Many triangulated odd-spheres

TL;DR

This paper constructs exponentially many triangulations of odd-dimensional spheres, significantly improving previous lower bounds and introducing new classes of geodesic triangulations and polytopes with complex facet structures.

Contribution

It provides new exponential lower bounds on the number of triangulations of odd-spheres and introduces geodesic triangulations and non-simplex facet-rich polytopes.

Findings

Constructed at least 2^{Ω(n^k)} triangulations of (2k-1)-spheres.

Developed 2^{Ω(n^{k-1+1/k})} geodesic triangulations.

Built 4-polytopes with Ω(n^{3/2}) non-simplex facets or edges of degree three.

Abstract

It is known that the $(2k-1)$-sphere has at most $2^{O(n^k \log n)}$ combinatorially distinct triangulations with $n$ vertices, for every $k\ge 2$. Here we construct at least $2^{\Omega(n^k)}$ such triangulations, improving on the previous constructions which gave $2^{\Omega(n^{k-1})}$ in the general case (Kalai) and $2^{\Omega(n^{5/4})}$ for $k=2$ (Pfeifle-Ziegler). We also construct $2^{\Omega\left(n^{k-1+\frac{1}{k}}\right)}$ geodesic (a.k.a. star-convex) $n$-vertex triangualtions of the $(2k-1)$-sphere. As a step for this (in the case $k=2$) we construct $n$-vertex $4$-polytopes containing $\Omega(n^{3/2})$ facets that are not simplices, or with $\Omega(n^{3/2})$ edges of degree three.