Standard Projective Simplicial Kernels and the Second Abelian Cohomology of Topological Groups

TL;DR

This paper provides a new interpretation of the second cohomology of topological groups with coefficients in an abelian module, and proves that projective topological groups have trivial second cohomology.

Contribution

It introduces a novel interpretation of $H^{2}(G,A)$ and establishes that projective topological groups have vanishing second cohomology for all abelian modules.

Findings

Second cohomology $H^{2}(G,A)$ has a new interpretative framework.

If $P$ is a projective topological group, then $H^{2}(P,A)=0$ for all abelian modules $A$.

Provides insights into the structure of topological groups via cohomology.

Abstract

Let $A$ be an abelian topological $G$-module. We give an interpretion for the second cohomology, $H^{2}(G,A)$, of $G$ with coefficients in $A$. As a result we show that if $P$ is a projective topological group, then $H^{2}(P,A)=0$ for every abelian topological $P$-module $A$.