On graphs, homology bases, and triangulated homology spheres
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912576047153152 |
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| author | Adiprasito, Karim Lackenby, Marc Souto, Juan Yashfe, Geva |
| author_facet | Adiprasito, Karim Lackenby, Marc Souto, Juan Yashfe, Geva |
| contents | We describe a construction that takes as input a graph and a basis for its first homology, and returns a triangulation of a 3-dimensional homology sphere. This makes precise an idea of M. Gromov and A. Nabutovski. The immediate application, essentially described by Gromov, is to translate problems about asymptotics of homology sphere triangulations to asymptotic counting problems for constant-degree graphs with "short" homology bases. We construct families of 3- sphere triangulations with dual graphs that are expanders, answering a relaxation of a question asked by G. Kalai. Our results also imply that if the number of d-dimensional triangulated homology spheres with n facets is superexponential in n for some d then the same holds for d = 3. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06275 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On graphs, homology bases, and triangulated homology spheres Adiprasito, Karim Lackenby, Marc Souto, Juan Yashfe, Geva Combinatorics 05C10, 57Q15 We describe a construction that takes as input a graph and a basis for its first homology, and returns a triangulation of a 3-dimensional homology sphere. This makes precise an idea of M. Gromov and A. Nabutovski. The immediate application, essentially described by Gromov, is to translate problems about asymptotics of homology sphere triangulations to asymptotic counting problems for constant-degree graphs with "short" homology bases. We construct families of 3- sphere triangulations with dual graphs that are expanders, answering a relaxation of a question asked by G. Kalai. Our results also imply that if the number of d-dimensional triangulated homology spheres with n facets is superexponential in n for some d then the same holds for d = 3. |
| title | On graphs, homology bases, and triangulated homology spheres |
| topic | Combinatorics 05C10, 57Q15 |
| url | https://arxiv.org/abs/2509.06275 |