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

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.

10 pages, corrected some indices $n\ge 3$ that should have said $n\ge 4$

References in corpus (3)

• Realisability problem in arrow categories
• Homotopically rigid Sullivan algebras and Their applications
• The group of self-homotopy equivalences of $A_n^2$-polyhedra