Group homology

Summary

Group homology attaches to a group $G$ and a $G$-module $M$ a sequence of abelian groups $H_n(G, M)$, for $n \ge 0$. It is defined with the Tor functors over the group ring $\Z[G]$, and it is dual to group cohomology. In 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. Topologically, $H_n(G, \Z)$ is the singular homology of a classifying space of $G$.