Small Triangulations of $4$-Manifolds: Introducing the $4$-Manifold Census
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914561915879424 |
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| author | Burke, Rhuaidi Antonio Burton, Benjamin A. Spreer, Jonathan |
| author_facet | Burke, Rhuaidi Antonio Burton, Benjamin A. Spreer, Jonathan |
| contents | We present a framework to classify PL-types of large censuses of triangulated $4$-manifolds, which we use to classify the PL-types of all triangulated $4$-manifolds with up to six pentachora. This is successful except for triangulations homeomorphic to the $4$-sphere, $\mathbb{C}P^2$, and the rational homology sphere $QS^4(2)$, where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations -- which we deem interesting in their own rights. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_04768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Small Triangulations of $4$-Manifolds: Introducing the $4$-Manifold Census Burke, Rhuaidi Antonio Burton, Benjamin A. Spreer, Jonathan Geometric Topology 57-04, 57-08, 57-11, 57Q15, 57Q05, 57Q25, 57K40, 57Q70, 55U10, 57R05, 57K41 We present a framework to classify PL-types of large censuses of triangulated $4$-manifolds, which we use to classify the PL-types of all triangulated $4$-manifolds with up to six pentachora. This is successful except for triangulations homeomorphic to the $4$-sphere, $\mathbb{C}P^2$, and the rational homology sphere $QS^4(2)$, where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations -- which we deem interesting in their own rights. |
| title | Small Triangulations of $4$-Manifolds: Introducing the $4$-Manifold Census |
| topic | Geometric Topology 57-04, 57-08, 57-11, 57Q15, 57Q05, 57Q25, 57K40, 57Q70, 55U10, 57R05, 57K41 |
| url | https://arxiv.org/abs/2412.04768 |