The group of self-homotopy equivalences of $A_n^2$-polyhedra

TL;DR

This paper investigates the structure of self-homotopy equivalences of finite type $A_n^2$-polyhedra, revealing limitations on the groups that can be realized as their automorphism groups or their quotients.

Contribution

It characterizes which groups can be realized as the group of self-homotopy equivalences or their quotients for $A_n^2$-polyhedra, especially for $n=2$ and $n eq 2$ cases.

Findings

Not all groups can be realized as $\\mathcal{E}(X)$ or its quotient for $A_n^2$-polyhedra.

Specific results are provided for the case when $n=2$.

The structure of the quotient group $\\mathcal{E}(X)/\\mathcal{E}_*(X)$ is constrained by the properties of the polyhedron.

Abstract

Let $X$ be a finite type $A_n^2$-polyhedron, $n \geq 2$. In this paper we study the quotient group $\mathcal{E}(X)/\mathcal{E}_*(X)$, where $\mathcal{E}(X)$ is the group of self-homotopy equivalences of $X$ and $\mathcal{E}_*(X)$ the subgroup of self-homotopy equivalences inducing the identity on the homology groups of $X$. We show that not every group can be realised as $\mathcal{E}(X)$ or $\mathcal{E}(X)/\mathcal{E}_*(X)$ for $X$ an $A_n^2$-polyhedron, $n\ge 3$, and specific results are obtained for $n=2$.