Fundamental groups of small simplicial complexes
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915596837322752 |
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| author | Govc, Dejan Marzantowicz, Wacław Michalak, Łukasz Patryk Pavešić, Petar |
| author_facet | Govc, Dejan Marzantowicz, Wacław Michalak, Łukasz Patryk Pavešić, Petar |
| contents | The number of nonisomorphic simplicial complexes with up to $n$ vertices increases super-exponentially with $n$, which makes exhaustive computation of invariants associated with such complexes a daunting task. In this paper we provide a complete list of groups that arise as fundamental groups of simplicial complexes with at most $8$ vertices. In addition we give many examples of fundamental groups of complexes with $9$ vertices although the complete classification seems to be beyond reach at the moment.
Our results lead to many applications, including progress on the Björner-Lutz conjecture regarding vertex-minimal triangulations of the Poincaré homology sphere, improved recognition criteria for PL triangulations of manifolds and computation of the Karoubi-Weibel invariant for many groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02586 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fundamental groups of small simplicial complexes Govc, Dejan Marzantowicz, Wacław Michalak, Łukasz Patryk Pavešić, Petar Algebraic Topology 55U10, 20F65 (Primary) 55M30, 57-04, 57Q15 (Secondary) The number of nonisomorphic simplicial complexes with up to $n$ vertices increases super-exponentially with $n$, which makes exhaustive computation of invariants associated with such complexes a daunting task. In this paper we provide a complete list of groups that arise as fundamental groups of simplicial complexes with at most $8$ vertices. In addition we give many examples of fundamental groups of complexes with $9$ vertices although the complete classification seems to be beyond reach at the moment. Our results lead to many applications, including progress on the Björner-Lutz conjecture regarding vertex-minimal triangulations of the Poincaré homology sphere, improved recognition criteria for PL triangulations of manifolds and computation of the Karoubi-Weibel invariant for many groups. |
| title | Fundamental groups of small simplicial complexes |
| topic | Algebraic Topology 55U10, 20F65 (Primary) 55M30, 57-04, 57Q15 (Secondary) |
| url | https://arxiv.org/abs/2511.02586 |