A minimal triangulation of the quaternionic projective plane

Городков, Денис Александрович

doi:10.1070/rm9726

Cited by 5 publications

(2 citation statements)

References 6 publications

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“…Nevertheless, they could not decide whether these combinatorial manifolds are homeomorphic to the quaternionic projective plane HP 2 . This problem remained open for a long time until Gorodkov proved that the Brehm-Kühnel complexes are indeed PL homeomorphic to HP 2 , see [20], [21].…”

Section: Introductionmentioning

confidence: 99%

“…He showed that in dimension 8 such manifolds are distinguished by their Pontryagin numbers, and in dimension 16 by their Pontryagin numbers and certain exotic PL characteristic number. Gorodkov's result in the quaternionic case (see [20], [21]) was based on the computation of the first Pontryagin class of the Brehm-Kühnel combinatorial manifolds using an explicit combinatorial formula due to the author of the present paper, cf. [14], [15], [17], [19].…”

Section: Introductionmentioning

confidence: 99%

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634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

Gaifullin¹

2022

Preprint

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In 1987 Brehm and Kühnel 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 ∈ {2, 4, 8, 16} only. There exist a unique 6-vertex triangulation of RP 2 , a unique 9-vertex triangulation of CP 2 , and at least three 15-vertex triangulations of 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 C 3 3 ⋊ C 13 of order 351, and the other 630 have symmetry group 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 C 3 , C 2 3 , and C 13 . We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane OP 2 . Nevertheless, we have no proof of this fact so far.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

Gaifullin¹

2022

Preprint

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A 15-Vertex Triangulation of the Quaternionic Projective Plane

Городков

2019

Discrete Comput Geom

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In 1992, Brehm and Kühnel constructed a 8-dimensional simplicial complex M 8 15 with 15 vertices as a candidate to be a minimal triangulation of the quaternionic projective plane. They managed to prove that it is a manifold "like a projective plane" in the sense of Eells and Kuiper. However, it was not known until now if this complex is PL homeomorphic (or at least homeomorphic) to HP 2 . This problem was reduced to the computation of the first rational Pontryagin class of this combinatorial manifold. Realizing an algorithm due to Gaifullin, we compute the first Pontryagin class of M 8 15 . As a result, we obtain that it is indeed a minimal triangulation of HP 2 .

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634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

Gaifullin

2024

Известия Российской академии наук. Серия математическая

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In 1987 Brehm and Kühnel 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. Bibliography: 52 titles.

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A minimal triangulation of the quaternionic projective plane

Cited by 5 publications

References 6 publications

634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

A 15-Vertex Triangulation of the Quaternionic Projective Plane

634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

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