Generalized Periodicity in Group Cohomology

TL;DR

This paper introduces the concept of encoding pairs in group cohomology to generalize periodicity, extending classical results about finite groups acting freely on spheres.

Contribution

It defines encoding pairs and uses them to generalize the theory of periodic cohomology for finite groups, broadening the scope of Swan's theorem.

Findings

Generalization of periodic cohomology via encoding pairs

Extension of Swan's theorem to new classes of groups

Framework for analyzing group actions on topological spaces

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 $H^\bullet(G,\mathrm{Hom}_\mathbb Z(M,-))$ and $H^\bullet(G,\mathrm{Hom}_\mathbb Z(M',-))$. 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.