Papers › 634 vertex-transitive and more than 10¹⁰³ non-vertex-transitive 27-vertex...
634 vertex-transitive and more than 10¹⁰³ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane
Alexander A. Gaifullin
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In 1987 Brehm and K\"uhnel 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 ℝℙ², a unique $9$-vertex triangulation of ℂℙ², and at least three $15$-vertex triangulations of ℍℙ². 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 C₃³⋊C₁₃ of order $351$, and the other $630$ have symmetry group C₃³ of order $27$. Further, we construct more than 10¹⁰³ 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 C₃, C₃², and C₁₃. We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane 𝕆ℙ². Nevertheless, we have no proof of this fact so far.
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