Kunneth formula for cohomology

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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)