Trees of homotopy types of 2-dimensional $\textrm {CW}$ complexes. II

Abstract:

A $\pi$-complex is a finite, connected $2$-dimensional CW complex with fundamental group $\pi$. The tree $\text {HT} (\pi )$ of homotopy types of $\pi$-complexes has width $\leq N$ if there is a root $Y$ of the tree such that, for any $\pi$-complex $X,X \vee ( \vee _{i = 1}^NS_i^2)$ lies on the stalk generated by $Y$. Let $\pi$ be a finite abelian group with torsion coefficients ${\tau _1}, \cdots ,{\tau _n}$. The main theorem of this paper asserts that width $\text {HT} (\pi ) \leq n(n - 1)/2$. This generalizes the results of [4].