group homology in nLab

Contents

Contents

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 GG is equivalently the ordinary homology of its classifying space BGB G (the Eilenberg-MacLane space K(G,1)K(G,1)):

H • grp(G)≃H •(BG). 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]