Positivity of simplicial volume for closed nonpositively curved four-manifolds with nonzero Euler characteristic
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918181110546432 |
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| author | Kim, Inkang Wan, Xueyuan |
| author_facet | Kim, Inkang Wan, Xueyuan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_09524 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positivity of simplicial volume for closed nonpositively curved four-manifolds with nonzero Euler characteristic Kim, Inkang Wan, Xueyuan Geometric Topology Differential Geometry 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. |
| title | Positivity of simplicial volume for closed nonpositively curved four-manifolds with nonzero Euler characteristic |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2506.09524 |