{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/new-examples-and-partial-classification-of-15","title":"New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane","arxiv_id":"2311.11309","date":"2023-11-19","proceeding":null,"authors":["Alexander A. Gaifullin"],"abstract":"Brehm and K\\\"uhnel (1992) constructed three 15-vertex combinatorial 8-manifolds `like the quaternionic projective plane' with symmetry groups $\\mathrm{A}_5$, $\\mathrm{A}_4$, and $\\mathrm{S}_3$, respectively. Gorodkov (2016) proved that these three manifolds are in fact PL homeomorphic to $\\mathbb{HP}^2$. Note that 15 is the minimal number of vertices of a combinatorial 8-manifold that is not PL homeomorphic to $S^8$. In the present paper we construct a lot of new 15-vertex triangulations of $\\mathbb{HP}^2$. A surprising fact is that such examples are found for very different symmetry groups, including those not in any way related to the group $\\mathrm{A}_5$. Namely, we find 19 triangulations with symmetry group $\\mathrm{C}_7$, one triangulation with symmetry group $\\mathrm{C}_6\\times\\mathrm{C}_2$, 14 triangulations with symmetry group $\\mathrm{C}_6$, 26 triangulations with symmetry group $\\mathrm{C}_5$, one new triangulation with symmetry group $\\mathrm{A}_4$, and 11 new triangulations with symmetry group $\\mathrm{S}_3$. Further, we obtain the following classification result. We prove that, up to isomorphism, there are exactly 75 triangulations of $\\mathbb{HP}^2$ with 15 vertices and symmetry group of order at least 4: the three Brehm-K\\\"uhnel triangulations and the 72 new triangulations listed above. On the other hand, we show that there are plenty of triangulations with symmetry groups $\\mathrm{C}_3$ and $\\mathrm{C}_2$, as well as the trivial symmetry group.","url_abs":"https://arxiv.org/abs/2311.11309v2","url_pdf":"https://arxiv.org/pdf/2311.11309v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"new-examples-and-partial-classification-of-15","repo_url":"https://github.com/agaif/triangulations-like-op2","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}