Finite groups as homotopy self-equivalences of finite spaces

TL;DR

This paper constructs finite spaces with a prescribed finite group as their homotopy self-equivalence group, advancing understanding of the realization problem in algebraic topology.

Contribution

It introduces a novel construction method for finite spaces whose self-homotopy equivalence groups are any given finite group.

Findings

Successfully realizes any finite group as a homotopy self-equivalence group

Develops a new construction using asymmetric spaces

Provides examples of finite spaces with prescribed symmetry groups

Abstract

We study the realization problem of finite groups as the group of homotopy classes of self-homotopy equivalences of finite spaces. Let $G$ be a finite group. Using an infinite family of pairwise non weakly homotopic asymmetric spaces we present a new construction of a finite space whose group of homotopy classes of self-homotopy equivalences is isomorphic to $G$.