Finite group actions on Kan complexes

TL;DR

This paper investigates how finite groups act on Kan complexes, demonstrating realizability of certain group extensions and computing cohomology of fixed points in aspherical cases.

Contribution

It shows that semi-direct products of the fundamental group with finite groups can be realized as simplicial actions and calculates fixed point cohomology for finite p-groups.

Findings

Semi-direct products of fundamental groups with finite groups are realizable as simplicial actions.

Cohomology of fixed point sets under finite p-group actions on aspherical Kan complexes is computed.

Provides new insights into group actions on simplicial complexes and their topological invariants.

Abstract

We study simplicial action of groups on one vertex Kan complexes. We show that every semi-direct product of the fundamental group of an one vertex Kan complex with a finite group can be simplicially realized. We also calculate the cohomology of the fixed point set of a finite $p-$group action on an one vertex aspherical Kan complex.