Kunneth formula for group homology

Statement

For trivial group action

Suppose $G_1,G_2$ are groups and $M$ is an abelian group. We have the following formula for the homology groups for trivial group action of $G_1 \times G_2$ on $M$ in terms of the homology groups for trivial group action of $G_1$ and $G_2$ respectively on $M$:

$H_p(G_1 \times G_2; M) \cong \left(\sum_{i+j = p} H_i(G_1;M) \otimes H_j(G_2;M) \right) \oplus \left(\sum_{u + v = p - 1} \operatorname{Tor}^1_{\mathbb{Z}}(H_u(G_1;M),H_v(G_2;M))\right)$

Related facts

• Kunneth formula for group cohomology
• Cohomology group for trivial group action commutes with direct product in second coordinate