Abstract:
In 1987, Brehm and Kühnel proved that the only $d$-manifold that admits a triangulation with fewer than $3d/2+3$ vertices is the $d$-dimensional sphere $S^d$. Moreover, if a $d$-manifold $M^d$ different from $S^d$ admits a triangulation with exactly $3d/2+3$ vertices, then d is one of the four numbers $2, 4, 8$, and $16$, and $M^d$ is a manifold like a projective plane. This is a remarkable class of manifolds that appeared in the 1950s in the works of Milnor, Shimada, Eells, and Kuiper. It consists of manifolds that are very close in a number of their properties to the true projective planes $\mathbb{RP}^2, \mathbb{CP}^2, \mathbb{HP}^2$ and $\mathbb{OP}^2$ corresponding to the four classical division algebras $\mathbb R, \mathbb C, \mathbb H$ and $\mathbb O$. Thus, the construction and study of $(3d/2+3)$-vertex triangulations of $d$-manifolds like projective planes is of particular interest. Until recently, very few such triangulations were known: one in each of the dimensions $2$ and $4$ and six in dimension $8$. No examples in dimension $16$ were known.
I will present a series of my recent results on this problem. The main one is the solution of the problem on existence of $27$-vertex triangulations of $16$-dimensional manifolds like the octonion projective plane. A huge number (more than $10^{103}$) of such triangulations have been constructed. The four most symmetric triangulations were found using a specially developed computer algorithm; the others were obtained from these four using special moves called triple flips. Further, I will talk on the construction of a large number of new examples of $15$-vertex triangulations of the quaternionic projective plane $\mathbb{HP}^2$, and their partial classification, as well as about results on possible symmetry groups of $27$-vertex $16$-dimensional and $15$-vertex $8$-dimensional simplicial manifolds like projective planes.
