Two remarks on Wall's D2 problem

TL;DR

This paper discusses conditions under which certain finite complexes with Wall's D2-property can be simplified to finite 2-complexes, focusing on the role of the fundamental group and Euler characteristic.

Contribution

It provides new insights into Wall's D2 problem by establishing when complexes can be simplified based on the fundamental group and Euler characteristic constraints.

Findings

Finite groups isomorphic to subgroups of SO(3) have the D2-property.

Under certain conditions, complexes with Wall's D2-conditions are homotopy equivalent to finite 2-complexes.

The simple homotopy type depends only on the fundamental group and Euler characteristic.

Abstract

If a finite group $G$ is isomorphic to a subgroup of $SO(3)$, then $G$ has the D2-property. Let $X$ be a finite complex satisfying Wall's D2-conditions. If $\pi_1(X)=G$ is finite, and $\chi(X) \geq 1-Def(G)$, then $X \vee S^2$ is simple homotopy equivalent to a finite $2$-complex, whose simple homotopy type depends only on $G$ and $\chi(X)$.