Murai and Nevo completed the generalized lower bound conjecture (GLBC) of McMullen and Walkup by proving that if a simplicial d-polytope P with boundary complex Δ and h-vector (h_0, …, h_d) satisfies hr−1=hrh_{r-1} = h_r for some r≤d/2r ≤ d/2, then the combinatorially defined complex Δ(d−r)Δ(d-r) is the unique (r−1)(r-1)-stacked triangulation of P, and they extended the statement to homology spheres having the weak Lefschetz property (WLP). The paper explicitly left open the fine structure of this canonical stacked filling: Conjecture 6.4 asks whether the (r−1)(r-1)-stacked triangulation of an (r−1)(r-1)-stacked d-polytope with r≤(d+1)/2r ≤ (d+1)/2 is regular (a conjecture of McMullen) and shellable (a conjecture of Bagchi and Datta); Question 6.5 asks whether the boundary of every shellable 2-stacked triangulation of a d-ball is polytopal; and Question 6.6 asks whether Δ(d−r)Δ(d-r) is a triangulation of a ball for every triangulated sphere with hr−1=hrh_{r-1} = h_r, the PL case being open even to the authors. The original blocking hypothesis for the sphere statement — the WLP for every homology sphere in characteristic 0, equivalent to the g-conjecture — was resolved affirmatively by Adiprasito in 2018 and by independent characteristic-2 proofs, so the filling statement is now known, as a homology-ball statement, for all homology spheres.

Background

A simplicial d-polytope is a convex polytope of dimension d all of whose facets are simplices; its boundary complex Δ is the simplicial complex of vertex sets of its faces. The face numbers of Δ (the f-vector) are encoded by the h-vector (h0,h1,…,hd)(h_0, h_1, …, h_d), an invertible linear transform of the f-vector. In 1971 McMullen and Walkup posed the generalized lower bound conjecture (GLBC): for a simplicial d-polytope, 1=h0≤h1≤…≤h⌊d/2⌋1 = h_0 ≤ h_1 ≤ … ≤ h_{⌊d/2⌋}, and if hr−1=hrh_{r-1} = h_r for some 1≤r≤d/21 ≤ r ≤ d/2, then the polytope admits a triangulation with no interior faces of dimension below d−r+1d-r+1; such a triangulation is called (r−1)(r-1)-stacked (a triangulation of the polytope is a simplicial complex homeomorphic to a d-ball whose boundary is Δ, and its interior faces are those not lying in Δ), and the polytope is (r−1)(r-1)-stacked if its boundary admits one. The inequality part follows from the g-theorem for f-vectors of simplicial polytopes, conjectured by McMullen (1971), with sufficiency proved by Billera and Lee (1981) and necessity by Stanley (1980) via the hard Lefschetz theorem for projective toric varieties. Murai and Nevo (2013) proved the equality part: if hr−1=hrh_{r-1} = h_r, then the complex Δ(d−r)Δ(d-r) is the unique geometric (r−1)(r-1)-stacked triangulation of P, where for a complex Δ on vertex set V and an integer i≥1i ≥ 1, Δ(i)Δ(i) is the subcomplex of the full simplex on V whose faces are the subsets F of V such that every subset of F of size at most i+1i+1 is a face of Δ; the boundary of Δ(i)Δ(i) always contains Δ, so Δ(i)Δ(i) is a candidate filling. They also proved (their Theorem 1.3) that the same conclusion in the weaker sense — Δ(d−r)Δ(d-r) is a homology d-ball bounded by Δ — holds for every homology sphere satisfying the weak Lefschetz property (WLP) over a field of characteristic 0: after replacing the Stanley–Reisner ring k[Δ] (the quotient of the polynomial ring over k in the vertex variables by the ideal of non-faces) by an Artinian reduction modulo a linear system of parameters, multiplication by a general linear form has maximal rank in every degree. The WLP hypothesis was needed because the algebraic form of the g-conjecture — every homology sphere over characteristic 0 has the WLP (recorded as Conjecture 4.22 in Swartz's 2009 survey) — was open in 2013. That hypothesis has since been discharged: Adiprasito (2018) proved the generic hard Lefschetz theorem for the face rings of triangulated rational homology spheres (his Theorem 9.1), resolving the g-conjecture in full generality, and further proofs followed via characteristic-2 anisotropy by Papadakis and Petrotou and by Karu and Xiao, with Adiprasito, Papadakis and Petrotou establishing anisotropy for pseudomanifolds in arbitrary characteristic. Hard Lefschetz implies the WLP, so the sphere statement is now unconditional: for every homology sphere with hr−1=hrh_{r-1} = h_r, Δ(d−r)Δ(d-r) is a homology d-ball bounded by Δ (with the case r=2r = 2 going back to Kalai's rigidity-theoretic work, which never needed the hypothesis); Novik and Zheng (2026) apply this unconditional form as a standard tool. What remains open is the fine structure of the canonical filling, formulated with four standard notions. A triangulation T of a finite point configuration A in RdR^d is regular (or coherent) if T is the projection of the lower faces of the convex hull of the lifted points (a,w(a))(a, w(a)) for some height function w:A→Rw : A → R; regular triangulations are the objects of secondary-polytope theory. A pure d-dimensional complex is shellable if its facets can be ordered so that each facet meets the union of its predecessors in a nonempty union of codimension-one faces of that facet; shellable complexes admit inductive treatments unavailable in general. A simplicial (d−1)(d-1)-sphere is polytopal if it is isomorphic to the boundary complex of a simplicial d-polytope, a strong and rare property. A complex is a PL sphere or PL ball if it has a common subdivision with the boundary of a simplex or with a simplex, respectively, which is finer than being homeomorphic to a sphere or ball. Murai and Nevo record two conjectures: McMullen's conjecture (their 6.4(i), originating in McMullen's 2004 paper on triangulations of simplicial polytopes) that for an (r−1)(r-1)-stacked d-polytope with r≤(d+1)/2r ≤ (d+1)/2 the canonical triangulation Δ(d−r)Δ(d-r) is regular, and Bagchi and Datta's conjecture (their 6.4(ii), appearing as Conjecture 3.3 in Bagchi and Datta's paper On k-stellated and k-stacked spheres, phrased there as: for d≥2kd ≥ 2k every k-stacked polytopal d-sphere is k-stellated, i.e., obtainable from the boundary of a simplex by a sequence of bistellar moves of index less than kk — a bistellar move being a local re-triangulation that replaces one collection of faces by another, its index being the dimension of the face it inserts) that Δ(d−r)Δ(d-r) is shellable; they note that regularity would imply shellability, and that the case r=2r = 2 of shellability is classical. Both blocking phenomena occur in neighboring regimes: joining Rudin's non-shellable 3-ball with a simplex of dimension at least 2 yields a non-shellable 2-stacked ball with non-polytopal boundary (their Example 6.2), and Kalai's squeezed spheres provide shellable (r−1)(r-1)-stacked balls with non-polytopal boundary for r≥4r ≥ 4 (their Example 6.3, resting on Kalai's 1988 construction of very many triangulated spheres). The paper's Questions 6.5 and 6.6 probe the two remaining gaps: polytopality of the boundary in the shellable 2-stacked case, and topological and PL ballness of Δ(d−r)Δ(d-r) for general triangulated spheres. The closest published result to the regularity question is Gonska and Ziegler's characterization of inscribable stacked polytopes, which concerns inscribability of stacked polytopes (equivalently, Delaunay realizability of stellar triangulations of the simplex) rather than regularity of the canonical stacked triangulations of general stacked polytopes.

Problem Statement

Let Δ be a simplicial complex on vertex set V, and for an integer i≥1i ≥ 1 let Δ(i)Δ(i) be the subcomplex of the full simplex on V whose faces are the subsets F of V such that every subset of F of size at most i+1i+1 is a face of Δ. Recall that a triangulation of a d-ball is (r−1)(r-1)-stacked if it has no interior faces of dimension below d−r+1d-r+1, and that a simplicial d-polytope is (r−1)(r-1)-stacked if its boundary complex admits such a triangulation. Murai and Nevo proved that if the h-vector of a simplicial d-polytope P with boundary complex Δ satisfies hr−1=hrh_{r-1} = h_r for some 1≤r≤d/21 ≤ r ≤ d/2, then Δ(d−r)Δ(d-r) is the unique (r−1)(r-1)-stacked triangulation of P, and the corresponding homology-ball statement now holds for all homology spheres. The problem is to settle the following four structural questions about the canonical stacked filling, each in its stated generality, by proof or counterexample. (1) Regularity (McMullen; Murai–Nevo Conjecture 6.4(i)): for every (r−1)(r-1)-stacked d-polytope with r≤(d+1)/2r ≤ (d+1)/2, is the (r−1)(r-1)-stacked triangulation Δ(d−r)Δ(d-r) a regular triangulation of the polytope, i.e., induced by some height function on its vertices? (2) Shellability (Bagchi–Datta; Murai–Nevo Conjecture 6.4(ii)): for every (r−1)(r-1)-stacked d-polytope with r≤(d+1)/2r ≤ (d+1)/2, is Δ(d−r)Δ(d-r) shellable? Murai and Nevo note that a positive answer to (1) implies a positive answer to (2); equivalently, in Bagchi–Datta's formulation, for d≥2kd ≥ 2k every k-stacked polytopal d-sphere is k-stellated. (3) Polytopality (Murai–Nevo Question 6.5): if Δ is a shellable 2-stacked triangulation of a d-ball (no interior faces of dimension below d−2d-2), must the boundary ∂Δ be polytopal? Negative answers are known in the neighboring regimes of non-shellable 2-stacked balls and shellable (r−1)(r-1)-stacked balls with r≥4r ≥ 4; this low-stackedness shellable case is the remaining gap. (4) Topological ballness (Murai–Nevo Question 6.6): if Δ is a triangulation of a (d−1)(d-1)-sphere whose h-vector satisfies hr−1=hrh_{r-1} = h_r for some 1≤r≤d/21 ≤ r ≤ d/2, must Δ(d−r)Δ(d-r) be a triangulation of a d-ball? In particular, if Δ is a PL sphere, must Δ(d−r)Δ(d-r) be a PL ball? The homology-ball version is known for all homology spheres, but the topological and PL statements are open, even for PL spheres. The four questions are independent of one another.

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