Minimal triangulations of manifolds like projective planes, and their symmetry groups

A theorem by Brehm and Kühnel (1987) says that a $d$-dimensional combinatorial manifold $K$ (without boundary) with $n$ vertices is PL homeomorphic to the sphere, provided that $n$ is less than $3d/2+3$. Moreover, if n is equal to $3d/2+3$, then $K$ is PL homeomorphic to either the sphere or a manifold like a projective plane, which exist in dimensions 2, 4, 8, and 16 only. There exists a 6-vertex triangulation of the real projective plane (the quotient of the boundary of regular icosahedron by the antipodal involution), a 9-vertex triangulation of the complex projective plane (Kühnel, 1983) and 15-vertex triangulations of the quaternionic projective plane (Brehm and Kühnel, 1992). Recently the speaker has constructed first examples of 27-vertex triangulations of manifolds like the octonionic projective plane and a lot of new 15-vertex triangulations of the quaternionic projective plane. I will speak about these results and also about symmetry groups of these traingulations.