Positivity of simplicial volume for closed nonpositively curved four-manifolds with nonzero Euler characteristic

TL;DR

This paper proves that closed nonpositively curved four-manifolds with nonzero Euler characteristic have positive simplicial volume, confirming conjectures relating curvature, Euler characteristic, and simplicial volume in four dimensions.

Contribution

It establishes the positivity of simplicial volume for certain four-manifolds using the Gauss-Bonnet theorem, partially resolving longstanding conjectures.

Findings

Positivity of simplicial volume for nonpositively curved 4-manifolds with nonzero Euler characteristic

Confirmation of Gromov's conjecture on positive simplicial volume for negatively Ricci curved 4-manifolds

Application of Gauss-Bonnet theorem to relate curvature and simplicial volume in four dimensions

Abstract

In this paper, by employing the Gauss-Bonnet theorem for Riemannian simplices due to Allendoerfer and Weil, we show that if a closed nonpositively curved $4$-manifold has nonzero Euler characteristic, then its simplicial volume is necessarily positive. This result partially resolves conjectures posed by Connell-Ruan-Wang and Gromov concerning the relationship between the simplicial volume and the Euler characteristic for four-dimensional manifolds. As an application, we show that if a closed nonpositively curved $4$-manifold has negative Ricci curvature, then its simplicial volume is positive, thereby confirming in dimension four another conjecture of Gromov on the positivity of simplicial volume.