{"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/634-vertex-transitive-and-more-than-10-103","title":"634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane","arxiv_id":"2207.08507","date":"2022-07-18","proceeding":null,"authors":["Alexander A. Gaifullin"],"abstract":"In 1987 Brehm and K\\\"uhnel showed that any combinatorial $d$-manifold with less than $3d/2+3$ vertices is PL homeomorphic to the sphere and any combinatorial $d$-manifold with exactly $3d/2+3$ vertices is PL homeomorphic to either the sphere or a manifold like a projective plane in the sense of Eells and Kuiper. The latter possibility may occur for $d\\in\\{2,4,8,16\\}$ only. There exist a unique $6$-vertex triangulation of $\\mathbb{RP}^2$, a unique $9$-vertex triangulation of $\\mathbb{CP}^2$, and at least three $15$-vertex triangulations of $\\mathbb{HP}^2$. However, until now, the question of whether there exists a $27$-vertex triangulation of a manifold like the octonionic projective plane has remained open. We solve this problem by constructing a lot of examples of such triangulations. Namely, we construct $634$ vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Four of them have symmetry group $\\mathrm{C}_3^3\\rtimes \\mathrm{C}_{13}$ of order $351$, and the other $630$ have symmetry group $\\mathrm{C}_3^3$ of order $27$. Further, we construct more than $10^{103}$ non-vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Most of them have trivial symmetry group, but there are also symmetry groups $\\mathrm{C}_3$, $\\mathrm{C}_3^2$, and $\\mathrm{C}_{13}$. We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane $\\mathbb{OP}^2$. Nevertheless, we have no proof of this fact so far.","url_abs":"https://arxiv.org/abs/2207.08507v4","url_pdf":"https://arxiv.org/pdf/2207.08507v4.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":"634-vertex-transitive-and-more-than-10-103","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":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}