The Homology of Invariant Group Chains

Let Q be a finite group acting on a group G as a group of automorphisms. This action induces an action of Q on the standard bar complex C•(G) for computing the homology of G; denote the subcomplex of invariants by C•(G) Q. In this paper we study the homology of the complex C•(G) Q. The resulting homology groups are denoted H • (G). Before summarizing the results, a brief history is in order. A quick literature search and consultation with experts in equivariant topology revealed nothing. Indeed, everyone I asked was more or less alarmed that I wanted to know about invariants of a chain complex (instead of a cochain complex). So I forged ahead with the cumbersome calculations using the bar complex and obtained a few results. For example, if A is an abelian group with trivial Gand Q-actions, then we can define homology groups with coefficients H • (G;A). The inclusion of complexes C•(G;A) Q → C•(G;A) induces a homomorphism H • (G;A) → H•(G;A) whose image clearly lies in the subgroup H•(G;A) Q of Q-invariant homology classes. The following is easily proved.