nLab group homology

Idea

Group homology is the homology dual of group cohomology.

See for instance at group cohomology – In terms of homological algebra and replace Ext by Tor.

Accordingly, the group homology of a discrete group $G$ is equivalently the ordinary homology of its classifying space $B G$ (the Eilenberg-MacLane space $K(G,1)$):

$$H^{grp}_\bullet(G) \;\simeq\; H_\bullet\big( B G \big) \,.$$

(eg. Brown 1982, §4.1).

References

• Kenneth Brown, The Homology of a Group, Chapter II of: Cohomology of Groups, Graduate Texts in Mathematics, 87, Springer (1982) [doi:10.1007/978-1-4684-9327-6]