Künneth Formula

Cross product on Cohomology

Define $x:H^i(X;R)\times H^j(Y;R)\to H^{i+j}(X\times Y;R)$ with $(\alpha ,\beta )\to {\mathrm{pr}}_x^{\ast }(\alpha )⌣{\mathrm{pr}}_y^{\ast }(\beta )$. ${\mathrm{pr}}_x:X\times Y\to X$,${\mathrm{pr}}_x^{\ast }:H^i(X)\to H^i(X\times Y;R)$ Moreover, it gives a map to ring homomorphism $\mu :H^{\ast }(X;R){\otimes }_R{H}^{\ast }(Y;R)\to H^{\ast }(X\times Y;R)$ with multiplication defined as $(a\otimes b)(c\otimes d)=(-1{)}^{|b||c|}ac\otimes bd$

Theorem

The cross product $x^{\ast }:H^{\ast }(X;R){\otimes }_R{H}^{\ast }(Y;R)\to H^{\ast }(X\times Y;R)$ is an isomorphism of rings if $X$ and $Y$ are CW complexes and $H^k(Y;R)$ is a finitely generated free $R$ module for all k

$H^k(Y;R)$ or $H^k(X;R)$ to be a finitely generated free $R$ module for all k

Here we actually just require either

Theorem

$H^{\ast }(X,A;R){\otimes }_R{H}^{\ast }(Y,B;R)\stackrel{\times }{\to }{H}^{\ast }(X\times Y,A\times Y\cup X\times B;R)$ for CW pairs $(X,A)$ and $(Y,B)$, defined just as in the absolute case by $a\times b=p_1^{\ast }(a)\cup p_2^{\ast }(b)$ where $p_1^{\ast }(a)\in H^{\ast }(X\times Y,A\times Y;R)$ and $p_2^{\ast }(b)\in H^{\ast }(X\times Y,X\times B;R)$.

For CW pairs $(X,A)$ and $(Y,B)$ the cross product homomorphism $H^{\ast }(X,A;R){\otimes }_R{H}^{\ast }(Y,B;R)\to H^{\ast }(X\times Y,A\times Y\cup X\times B;R)$ is an isomorphism of rings if $H^k(Y,B;R)$ is a finitely generated free $R$-module for each $k$.

Theorem

If $R$ is a principal ideal domain and the $R$-modules $C_i$ are free, then for each $n$ there is a natural short exact sequence $0\to {⨁}_i(H_i(C){\otimes }_R{H}_{n-i}(C^{\prime }))\to H_n(C{\otimes }_R{C}^{\prime })\to {⨁}_i{\text{Tor}}_R(H_i(C),H_{n-i-1}(C^{\prime }))\to 0$ and this sequence splits.

Theorem

If $X$ and $Y$ are CW complexes and $R$ is a principal ideal domain, then there are natural short exact sequences $0\to {⨁}_i(H_i(X;R){\otimes }_R{H}_{n-i}(Y;R))\to H_n(X\times Y;R)\to {⨁}_i{\text{Tor}}_R(H_i(X;R),H_{n-i-1}(Y;R))\to 0$