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On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane

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 ca...

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Published in:arXiv.org 2024-04
Main Author: Gaifullin, Alexander A
Format: Article
Language:English
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2310.16679