On the combinatorics of Murai spheres and its applications

TL;DR

This paper classifies Murai spheres in low dimensions, explores their convex realizations, describes specific chordal cases, and determines key combinatorial invariants like Buchstaber and chromatic numbers.

Contribution

It provides a complete classification of Murai spheres in dimensions 1 and 2, including their convex realizations and combinatorial invariants, advancing understanding of their structure.

Findings

Murai spheres in dimensions 1 and 2 are classified.

Convex simple polytopes associated with these spheres have Delzant realizations.

All possible Buchstaber and chromatic numbers for Murai spheres are determined.

Abstract

We classify the combinatorial types of Murai spheres in dimensions $1$ and $2$, thereby showing that the corresponding convex simple polytopes have Delzant realizations. Then we describe all chordal Murai spheres $\mathrm{Bier}_c(M)$ with $c\in\mathbb N^m$ and $m\leq 2$. Finally, we find all possible values for the Buchstaber and chromatic numbers of arbitrary Murai spheres.