Two results on cohomology of groups adapted to cochains

TL;DR

This paper provides explicit cochain-level formulas demonstrating the triviality of group actions on cohomology and the graded commutativity of the cup product, enhancing the understanding of cohomological structures in group theory.

Contribution

It introduces explicit cochain maps that realize the triviality of the group action and the graded commutativity of the cup product at the cochain level.

Findings

Explicit cochain homotopies for group action triviality.

Explicit cochain homotopies for cup product commutativity.

Enhanced understanding of cohomological operations at the cochain level.

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 $sa-a=(h_sd+dh_s)(a)$ $\forall a\in C^n(G,M)$, 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. $[a]\cup [b]=(-1)^{pq}t_*([b]\cup [a])$ $\forall [a]\in H^p(G,N)$, $[b]\in H^q(G,M)$. (Here $t:N\otimes M\to M\otimes N$ is the natural bijection.) Adapted to cochains, we prove that $(-1)^{pq}t_*(b\cup a)-a\cup b=(hd+dh)(a\otimes b)$ $\forall a\in C^p(G,M)$, $b\in C^q(G,N)$, for some explicit map $h:C(G,M)\otimes C(G,N)\to C(G,M\otimes N)[-1]$.