A note on rational homology vanishing theorem for hypersurfaces in aspherical manifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909319483621376 |
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| author | He, Shihang Zhu, Jintian |
| author_facet | He, Shihang Zhu, Jintian |
| contents | In this note, we generalize Gromov's reduction \cite{Gro20} from the aspherical conjecture to the generalized filling radius conjecture to the smooth $\mathbb Q$-homology vanishing conjecture for hypersurface. In particular, we can show that any continuous map from a closed $4$-manifold admitting positive scalar curvature to an aspherical $5$-manifold induces zero map in $H_4(\cdot,\mathbb Q)$. As a corollary, we obtain the following splitting theorem: if a complete aspherical $5$-manifold has nonnegative scalar curvature and two ends, then it splits into the Riemannian product of a closed flat manifold and the real line. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_14008 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on rational homology vanishing theorem for hypersurfaces in aspherical manifolds He, Shihang Zhu, Jintian Differential Geometry In this note, we generalize Gromov's reduction \cite{Gro20} from the aspherical conjecture to the generalized filling radius conjecture to the smooth $\mathbb Q$-homology vanishing conjecture for hypersurface. In particular, we can show that any continuous map from a closed $4$-manifold admitting positive scalar curvature to an aspherical $5$-manifold induces zero map in $H_4(\cdot,\mathbb Q)$. As a corollary, we obtain the following splitting theorem: if a complete aspherical $5$-manifold has nonnegative scalar curvature and two ends, then it splits into the Riemannian product of a closed flat manifold and the real line. |
| title | A note on rational homology vanishing theorem for hypersurfaces in aspherical manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2311.14008 |