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In this paper, we propose a new method to the homotopy classification of mapsbetween $(n-1)$-connected polyhedra $X,Y$ of dimension at most $n+2$, $n\geq4$, and determine appropriate generating sets of the Abelian groups $[X,Y]$ ofhomotopy classes of based maps from polyhedra $X$ to $Y$. The group structuresof $[X,Y]$ with generators are used to study certain (sub)groups ofself-homotopy equivalences of certain Chang-complexes.

Homotopy classification of maps between $\mathbf{A}_n^2$-complexes and applications in self-homotopy equivalences