Spaces dominated by two-complexes

Abstract:

We say a group $G$ is of geometric dimension $\leqslant 2$ if there is an aspherical $2$-dimensional CW-complex $P$ with fundamental group isomorphic to $G$. In this note, we study the following problem: Suppose $G$ is a group of geometric dimension $\leqslant 2$ with associated aspherical $2$-dimensional CW-complex $P$. Suppose further that $X$ is a CW-complex having fundamental group isomorphic to $G$ and that $X$ is dominated by a $2$-complex. If the Wall invariant ${\text {W}}{{\text {a}}_2}[X] \in {\tilde K_0}(ZG)$ vanishes, does $X$ have the same homotopy type as $P \vee k{S^2}$ where $k{S^2}$ denotes the sum of $k$ copies of the $2$-sphere ${S^2}$?