The topological and smooth Hausmann-Weinberger invariants disagree

TL;DR

This paper demonstrates that the minimal Euler characteristic for a given finitely presented group differs when considering topological versus smooth 4-manifolds, highlighting a fundamental distinction in their invariants.

Contribution

It establishes that the Hausmann-Weinberger invariant varies between topological and smooth categories for certain groups, revealing a key difference in 4-manifold invariants.

Findings

The minimal Euler characteristic differs between topological and smooth 4-manifolds.

The invariants are not always equal, indicating a fundamental discrepancy.

This result impacts the understanding of 4-manifold classification and invariants.

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.