Group 1-cohomology is complemented

TL;DR

This paper demonstrates that for certain cohomology groups with coefficients in uniformly convex Banach spaces, the reduced cohomology is a complemented subspace of cocycles, revealing a structural property.

Contribution

It establishes that reduced first cohomology groups are complemented subspaces within the space of cocycles for isometric representations on uniformly convex Banach spaces.

Findings

Reduced cohomology groups are complemented in cocycles

The complement of the cohomology group is the space of coboundaries

Structural property holds for isometric representations on uniformly convex Banach spaces

Abstract

We show a structural property of cohomology with coefficients in an isometric representation on a uniformly convex Banach space: if the cohomology group $H^1(G,\pi)$ is reduced, then, up to an isomorphism, it is a closed complemented, subspace of the space of cocycles and its complement is the subspace of coboundaries.