Universal spaces for finite group actions on spaces of type $K(\pi,1)$

TL;DR

The paper constructs a universal space of type $K(,1)$ that encapsulates all finite group actions on such spaces up to homotopy, providing a comprehensive framework for understanding these symmetries.

Contribution

It introduces a universal $K(,1)$ space that models all finite group actions on spaces of the same homotopy type, unifying their study.

Findings

Universal space exists for finite group actions on $K(,1)$ spaces.

Any finite group action on a $K(,1)$ space can be realized on this universal space.

The universal space covers all possible actions up to homotopy conjugation.

Abstract

In this paper author proposes a construction of a universal space of type $K(\pi,1)$ such that any action (up to homotopy conjugation) of a given finite group $G$ on spaces of the same homotopy type is presented on the constructed space. Moreover, any action of $G$ on any space of type $K(\pi,1)$ is covered by some action of $G$ on this universal space.