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

Abstract

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 ${\mathrm{C}}_3^3⋊{\mathrm{C}}_{13}$ 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.

arXiv comment45 pages, 19 tables; minor changes

References in corpus (9)

• Triangulated Manifolds with Few Vertices: Combinatorial Manifolds
• Local formulae for combinatorial Pontrjagin classes
• Triangulated Manifolds with Few Vertices: Vertex-Transitive Triangulations I
• A 15-vertex triangulation of the quaternionic projective plane
• Construction of combinatorial manifolds with the prescribed sets of links of vertices
• Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class
• On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane
• Triangulations and Severi varieties
• A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices

Cited by in corpus (2)

• On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane
• New examples and partial classification of 15-vertex triangulations of the quaternionic projective plane