On the combinatorics of Murai spheres and its applications

TL;DR

This work studies Murai spheres, a generalization of Bier spheres, focusing on their combinatorial structure and toric-topological invariants. It provides a complete classification of Murai spheres in dimensions $1$ and $2$, proving Delzant realizations for the associated polytopes, and classifies chordal Murai spheres for $m\le 2$ with explicit exceptional cases. The authors analyze Buchstaber and chromatic numbers, showing Murai spheres are either maximal or one less, and present explicit constructions (including cyclic polytopes) that realize minimal or near-minimal invariants. They also connect Murai spheres to truncation and stacked polytopes, and pose open problems on neighborliness, maximal Buchstaber numbers, and non-polytopal Murai spheres, highlighting rich interactions between combinatorics and toric geometry.

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.

Figures (3)

• Figure 1: The two-dimensional Murai spheres $\mathcal{S}_i$ with $c=(1,1,1,1)$ and the corresponding simple polytopes $\mathcal{P}_i$ such that $\mathcal{S}_i=\partial\mathcal{P}_i^{*}$ for $1\leq i\leq 13$.
• Figure 2: Truncation polytopes $P$ and $Q$ corresponding to the two exceptional Murai spheres in dimension two.
• Figure 3: Those are up to symmetry all possible cases for 4-cycle in a Murai sphere. In cases 2, 3 and 4, we have $\alpha_1<\alpha_2<\alpha_3$.

Theorems & Definitions (45)

• Definition 2.1
• Remark
• Definition 2.2
• Proposition 2.3: LZ
• Definition 2.4
• Remark
• Definition 2.5
• Theorem 2.6
• Example 2.7
• Example 2.8