Generalized Periodicity in Group Cohomology

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