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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2401.02560 |
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| _version_ | 1866914028029214720 |
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| author | Peruyero, H. Contreras Suárez-Serrato, P. |
| author_facet | Peruyero, H. Contreras Suárez-Serrato, P. |
| contents | We show that the fundamental groups of smooth $4$-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to 4 when aspherical. We also show that closed $3$-manifold groups have asymptotic dimension at most 3. Our proof method yields that the asymptotic dimension of closed $3$-dimensional Alexandrov spaces is at most 3. We thus obtain that the Novikov conjecture holds for closed $4$-manifolds with such a geometric decomposition and closed $3$-dimensional Alexandrov spaces. Consequences of these results include a vanishing result for the Yamabe invariant of certain $0$-surgered geometric $4$-manifolds and the existence of zero in the spectrum of aspherical smooth $4$-manifolds with a geometric decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02560 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic dimension and geometric decompositions in dimensions 3 and 4 Peruyero, H. Contreras Suárez-Serrato, P. Geometric Topology Differential Geometry Group Theory We show that the fundamental groups of smooth $4$-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to 4 when aspherical. We also show that closed $3$-manifold groups have asymptotic dimension at most 3. Our proof method yields that the asymptotic dimension of closed $3$-dimensional Alexandrov spaces is at most 3. We thus obtain that the Novikov conjecture holds for closed $4$-manifolds with such a geometric decomposition and closed $3$-dimensional Alexandrov spaces. Consequences of these results include a vanishing result for the Yamabe invariant of certain $0$-surgered geometric $4$-manifolds and the existence of zero in the spectrum of aspherical smooth $4$-manifolds with a geometric decomposition. |
| title | Asymptotic dimension and geometric decompositions in dimensions 3 and 4 |
| topic | Geometric Topology Differential Geometry Group Theory |
| url | https://arxiv.org/abs/2401.02560 |