On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane
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arXiv
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2023
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| _version_ | 1866929597346480128 |
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| author | Gaifullin, Alexander A. |
| author_facet | Gaifullin, Alexander A. |
| contents | In 1987 Brehm and Kühnel 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_16679 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane Gaifullin, Alexander A. Combinatorics Group Theory Geometric Topology 57Q15, 57Q70, 05E45, 55M35 In 1987 Brehm and Kühnel 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. |
| title | On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane |
| topic | Combinatorics Group Theory Geometric Topology 57Q15, 57Q70, 05E45, 55M35 |
| url | https://arxiv.org/abs/2310.16679 |