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

TL;DR

The paper proves a sharp bound for the simplicial volume of closed nonpositively curved $4$-manifolds in terms of their Euler characteristic: $\|M\| = \frac{1}{11}|\chi(M)|$ under the given curvature hypothesis, in particular yielding $\|M\|>0$ whenever $\chi(M)\neq 0$. The method combines the Gauss–Bonnet theorem for Riemannian simplices (Allendoerfer–Weil) with a geodesic simplex decomposition in the universal cover, relating topological data to curvature integrals via $\Psi_r$ integrands. As an application, the result implies positivity of simplicial volume for closed nonpositively curved $4$-manifolds with negative Ricci curvature, addressing a four-dimensional case of Gromov’s conjecture and contributing to the broader dialogue on how curvature constraints control $\|M\|$. The approach also yields corollaries on connected sums and products, and informs the existence of manifolds with prescribed sign of $\|M\|$.

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.

Theorems & Definitions (10)

• Conjecture 1.1
• Conjecture 1.2
• Conjecture 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Definition 2.1
• Definition 2.2
• Remark 2.3
• proof : Proof of Theorem \ref{['main thm']}