Kunneth formula for group homology: Difference between revisions

Latest revision as of 22:52, 12 January 2013

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 $M$ :

$H_{p} (G_{1} \times G_{2}; M) ≅ (\sum_{i + j = p} H_{i} (G_{1}; M) \otimes H_{j} (G_{2}; M)) \oplus (\sum_{u + v = p - 1} T_{o}^{r} (H_{u} (G_{1}; M), H_{v} (G_{2}; M)))$

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 Suppose <math>G_1,G_2</math> are [[group]]s and <math>M</math> is an [[abelian group]]. We have the following formula for the [[fact about::homology group for trivial group action;1| ]][[homology group for trivial group action|homology groups for trivial group action]] of <math>G_1 \times G_2</math> on <matH>M</math> in terms of the [[homology group for trivial group action|homology groups for trivial group action]] of <math>G_1</math> and <math>G_2</math> respectively on <math>M</math>:
-<math>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)</math>
+<math>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)</math>
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