Symmetric cohomology of groups and Poincar\'e duality

TL;DR

This paper introduces new groups related to symmetric cohomology of finite groups, establishing duality relations with existing cohomology theories and identifying conditions for isomorphisms with Tate cohomology.

Contribution

It constructs groups linking symmetric and Tate cohomology, providing duality results and conditions for isomorphisms, extending the understanding of group cohomology.

Findings

Established isomorphisms between constructed groups and symmetric cohomology.

Derived duality relations connecting different cohomology theories.

Identified conditions under which transformations from Tate cohomology are isomorphisms.

Abstract

Let $G$ be a finite group of order $n$ and let $M$ be a $G$-module. We construct groups $H_*^\varkappa(G,M)$ for which $H_k^\varkappa (G,M^{tw}) \cong H^{n-k-1}_\lambda(G,M),$ where $M^{tw}$ is a twisting of a $G$-module $M$ defined in Section $5$ and $H^{*}_\lambda(G,M)$ is a variation of the group cohomology introduced by Zarelua, which in many cases is isomorphic to the symmetric cohomology of groups defined by Staic. The groups $H_*^\varkappa(G,M)$ come together with transformations from Tate cohomology. We find conditions under which these transformations are isomorphisms.