Homotopy classification of based maps between $\mathbf{A}_n^2$-complexes

TL;DR

This paper provides elementary proofs for the abelian group structure of homotopy classes of maps between certain CW-complexes and explicitly determines generators, aiding in understanding self-homotopy equivalences.

Contribution

It offers new elementary proofs of known abelian group structures and explicitly identifies generators for homotopy classes of maps between specific complexes.

Findings

Confirmed the abelian group structure of $[X,Y]$

Explicitly determined generators for $[X,Y]$

Computed subgroups of self-homotopy equivalences

Abstract

Let $X,Y$ be $(n-1)$-connected finite pointed CW-complexes of dimension at most $n+2$, $n\geq 3$. In this paper we give elementary proofs of the abelian group structure of $[X,Y]$ of homotopy classes of based maps from $X$ to $Y$, which was due to Baues and Schmidt. Furthermore, we determine the explicit generators associated to $[X,Y]$. As an application, we compute certain (sub)groups of self-homotopy equivalences of certain Chang complexes.