The topological and smooth Hausmann-Weinberger invariants disagree

Mike Miller Eismeier

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arXiv · 2601.07040

The topological and smooth Hausmann-Weinberger invariants disagree

Mike Miller Eismeier

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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.

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Mike Miller Eismeier. 2026-01-11. The topological and smooth Hausmann-Weinberger invariants disagree. https://arxiv.org/abs/2601.07040

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