For every proper simplicial complex KK on [m][m], the deleted join of KK with its combinatorial Alexander dual — the Bier sphere Bier⁡(K)\operatorname{Bier}(K) — is an (m−2)(m-2)-dimensional PL sphere on at most 2m2m vertices. Counting results show that asymptotically almost all Bier spheres cannot be boundaries of convex polytopes, yet after more than three decades of study not a single explicit non-polytopal Bier sphere has been constructed: all Bier spheres with at most 12 vertices are now known to be polytopal, and every structured subfamily examined so far (threshold, flag, chordal, and neighborly Bier/Murai spheres) is polytopal. The problem asks for an explicit simplicial complex KK such that Bier⁡(K)\operatorname{Bier}(K) is provably not combinatorially equivalent to the boundary complex of any convex polytope — posed as Problem 4.7 of Limonchenko–Sergeev (2024) and restated as Problem 5.3 of Holleben–Yang (2026) — and, more broadly, for a classification of which Bier spheres are polytopal. The problem remains open as of August 2026.

Background

Let [m]={1,…,m}[m]=\{1,\dots,m\}. An (abstract) simplicial complex KK on [m][m] is a collection of subsets of [m][m], called faces, closed under taking subsets. Its combinatorial Alexander dual is the complex K∨={σ⊆[m]:[m]∖σ∉K}K^\vee=\{\sigma\subseteq[m] : [m]\setminus\sigma\notin K\}. The deleted join of KK and K∨K^\vee is the complex Bier⁡(K)=K∗ΔK∨={σ⊎τ:σ∈K, τ∈K∨, σ∩τ=∅}\operatorname{Bier}(K)=K\ast_\Delta K^\vee=\{\sigma\uplus\tau : \sigma\in K,\ \tau\in K^\vee,\ \sigma\cap\tau=\varnothing\} on two disjoint copies of the vertex set [m][m]. Thomas Bier observed in a 1992 note, building on a construction studied algebraically by Villarreal in 1990, that for every proper complex K⊊2[m]K\subsetneq 2^{[m]} this deleted join is always a sphere: Bier⁡(K)\operatorname{Bier}(K) is an (m−2)(m-2)-dimensional PL sphere — a complex piecewise-linearly homeomorphic to Sm−2S^{m-2} — with at most 2m2m vertices (unused "ghost" vertices may occur). Bier spheres form one of the largest and best-studied families of simplicial spheres: they are shellable (their facets admit an ordering in which each new facet intersects the previous ones along a nonempty union of boundary faces), their face numbers, chromatic numbers and Buchstaber numbers (a toric-topological invariant measuring the dimension of the torus acting freely on the associated moment-angle complex) are known in closed form, and they are a standard testing ground in topological combinatorics and toric topology, where a polytopal Bier sphere yields quasitoric manifolds and small covers — two standard classes of manifolds constructed over simple polytopes in toric topology — over the dual simple polytope.

A simplicial (d−1)(d-1)-sphere is called polytopal if it is combinatorially equivalent to the boundary complex of a convex simplicial dd-polytope. The first non-polytopal sphere was found by Brückner in 1910; by Steinitz's theorem every triangulated 2-sphere is polytopal; and counting results of Kalai (1988) and Pfeifle–Ziegler (2004) show that most triangulated spheres are non-polytopal in dimensions 3 and higher. Explicit non-polytopal spheres nevertheless remain rare, and the Bier family sits exactly at this tension between asymptotic abundance and the lack of explicit examples. Björner–Paffenholz–Sjöstrand–Ziegler proved by counting that there are asymptotically far more Bier spheres on 2n2n vertices than the 28n3+O(n2)2^{8n^3+O(n^2)} combinatorial types of simplicial polytopes on 2n2n vertices, so most Bier spheres must be non-polytopal. Yet every Bier sphere ever analyzed is polytopal: Bier spheres of threshold complexes (complexes of the form {σ:∑i∈σwi<q}\{\sigma : \sum_{i\in\sigma}w_i<q\} for nonnegative weights wiw_i and a quota qq) are polytopal, every Bier sphere admits at least a star-shaped realization, flag Bier spheres are polytopal and completely classified, chordal and neighborly Bier/Murai spheres are polytopal, and — the current frontier — all Bier spheres with at most 12 vertices are polytopal with rational realizations.

Two further facts shape how the problem can be attacked. First, since the gg-conjecture for homology spheres is now a theorem (proved by combinatorial Lefschetz methods), no face-number inequality can separate Bier spheres from polytopal spheres; non-polytopality must be certified geometrically. The standard route is via oriented matroids: the vertex set of a simplicial dd-polytope in general position determines a realizable uniform oriented matroid of rank d+1d+1 — encoded by a chirotope, an alternating sign function on the (d+1)(d+1)-element vertex subsets recording the orientation of each simplex — whose face lattice recovers the boundary complex. A sphere is polytopal only if such an oriented matroid structure is realizable over R\mathbb{R}, and finite algebraic certificates of non-realizability, such as biquadratic final polynomials, are the classical tool for proving it. Second, deciding polytopality of a given simplicial sphere is NP-hard already for 3-spheres, so there is no general feasible decision procedure; known realization-search algorithms (bistellar flips with radial adjustment, nonlinear optimization) currently exhaust the Bier family only up to 12 vertices.

Problem Statement

Exhibit an explicit finite simplicial complex K⊊2[m]K\subsetneq 2^{[m]} and prove that its Bier sphere Bier⁡(K)=K∗ΔK∨\operatorname{Bier}(K)=K\ast_\Delta K^\vee is not combinatorially equivalent to the boundary complex of any convex simplicial polytope. This is Problem 4.7 of Limonchenko–Sergeev, restated as Problem 5.3 of Holleben–Yang; by current knowledge the Bier sphere of any example must have at least 13 vertices, since every Bier sphere with at most 12 vertices is polytopal. The claim must rule out all polytopal realizations at once — exhibiting the failure of a particular construction (such as the star-shaped realization that every Bier sphere possesses) is not sufficient.

The broader objective, of which this is the central missing piece, is the polytopal classification of the Bier family: determine, in terms of KK, which Bier spheres are polytopal. In particular, Problems 4.8–4.9 of Limonchenko–Sergeev ask to classify the Bier spheres that are combinatorially equivalent to nerve complexes of nestohedra and of generalized truncation polytopes — the nerve complex of a simple polytope being the simplicial complex on its facets in which a set of facets is a face whenever those facets share a common point, nestohedra being the simple polytopes associated with building sets of subsets of a finite set, and generalized truncation polytopes being the polytopes obtained from a simplex by iterated vertex cuts — and have been solved by Limonchenko–Živaljević only in the flag case, for flag Bier spheres and for flag Murai spheres (the weighted generalization of Bier spheres). Any complete classification must be consistent with the counting theorem that most Bier spheres are non-polytopal and must therefore exhibit explicit non-polytopal instances. The central missing piece is an explicit non-polytopal Bier sphere (Problem 4.7 / Problem 5.3); the polytopal classification of the Bier family, including Problems 4.8–4.9, is the motivating broader program.

References

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