Kunneth formula for cohomology

Statement

Suppose $X$ and $Y$ are topological spaces. We then have the following relation betwen the cohomology groups of $X$, $Y$, and the product space $X\times Y$.

For any $n\geq 0$ and any module $M$ over a principal ideal domain $R$, we have:

$\!H^{n}(X\times Y;M)\cong \left(\sum _{i+j=n}H^{i}(X;M)\otimes H^{j}(Y;M)\right)\oplus \left(\sum _{p+q=n+1}\operatorname {Tor} (H^{p}(X;M),H^{q}(Y;M))\right)$

Here, $\operatorname {Tor}$ is torsion of modules over the ring $R$.

Particular cases

Case of free modules

If all the cohomology groups $H^{i}(X;M)$ are free (or more generally, torsion-free) modules over $R$, and/or all the cohomology groups $H^{j}(Y;M)$ are free (or more generally, torsion-free) modules over $R$, then all the torsion part vanishes and we simply get:

$\!H^{n}(X\times Y;M)\cong \left(\sum _{i+j=n}H^{i}(X;M)\otimes H^{j}(Y;M)\right)$