z}}(y)={\Lambda }_{\mathbb{Z}}(x,y)$
• $H^{\ast }({\text{RP}}^{\infty }\times {\text{RP}}^{\infty };\mathbb{Z}/2)\cong \mathbb{Z}/2[x]{\otimes }_{\mathbb{Z}/2}\mathbb{Z}/2[y]\cong \mathbb{Z}/2[x,y]$
• $H^{\ast }({\text{CP}}^{\infty }\times {\text{CP}}^{\infty };\mathbb{Z})\cong \mathbb{Z}[x]{\otimes }_{\mathbb{Z}}\mathbb{Z}[y]\cong \mathbb{Z}[x,y]$
  

The general Künneth formula relates the (co)homology of a product $X\times Y$ to the (co)homology of the factors, giving a tensor-product term plus a torsion correction via ⁡$\mathrm{Tor}$ over the chosen coefficient ring $R$.