Symmetric cohomology of groups and Poincaré duality

Abstract

Let $G$ be a finite group of order $n$ and let $M$ be a $G$-module. We construct groups $H_{\ast }^{\varkappa }(G,M)$ for which $H_k^{\varkappa }(G,M^{tw})\cong H_{\lambda }^{n-k-1}(G,M),$ where $M^{tw}$ is a twisting of a $G$-module $M$ defined in Section $5$ and $H_{\lambda }^{\ast }(G,M)$ 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 $H_{\ast }^{\varkappa }(G,M)$ come together with transformations from Tate cohomology. We find conditions under which these transformations are isomorphisms.