Invariant chains and the homology of quotient spaces

Abstract

For a finite group G and a finite G-CW-complex X, we construct groups H_\bullet(G,X) as the homology groups of the G-invariants of the cellular chain complex C_\bullet(X). These groups are related to the homology of the quotient space X/G via a norm map, and therefore provide a mechanism for calculating H_\bullet(X/G). We compute several examples and provide a new proof of ``Smith theory": if G=Z/p and X is a mod p homology sphere on which G acts, then the subcomplex X^G is empty or a mod $p$ homology sphere. We also get a new proof of the Conner conjecture: If G=Z/p acts on a Z-acyclic space X, then X/G is Z-acyclic.