The topological and smooth Hausmann-Weinberger invariants disagree

Abstract

For $\pi$ a finitely presented group, Hausmann and Weinberger defined $q(\pi) \in \mathbb Z$ to be the minimum Euler characteristic over all closed, oriented $4$-manifolds with fundamental group $\pi$. This short note establishes that this minimum value in general differs depending on whether one minimizes over topological manifolds or only those admitting a smooth structure.