Minimal Euler Characteristics for Even-Dimensional Manifolds with Finite Fundamental Group

TL;DR

This paper investigates the minimal Euler characteristics of even-dimensional manifolds with specific fundamental groups, extending previous bounds and applying results to restrict possible fundamental groups of rational homology 4-spheres.

Contribution

It introduces a generalized invariant for even-dimensional manifolds and provides new bounds based on cohomological invariants of the fundamental group.

Findings

Established new lower bounds for Euler characteristics.

Extended previous estimates to higher dimensions.

Applied results to restrict fundamental groups of rational homology 4-spheres.

Abstract

We consider the Euler characteristics $\chi(M)$ of closed orientable topological $2n$-manifolds with $(n-1)$-connected universal cover and a given fundamental group $G$ of type $F_n$. We define $q_{2n}(G)$, a generalized version of the Hausmann-Weinberger invariant for 4-manifolds, as the minimal value of $(-1)^n\chi (M)$. For all $n\geq 2$, we establish a strengthened and extended version of their estimates, in terms of explicit cohomological invariants of $G$. As an application we obtain new restrictions for non-abelian finite groups arising as fundamental groups of rational homology 4-spheres.