The Nevo--Santos--Wilson spheres are shellable

TL;DR

This paper proves that all spheres constructed by Nevo, Santos, and Wilson are shellable, leading to a large family of shellable simplicial spheres with many vertices, advancing understanding of their combinatorial properties.

Contribution

It demonstrates that the spheres from Nevo, Santos, and Wilson's construction are shellable, expanding the class of known shellable spheres.

Findings

All spheres from their method are shellable.

There are exponentially many shellable spheres in high dimensions.

The results unify previous findings on shellability of simplicial spheres.

Abstract

Nevo, Santos, and Wilson constructed $2^{\Omega(N^d)}$ combinatorially distinct simplicial $(2d-1)$-spheres with $N$ vertices. We prove that all spheres produced by one of their methods are shellable. Combining this with prior results of Kalai, Lee, and Benedetti and Ziegler, we conclude that for all $D \ge 3$, there are $2^{\Theta(N^{\lceil D/2 \rceil})}$ shellable simplicial $D$-spheres with $N$ vertices.