
New examples and partial classification of 15vertex triangulations of the quaternionic projective plane
A. A. Gaifullin^{abcd} ^{a} Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
^{b} Skolkovo Institute of Science and Technology
^{c} Lomonosov Moscow State University
^{d} Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow
Abstract:
Brehm and Kühnel (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–Kühnel 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.
Keywords:
minimal triangulation, quaternionic projective plane, manifold like a projective plane, Kühnel triangulation, vertextransitive triangulation, combinatorial manifold, transformation group, Smith theory, fixed point set, symmetry group
Received: November 19, 2023 Revised: May 19, 2024 Accepted: June 7, 2024
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