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 $n \ge 0$ 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:

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