Generalized periodicity in group cohomology

Abstract

Given a finite group G, we introduce “encoding pairs,” which are a pair of G-modules M and M ′ equipped with a shifted natural isomorphism between the cohomological functors H • ( G , Hom Z ( M , − ) ) and H • ( G , Hom Z ( M ′ , − ) ) . Studying these encoding pairs generalizes the theory of periodic cohomology for finite groups, allowing us to generalize the cohomological input of a theorem due to Swan that roughly says that a finite group with periodic cohomology acts feely on some sphere.