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The topological and smooth Hausmann-Weinberger invariants disagree

Mike Miller Eismeier

Abstract

For $π$ a finitely presented group, Hausmann and Weinberger defined $q(π) \in \mathbb Z$ to be the minimum Euler characteristic over all closed, oriented $4$-manifolds with fundamental group $π$. 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.

The topological and smooth Hausmann-Weinberger invariants disagree

Abstract

For a finitely presented group, Hausmann and Weinberger defined to be the minimum Euler characteristic over all closed, oriented -manifolds with fundamental group . 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.
Paper Structure (1 theorem, 1 equation)

This paper contains 1 theorem, 1 equation.

Key Result

Theorem 1

There exists a finitely-presented group $\pi$ for which $q^{\text{TOP}}(\pi)< q^{\text{DIFF}}(\pi)$.

Theorems & Definitions (3)

  • Theorem
  • proof
  • Remark