Generalized Periodicity in Group Cohomology

Abstract

Given a finite group $G$, we introduce "encoding pairs," which are a pair of $G$-modules $M$ and $M^{\prime }$ equipped with a shifted natural isomorphism between the cohomological functors $H^{\bullet }(G,{\mathrm{Hom}}_{\mathbb{Z}}(M,-))$ and $H^{\bullet }(G,{\mathrm{Hom}}_{\mathbb{Z}}(M^{\prime },-))$. 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.

18 pages. Comments welcome!

References in corpus (1)

• Groups with twisted p-periodic cohomology