Two results on cohomology of groups adapted to cochains

Abstract

Given a group $G$ and a $G$-module $M$, we denote by $(C(G,M),d)$ the corresponding cochain complex obtained from the standard resolution. An element of the cohomology $H(G,M)$ will be written as the class $[a]$ of some cocycle $a\in C(G,M)$. The first result involves the triviality of the action of $G$ on $H(G,M)$, i.e. $s[a]=[a]$ $\forall [a]\in H^n(G,M)$, $s\in G$. Adapted to cochains, we prove that $sa-a=(h_{s}d+d{h}_s)(a)$ $\forall a\in C^n(G,M)$, for some explicit map $h_s:C(G,M)\to C(G,M)[-1]$. The second result regards the commutativity of the cup product, i.e. $[a]\cup [b]=(-1{)}^{pq}{t}_{\ast }([b]\cup [a])$ $\forall [a]\in H^p(G,N)$, $[b]\in H^q(G,M)$. (Here $t:N\otimes M\to M\otimes N$ is the natural bijection.) Adapted to cochains, we prove that $(-1{)}^{pq}{t}_{\ast }(b\cup a)-a\cup b=(hd+dh)(a\otimes b)$ $\forall a\in C^p(G,M)$, $b\in C^q(G,N)$, for some explicit map $h:C(G,M)\otimes C(G,N)\to C(G,M\otimes N)[-1]$.