On polytopal upper bound spheres

TL;DR

This paper proves that odd-dimensional polytopal upper bound spheres are part of a specific class where all vertex links are k-stacked spheres, revealing an exception to a conjecture about neighborly members of this class.

Contribution

It establishes that odd-dimensional polytopal upper bound spheres belong to the generalized Walkup class, showing a surprising link between face-maximizing and face-minimizing properties.

Findings

Odd-dimensional polytopal upper bound spheres are in the generalized Walkup class.

This class's vertex links are all k-stacked spheres.

The result shows an exception to the conjecture about tightness for certain neighborly spheres.

Abstract

Generalizing a result (the case $k = 1$) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k + 1$ belongs to the generalized Walkup class ${\cal K}_k(2k + 1)$, i.e., all its vertex links are $k$-stacked spheres. This is surprising since the $k$-stacked spheres minimize the face-vector (among all polytopal spheres with given $f_0,..., f_{k - 1}$) while the upper bound spheres maximize the face vector (among spheres with a given $f_0$). It has been conjectured that for $d\neq 2k + 1$, all $(k + 1)$-neighborly members of the class ${\cal K}_k(d)$ are tight. The result of this paper shows that, for every $k$, the case $d = 2k +1$ is a true exception to this conjecture.