Centrally symmetric and balanced triangulations of ${\mathbb{S}}^2\times {\mathbb{S}}^{d-3}$ with few vertices

Abstract

A small triangulation of the sphere product can be found in lower dimensions by computer search and is known in few other cases: Klee and Novik constructed a centrally symmetric triangulation of ${\mathbb{S}}^i\times {\mathbb{S}}^{d-i-1}$ with $2d+2$ vertices for all $d\ge 3$ and $1\le i\le d-2$; they also proposed a balanced triangulation of ${\mathbb{S}}^1\times {\mathbb{S}}^{d-2}$ with $3d$ or $3d+2$ vertices. In this paper, we provide another centrally symmetric $(2d+2)$-vertex triangulation of ${\mathbb{S}}^2\times {\mathbb{S}}^{d-3}$. We also construct the first balanced triangulation of ${\mathbb{S}}^2\times {\mathbb{S}}^{d-3}$ with $4d$ vertices, using a sphere decomposition inspired by handle theory.

arXiv comment17 pages, accepted to the European Journal of Combinatorics