Equivariant Simplicial Cohomology With Local Coefficients and its Classification

TL;DR

This paper develops a new framework for equivariant simplicial cohomology with local coefficients, establishing an isomorphism with Bredon-Illman cohomology and proving a classification theorem for these structures.

Contribution

It introduces equivariant twisted cohomology for simplicial sets with group actions and proves a classification theorem, connecting it to existing cohomology theories.

Findings

Equivariant twisted cohomology is isomorphic to Bredon-Illman cohomology with local coefficients.

A classification theorem for equivariant simplicial cohomology with local coefficients is established.

The framework extends cohomology theories to include group actions and local coefficients.

Abstract

We introduce equivariant twisted cohomology of a simplicial set equipped with simplicial action of a discrete group and prove that for suitable twisting function induced from a given equivariant local coefficients, the simplicial version of Bredon-Illman cohomology with local coefficients is isomorphic to equivariant twisted cohomology. The main aim of this paper is to prove a classification theorem for equivariant simplicial cohomology with local coefficients.