Chromatic numbers, Buchstaber numbers and chordality of Bier spheres

TL;DR

This paper characterizes the chromatic numbers of Bier spheres, provides formulas for Buchstaber numbers based on the underlying complex, and classifies chordal Bier spheres with their geometric realizations.

Contribution

It offers a complete classification of Bier spheres by chromatic number, derives formulas for Buchstaber numbers, and classifies and realizes chordal Bier spheres as stacked polytopes.

Findings

All Bier spheres of dimension d have chromatic number d+1 or d+2.

Formulas for Buchstaber numbers in terms of the f-vector of the underlying complex.

Classification and geometric realization of chordal Bier spheres.

Abstract

We describe all the Bier spheres of dimension $d$ with chromatic number equal to $d+1$ and prove that all other $d$-dimensional Bier spheres have chromatic number equal to $d+2$, for any integer $d\geq 0$. Then we prove a general formula for complex and mod $p$ Buchstaber numbers of a Bier sphere $\mathrm{Bier}(K)$, for each prime $p\in\mathbb{N}$ in terms of the $f$-vector of the underlying simplicial complex $K$. Finally, we classify all chordal Bier spheres and obtain their canonical realizations as boundaries of stacked polytopes.