Every group is the group of self homotopy equivalences of finite dimensional CW-complex

TL;DR

This paper demonstrates that any group can be realized as the group of self-homotopy equivalences of a finite-dimensional CW-complex, extending previous results limited to finite groups and rational elliptic spaces.

Contribution

It generalizes the known theorem by Costoya and Viruel to show that every group, finite or infinite, can be represented as the self-homotopy equivalence group of some finite-dimensional CW-complex.

Findings

Any group G can be realized as E(X) for some finite-dimensional CW-complex X.

Extends previous results from finite groups to all groups.

Provides a construction method for such CW-complexes.

Abstract

We prove that any group $G$ occurs as $\E(X)$, where $X$ is CW-complex of finite dimension and $\E(X)$ denotes its group of self-homotopy equivalence. Thus, we generalize a well know-theorem due to Costoya and Viruel \cite{CV} asserting that any finite group occurs as $\E(X)$, where $X$ is rational elliptic space.