On the combinatorics of Murai spheres and its applications

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.

💡 Research Summary

The paper investigates the combinatorial structure of Murai spheres, a class of simplicial spheres introduced as a generalization of Bier spheres to arbitrary multicomplexes. After recalling the definitions of c‑multicomplexes, their Alexander duality, and the associated Stanley–Reisner ideals, the authors present a key result (Theorem 2.6) that expresses the Stanley–Reisner ideal of a Murai sphere as the sum of three polarized ideals. This algebraic description underlies all subsequent combinatorial analyses.

The first major contribution is a complete classification of Murai spheres in dimensions 1 and 2. In dimension 1 the only possible c‑vectors are (3), (2,1) and (1,1,1); the corresponding spheres are cycles Z₃, Z₄ or Z₅, depending on the choice of the multicomplex M. In dimension 2 the situation is richer: the authors enumerate all admissible c‑vectors and multicomplexes, showing that every 2‑dimensional Murai sphere is either a join of two simplex boundaries ∂Δₐ * ∂Δ_b (with a,b≥0) or a more intricate join involving the boundary of a cyclic polytope. By analyzing the 1‑skeletons they prove that all such low‑dimensional Murai spheres correspond to simple polytopes that admit Delzant realizations, i.e., their normal fans are regular. This settles the Delzant‑realization problem for Murai spheres in dimensions 1 and 2.

The second focus is on chordality. A simplicial complex is chordal if its 1‑skeleton contains no induced cycles of length ≥4. Previous work showed that for Bier spheres of dimension >1, chordality coincides with being stacked. The authors extend the investigation to Murai spheres with at most two variables (m≤2). They list all c‑multicomplexes M for which Bier_c(M) is chordal, and demonstrate that some of these chordal Murai spheres are not stacked—specifically, joins of two simplex boundaries and certain cyclic‑polytope boundaries provide new non‑stacked chordal examples. Thus, chordality does not force a Murai sphere to be stacked, contrasting with the Bier case.

The third contribution concerns two toric‑topological invariants: the Buchstaber number s(K) and the chromatic number χ(K). The Buchstaber number measures the maximal rank of a torus subgroup acting freely on the moment‑angle complex Z_K; for a simplicial (d‑1)-sphere with v vertices one always has s(K) ≤ v − d. For Bier spheres the bound is attained. The authors prove that for any Murai sphere K the Buchstaber number is either maximal (s(K)=v − d) or one less (s(K)=v − d − 1). Both possibilities occur, as illustrated by explicit families of Murai spheres. Regarding the chromatic number, they show that Murai spheres can realize all values permitted by the underlying graph, and they give a precise description of the attainable range in Theorem 4.3.

Finally, the paper constructs, for every even dimension d≥6, Murai spheres that do not admit regular (Delzant) realizations, thereby demonstrating that the positive Delzant result in low dimensions does not extend to higher dimensions. The authors conclude with several open problems, including a full classification of d‑dimensional Murai spheres whose 1‑skeleton is a complete graph, a deeper understanding of the relationship between chordality and stackedness for general m, and tighter bounds for Buchstaber and chromatic numbers in higher dimensions. Overall, the work significantly advances the theory of Murai spheres, linking combinatorial, geometric, and topological aspects.