{"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/on-possible-symmetry-groups-of-27-vertex","title":"On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane","arxiv_id":"2310.16679","date":"2023-10-25","proceeding":null,"authors":["Alexander A. Gaifullin"],"abstract":"In 1987 Brehm and K\\\"uhnel showed that any triangulation of a $d$-manifold (without boundary) that is not homeomorphic to the sphere has at least $3d/2+3$ vertices. Moreover, triangulations with exactly $3d/2+3$ vertices may exist only for `manifolds like projective planes', which can have dimensions $2$, $4$, $8$, and $16$ only. There is a $6$-vertex triangulation of the real projective plane $\\mathbb{RP}^2$, a $9$-vertex triangulation of the complex projective plane $\\mathbb{CP}^2$, and $15$-vertex triangulations of the quaternionic projective plane $\\mathbb{HP}^2$. Recently, the author has constructed first examples of $27$-vertex triangulations of manifolds like the octonionic projective plane $\\mathbb{OP}^2$. The four most symmetrical have symmetry group $\\mathrm{C}_3^3\\rtimes \\mathrm{C}_{13}$ of order $351$. These triangulations were constructed using a computer program after the symmetry group was guessed. However, it remained unclear why exactly this group is realized as the symmetry group and whether $27$-vertex triangulations of manifolds like $\\mathbb{OP}^2$ exist with other (possibly larger) symmetry groups. In this paper we find strong restrictions on symmetry groups of such $27$-vertex triangulations. Namely, we present a list of $26$ subgroups of $\\mathrm{S}_{27}$ containing all possible symmetry groups of $27$-vertex triangulations of manifolds like the octonionic projective plane. (We do not know whether all these subgroups can be realized as symmetry groups.) The group $\\mathrm{C}_3^3\\rtimes \\mathrm{C}_{13}$ is the largest group in this list, and the orders of all other groups do not exceed $52$. A key role in our approach is played by the use of Smith and Bredon's results on the topology of fixed point sets of finite transformation groups.","url_abs":"https://arxiv.org/abs/2310.16679v2","url_pdf":"https://arxiv.org/pdf/2310.16679v2.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":"on-possible-symmetry-groups-of-27-vertex","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}