The ring of stable homotopy classes of self-maps of $A_n^2$-polyhedra

TL;DR

This paper investigates which rings can be realized as the stable homotopy classes of self-maps of certain polyhedra, showing specific constructions and obstructions for realizability.

Contribution

It demonstrates that the sum of three endomorphism rings, with one free, can be realized as such a ring, and identifies non-realizability of _p^3 for finite type polyhedra.

Findings

Sum of three endomorphism rings is realizable as (X,X)

_p^3 is not realizable for finite type A_n^2-polyhedra

One endomorphism ring must be free for realizability

Abstract

We raise the problem of realisability of rings as $\{X,X\}$ the ring of stable homotopy classes of self-maps of a space $X$. By focusing on $A_n^2$-polyhedra, we show that the direct sum of three endomorphism rings of abelian groups, one of which must be free, is realisable as $\{X,X\}$ modulo the acyclic maps. We also show that $\mathbb{F}_p^3$ is not realisable in the setting of finite type $A_n^2$-polyhedra, for $p$ any prime.