Stellar subdivisions, wedges and Buchstaber numbers

A seed is a PL sphere that is not obtainable by a wedge operation from any other PL sphere. In this paper, we study two operations on PL spheres, known as the stellar subdivision and the wedge, that preserve the maximality of Buchstaber numbers and polytopality. We construct a new polytopal toric colorable seed from these two operations. As a corollary, we prove that the toric colorable seed inequality established by Choi and Park is tight.

💡 Research Summary

The paper investigates two fundamental operations on piecewise‑linear (PL) spheres—stellar subdivision and wedge—and shows that both preserve the maximality of Buchstaber numbers as well as polytopality. A “seed” is defined as a PL sphere that cannot be obtained by a wedge from any other PL sphere. The authors first recall the moment‑angle complex (Z_K=(D^2,S^1)_K) associated to a simplicial complex (K) on (m) vertices, and define the Buchstaber number (s(K)) as the maximal dimension of a subtorus of the (m)‑dimensional torus (T^m) acting freely on (Z_K). For a PL sphere of dimension (n-1) with (m) vertices, the Picard number is (\operatorname{Pic}(K)=m-n). When (s(K)=\operatorname{Pic}(K)) the sphere is called toric‑colorable (or Buchstaber‑maximal). Such spheres serve as the combinatorial backbone of toric manifolds, quasitoric manifolds, and topological toric manifolds.

The wedge operation duplicates a vertex (v) by adding a new vertex (v’) and a 1‑simplex (I); the resulting complex (Wed_v(K)) has one more vertex and one higher dimension, but its Picard number stays unchanged, and Proposition 2.3 proves that the Buchstaber number is also unchanged. Stellar subdivision at a face (\sigma) removes the star of (\sigma) and glues in a cone over (\partial\sigma * \operatorname{Lk}K(\sigma)) with a new vertex (v\sigma). This operation preserves the dimension while increasing the vertex count by one, thereby decreasing the Picard number by one; Proposition 2.2 guarantees that PL‑sphere and polytopal properties are retained.

The authors introduce the J‑construction: for a tuple (J=(j_1,\dots,j_m)) with each (j_i\ge1), the complex (K(J)) is obtained by applying wedge operations (j_i-1) times to each vertex (i). Lemma 2.1 shows that any “assembled face” of (K(J)) is indeed a face of the complex, a fact used repeatedly later.

The central result, Theorem 3.1, states that if (K) is a seed and (J) satisfies (j_v\le2) for all vertices, then for any assembled face (\sigma) of (K(J)) a new seed can be produced in two ways: (1) if (\sigma) consists of a single vertex (v), then the stellar subdivision of the wedge edge ({v,v’}) yields a suspended seed; (2) if (|\sigma|>1) and (K) is non‑suspended, then the stellar subdivision at (\sigma) yields a non‑suspended seed. The proof proceeds by assuming the existence of two vertices (x,y) that intersect every facet of the subdivided complex, and then exhaustively eliminating four possible configurations using Proposition 2.4 (characterising when every facet contains a given pair) and Lemma 2.5 (identifying suspended pairs). Each case leads to a contradiction with the non‑suspended hypothesis, establishing that the resulting complex cannot have such a universal pair and therefore is a seed.

Corollary 3.2 leverages Theorem 3.1 to construct, for any integer (p\ge3) and any (m\le2p-1) (with (p=m-n)), a polytopal ((n-1))-dimensional toric‑colorable seed (K) satisfying (s(K)=p). The construction proceeds by induction on (p). The base cases (p=3) are supplied by the pentagon, the cross‑polytope, and the boundary of the 4‑dimensional cyclic polytope (C_4(7)). For the inductive step, given a seed of Picard number (p), the authors either apply a wedge followed by a stellar subdivision (when the dimension is small) or, for larger dimensions, take a non‑suspended seed with the maximal allowed number of vertices, duplicate a suitable set of vertices (using a tuple (J) with entries 2), and then perform a stellar subdivision at a face (\sigma). This yields a new seed with Picard number (p+1) while preserving polytopality and toric‑colorability. Remark 3.3 notes that for (p\ge4) all constructed seeds can be made non‑suspended, further strengthening the result.

The paper also revisits the “toric‑colorable seed inequality” originally proved by Choi and Park: \