The Nevo–Petersen conjecture predicts that the γ-vector of every flag homology sphere — the coefficient vector of the unique expansion of its palindromic h-polynomial in the basis ti(1+t)d−2it^i(1+t)^{d-2i} — is itself the ff-vector of a balanced simplicial complex; equivalently, that it satisfies the Frankl–Füredi–Kalai inequalities characterizing balanced ff-vectors. The conjecture refines Gal's γ-nonnegativity conjecture and the Charney–Davis conjecture for flag spheres. It is confirmed only for restricted families: barycentric subdivisions (Nevo–Petersen–Tenner), Coxeter complexes and the dual complexes of associahedra and cyclohedra, flag spheres with at most 2d+32d+3 vertices, and flag Bier and Murai spheres. The source paper of this candidate (Park, arXiv:2410.08139) proves that for even dd the γ-vector of a (d−1)(d-1)-dimensional flag sphere is the ff-vector of a complex of dimension at most d/2−1d/2-1 whose 1-skeleton admits a proper coloring by at most dd colors — a weakening of balancedness — and leaves the full conjecture open. This record asks for a proof or refutation of the conjecture for all flag homology spheres.

Background

A finite simplicial complex Δ\Delta is a collection of subsets (faces) of a finite vertex set, closed under taking subsets; its dimension is one less than the maximum face size. The ff-vector (f−1,f0,f1,… )(f_{-1},f_0,f_1,\dots) records the number of faces of each size, fif_i being the number of ii-dimensional faces. The complex Δ\Delta is flag if every minimal non-face has exactly two elements; equivalently, Δ\Delta is the clique complex of its graph, meaning a set of vertices is a face exactly when it is a clique. A homology sphere is a simplicial complex with the same reduced homology as a sphere of the same dimension, and such that the link of every proper face has the corresponding sphere homology; every boundary complex of a simplicial polytope, and more generally every simplicial sphere (a complex piecewise-linearly homeomorphic to a sphere), is a homology sphere. Flag homology spheres play the role for flag complexes that arbitrary spheres play in the classical face-enumeration theory.

The hh-vector of a (d−1)(d-1)-dimensional complex is defined through its ff-vector by ∑i=0dhitd−i=∑i=0dfi−1(t−1)d−i.\sum_{i=0}^{d} h_i t^{d-i} = \sum_{i=0}^{d} f_{i-1}(t-1)^{d-i}. For homology spheres the Dehn–Sommerville relations force the hh-polynomial to be palindromic, hi=hd−ih_i = h_{d-i}. Every palindromic integer polynomial of degree dd then admits a unique expansion h(t)=∑i=0⌊d/2⌋γi ti(1+t)d−2i,h(t) = \sum_{i=0}^{\lfloor d/2 \rfloor} \gamma_i\, t^i (1+t)^{d-2i}, and the coefficient vector γ(Δ)=(γ0,γ1,…,γ⌊d/2⌋)\gamma(\Delta)=(\gamma_0,\gamma_1,\dots,\gamma_{\lfloor d/2\rfloor}) is the γ-vector, introduced by Gal as the flag-sphere counterpart of the gg-vector (the sequence of consecutive differences of the hh-vector), whose nonnegativity for simplicial polytopes is the conclusion of the classical gg-theorem. For example, γ1=f0−2d\gamma_1 = f_0 - 2d measures the vertex count in excess of that of the boundary of the dd-dimensional cross-polytope (the generalization of the octahedron to dd dimensions), which is the flag sphere with the fewest faces. The Charney–Davis conjecture, which arose from the study of Euler characteristics of nonpositively curved piecewise Euclidean manifolds, predicts γd/2(Δ)≥0\gamma_{d/2}(\Delta) \ge 0 for flag spheres when dd is even; Gal's conjecture predicts γi(Δ)≥0\gamma_i(\Delta) \ge 0 for all ii. Both remain open in general, though they are known for important subclasses such as Coxeter complexes (the order complexes attached to finite reflection groups) and barycentric subdivisions (the subdivision whose vertices are the faces of Δ\Delta and whose faces are the chains of faces).

A (d−1)(d-1)-dimensional complex is balanced if its vertices can be colored with dd colors so that the vertices of every face receive distinct colors, i.e. the coloring is a proper coloring of the graph. Frankl, Füredi, and Kalai characterized exactly which integer sequences occur as ff-vectors of balanced complexes, via colored analogues of the classical Kruskal–Katona shadow inequalities that characterize ff-vectors of arbitrary complexes; consequently a sequence is the ff-vector of a balanced complex if and only if it satisfies these Frankl–Füredi–Kalai inequalities. Frohmader proved that every ff-vector of a flag complex is the ff-vector of a balanced complex, tying the flag and balanced worlds together.

Nevo and Petersen proposed that the γ-vector of a flag homology sphere is not merely nonnegative but is itself an ff-vector: their conjecture states that γ(Δ)\gamma(\Delta) satisfies the Kruskal–Katona inequalities, indeed the stronger Frankl–Füredi–Kalai inequalities — equivalently, γ(Δ)\gamma(\Delta) is the ff-vector of some balanced simplicial complex — and they posed the stronger problem of realizing γ(Δ)\gamma(\Delta) as the ff-vector of a flag complex. Nevo, Petersen, and Tenner confirmed the conjecture for barycentric subdivisions of arbitrary simplicial complexes, and Nevo–Petersen verified it for Coxeter complexes, for the complexes dual to the associahedron and the cyclohedron, and for all flag (d−1)(d-1)-spheres with at most 2d+32d+3 vertices. The source paper of this candidate (Park, arXiv:2410.08139) approaches the conjecture through a simplicial complex Γ\Gamma with f(Γ)=h(Δ)f(\Gamma) = h(\Delta) (such a Γ\Gamma exists and may be taken balanced by results quoted in that paper): it proves that for even dd the γ-vector of a (d−1)(d-1)-dimensional flag sphere is the ff-vector of a complex SS of dimension at most d2−1\frac{d}{2}-1 whose 1-skeleton admits a proper coloring by at most dd colors, via a Boolean-type decomposition Γ={F∪G:F∈S,  G∈2[d−2∣F∣]}\Gamma = \{F \cup G : F \in S,\; G \in 2^{[d-2|F|]}\} reminiscent of the Lefschetz decomposition of the middle cohomology of a compact Kähler manifold (a complex manifold with a compatible symplectic structure) into primitive components. Since a complex of dimension at most d2−1\frac{d}{2}-1 would need a proper coloring by at most d2\frac{d}{2} colors to be balanced in the relevant sense, this falls short of the conjecture, which remains open.

Problem Statement

Prove or refute the following conjecture (Nevo–Petersen, Conjecture 6.3 in its equivalent balanced-complex form; restated for flag simplicial spheres as Conjecture 0.6 of the source paper): for every flag homology sphere Δ\Delta of dimension d−1d-1 — in particular for every flag simplicial sphere — with γ-vector γ(Δ)=(γ0,γ1,…,γ⌊d/2⌋)\gamma(\Delta)=(\gamma_0,\gamma_1,\dots,\gamma_{\lfloor d/2\rfloor}), there exists a balanced simplicial complex Γ\Gamma whose ff-vector is γ(Δ)\gamma(\Delta). Equivalently: γ(Δ)\gamma(\Delta) satisfies the Frankl–Füredi–Kalai inequalities that characterize ff-vectors of balanced complexes. A positive answer must hold for all flag homology spheres, with no restriction on dimension, number of vertices, polytopality, or shellability; a proof of the stronger variant in which Γ\Gamma may moreover be taken flag (Nevo–Petersen's Problem 6.4) also settles the problem. A negative answer is an explicit flag homology sphere whose γ-vector fails the Frankl–Füredi–Kalai inequalities (or, a fortiori, fails to be the ff-vector of any simplicial complex, or fails to be nonnegative).

References

  1. Soohyun Park, "ff-vectors of balanced simplicial complexes, flag spheres, and geometric Lefschetz decompositions", arXiv:2410.08139 (2024), DOI 10.48550/arxiv.2410.08139, https://arxiv.org/abs/2410.08139 (source paper of this candidate)
  2. Eran Nevo and T. Kyle Petersen, "On γ-vectors satisfying the Kruskal–Katona inequalities", Discrete & Computational Geometry 45 (2011), 503–521, DOI 10.1007/s00454-010-9243-6, arXiv:0909.0694, https://arxiv.org/abs/0909.0694
  3. Eran Nevo, T. Kyle Petersen, and Bridget E. Tenner, "The γ-vector of a barycentric subdivision", Journal of Combinatorial Theory, Series A 118 (2011), 1364–1380, DOI 10.1016/j.jcta.2011.01.001, https://doi.org/10.1016/j.jcta.2011.01.001
  4. Swiatoslaw R. Gal, "Real Root Conjecture Fails for Five- and Higher-Dimensional Spheres", Discrete & Computational Geometry 34 (2005), 269–284, DOI 10.1007/s00454-005-1171-5, https://doi.org/10.1007/s00454-005-1171-5
  5. Ruth Charney and Michael Davis, "The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold", Pacific Journal of Mathematics 171 (1995), 117–137, DOI 10.2140/pjm.1995.171.117, https://doi.org/10.2140/pjm.1995.171.117
  6. P. Frankl, Z. Füredi, and G. Kalai, "Shadows of colored complexes", Mathematica Scandinavica 63 (1988), 169–178, DOI 10.7146/math.scand.a-12231, https://doi.org/10.7146/math.scand.a-12231
  7. Andrew Frohmader, "Face vectors of flag complexes", Israel Journal of Mathematics 164 (2008), 153–164, DOI 10.1007/s11856-008-0024-3, https://doi.org/10.1007/s11856-008-0024-3
  8. Jean-Philippe Labbé and Eran Nevo, "Bounds for Entries of γ-Vectors of Flag Homology Spheres", SIAM Journal on Discrete Mathematics 31(3) (2017), 2064–2078, DOI 10.1137/17M1110298, arXiv:1612.01169, https://arxiv.org/abs/1612.01169
  9. Hailun Zheng, "Face enumeration on flag complexes and flag spheres", in Algebraic and Geometric Combinatorics on Lattice Polytopes, World Scientific (2019), 435–449, DOI 10.1142/9789811200489_0028, arXiv:1809.06835, https://arxiv.org/abs/1809.06835
  10. Soohyun Park, "Repeated Lefschetz-like decompositions for flag doubly Cohen–Macaulay simplicial complexes and gamma vectors of flag spheres", arXiv:2411.02390 (2024), DOI 10.48550/arxiv.2411.02390, https://arxiv.org/abs/2411.02390
  11. Soohyun Park, "Flag complex face structures and decompositions", arXiv:2409.16773 (2024), DOI 10.48550/arxiv.2409.16773, https://arxiv.org/abs/2409.16773
  12. Ivan Limonchenko and Rade Živaljević, "Bier spheres, Nevo-Petersen conjecture and polyhedral products", Results in Mathematics 81, No. 1, Paper No. 6 (2026), DOI 10.1007/s00025-025-02561-9, https://doi.org/10.1007/s00025-025-02561-9 (preprint: arXiv:2411.14036, https://arxiv.org/abs/2411.14036, posted under the title "Simplicial vs. cubical spheres, polyhedral products and the Nevo-Petersen conjecture")
  13. Basile Coron, Luis Ferroni, and Shiyue Li, "Matroid analogues of Gal's conjecture", arXiv:2604.04550 (2026), https://arxiv.org/abs/2604.04550
  14. Soohyun Park, "Gamma-positivity, PL homeomorphism types, and orthogonal polynomials", arXiv:2603.19202 (2026), https://arxiv.org/abs/2603.19202