Generalized Periodicity in Group Cohomology
Nir Elber
TL;DR
The paper generalizes periodic cohomology for finite groups by introducing $r$-encoding pairs $(X,X')$ that yield natural isomorphisms $\widehat{H}^i(G,\operatorname{Hom}_\mathbb Z(X,-))\cong\widehat{H}^{i+r}(G,\operatorname{Hom}_\mathbb Z(X',-))$. It develops shiftable functors and shifting natural transformations, showing that any natural transformation between corresponding hom-functors is given by a cup product with a unique encoding element $x\in\widehat{H}^r(G,\operatorname{Hom}_\mathbb Z(X',X))$, and introduces decoding data $x'$. Encoding pairs are constructed and analyzed via free resolutions, tensoring, duals, and various module operations, with equivalence notions defined through cohomological equivalence. The framework yields equivalent characterizations of encodings, includes duality and torsion-free criteria, and connects to the classical Swan theory by showing when a group’s cohomological input can be realized via tensor powers of the augmentation ideal $I_G^{\otimes r}$ (or its dual) to produce isomorphisms of cohomology functors. Overall, the approach extends periodic cohomology beyond cyclic groups, providing a robust toolkit for encoding cohomological information and studying generalized periodic phenomena in finite group actions. $\,$
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.