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  "body": "\n## Summary\n\nGroup homology attaches to a [[group|G5S1M9]] $G$ and a $G$-module $M$ a sequence of abelian groups $H_n(G, M)$, for $n \\ge 0$.\nIt is defined with the [[Tor functors|DDQRERAQCV]] over the group ring $\\Z[G]$, and it is dual to group cohomology.\nIn low degrees it has concrete meaning: $H_1(G, \\Z)$ is the abelianization of $G$, and $H_2(G, \\Z)$ is given by [[Hopf's formula|S31M1ZYJRK]].\nTopologically, $H_n(G, \\Z)$ is the [[singular homology|M0Q5CFJ7ZH]] of a [[classifying space|WA796ZCNRX]] of $G$.\n\n## Explanations\n\n### L1 human\n\nGroup homology is a list of abelian groups, one for each degree, that record algebraic information about a group.\nThe first homology group records what is left of the group when every two elements are forced to commute.\nFor the integers $\\Z$, the first homology is $\\Z$ and all higher ones vanish, matching the circle, whose loops are counted by $\\Z$.\nFor the group $\\Z^2$, the second homology is $\\Z$, matching the single two-dimensional hole of the torus.\n",
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