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

2025/06/11 by Inkang Kim, Xueyuan Wan, Kim, Inkang +1 · 1 citation
#math.GT #math.DG

paper · pdf · doi:10.48550/arxiv.2506.09524

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.

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