Nonnegativity of $\gamma_2$ for flag simplicial spheres: the first nontrivial case of Gal's conjecture

Gal's 2005 conjecture asserts that the $\gamma$-vector of every flag homology sphere is entrywise nonnegative — the flag analogue of the nonnegativity part of the $g$-theorem for simplicial polytopes. This problem is its first nontrivial case: prove or disprove that every flag simplicial $d$-sphere $\Delta$ satisfies $\gamma_2(\Delta) = f_1(\Delta) - (2d-1)f_0(\Delta) + (d-1)(2d+2) \ge 0$, an optimal linear edge-density floor for flag spheres attained by boundaries of cross-polytopes. The 3-dimensional case is the Charney–Davis conjecture in its settled form (Davis–Okun, via $\ell^2$-homology of right-angled Coxeter groups); the statement is also known for special classes (Coxeter complexes, i.e., chamber complexes of finite reflection groups; complexes dual to associahedra and cyclohedra; order complexes of Gorenstein* posets, i.e., graded posets whose order complex is a homology sphere) and, by the 2026 classification of Basak–Ghosh–Gupta, for all flag normal $d$-pseudomanifolds with at most $2d+7$ vertices. As of 2026 the problem remains open in all dimensions $d \ge 4$ without vertex restrictions.

Background

Face numbers and the $\gamma$-vector. Let $\Delta$ be a finite $d$-dimensional simplicial complex (a downward-closed family of subsets of a finite vertex set). Its face numbers $f_i(\Delta)$ count the $i$-dimensional faces ($f_{-1}=1$ for the empty face; $f_0$ is the number of vertices and $f_1$ the number of edges). The $h$-polynomial $h_\Delta(t)=\sum_{k=0}^{d+1} h_k\,t^{d+1-k}$ is defined by $h_\Delta(t)=f_\Delta(t-1)$, where $f_\Delta(x)=\sum_{i=0}^{d+1} f_{i-1}\,x^{d+1-i}$ is the $f$-polynomial; it is a standard repackaging of the same information. For complexes with the homology of a sphere (homology spheres, in particular all simplicial complexes triangulating a sphere), the Dehn–Sommerville relations force $h_\Delta$ to be palindromic ($h_k = h_{d+1-k}$), so it admits a unique expansion in the basis $t^i(1+t)^{d+1-2i}$:

$$h_\Delta(t)=\sum_{0\le i\le (d+1)/2} \gamma_i(\Delta)\, t^i (1+t)^{d+1-2i}.$$

The coefficients $\gamma_i(\Delta)$ form the $\gamma$-vector; nonnegativity of the related $g$-numbers $g_i = h_i - h_{i-1}$ is the celebrated $g$-theorem for simplicial polytopes, and Gal's conjecture is the flag analogue of that nonnegativity. The first entries work out to $\gamma_1(\Delta)=f_0(\Delta)-2(d+1)$ and $\gamma_2(\Delta)=f_1(\Delta)-(2d-1)f_0(\Delta)+(d-1)(2d+2)$. (In the common convention that indexes by the dimension $d-1$ of the sphere, the same quantities appear as $\gamma_1=f_0-2d$ and $\gamma_2=f_1-(2d-3)f_0+2d(d-2)$.)

Flag complexes and Gal's conjecture. $\Delta$ is flag if every clique of its 1-skeleton is a face, equivalently if every minimal non-face has cardinality 2; flag complexes are exactly the clique complexes of graphs. Flagness is the combinatorial shadow of nonpositive curvature: a piecewise-Euclidean complex built from right-angled simplices is nonpositively curved precisely when its links are flag, and Euler-characteristic statements for nonpositively curved spaces motivated the Charney–Davis conjecture (1995), which predicts $(-1)^m h_\Delta(-1) \ge 0$ for every flag homology $(2m-1)$-sphere. Gal (2005) proposed the far-reaching generalization now known as Gal's conjecture: the $\gamma$-vector of every flag homology sphere is entrywise nonnegative. In the same paper Gal showed that the strongest plausible strengthening — real-rootedness of $h_\Delta$ — fails for flag homology spheres of dimension at least 5, so $\gamma$-nonnegativity is the strongest surviving candidate property of these $h$-polynomials. A yet stronger conjecture of Nevo–Petersen predicts that the whole $\gamma$-vector of a flag homology sphere satisfies the Kruskal–Katona inequalities (the classical inequalities characterizing $f$-vectors of simplicial complexes), which would imply nonnegativity.

The $\gamma_2$ case. Since $\gamma_1 = f_0 - 2(d+1) \ge 0$ is the classical fact that a flag normal $d$-pseudomanifold (a pure complex, i.e., one whose maximal faces all have dimension $d$, in which every codimension-1 face lies in exactly two facets) has at least $2(d+1)$ vertices — with equality exactly at the boundary of the $(d+1)$-dimensional cross-polytope, the join of $d+1$ pairs of antipodal vertices — the entry $\gamma_2$ is the first case of Gal's conjecture not settled by classical bounds. Unpacked, $\gamma_2 \ge 0$ is the sharp linear edge-density inequality $f_1(\Delta) \ge (2d-1)f_0(\Delta) - (d-1)(2d+2)$, with equality at the cross-polytope boundary. The statement is genuinely flag-specific: non-flag spheres can be edge-sparse (stacked spheres, built by repeatedly coning a new vertex onto a facet, violate the inequality already in dimension 3), so no argument that ignores missing-face structure can succeed. In dimensions 3 and 4, $\gamma_2$ is the top $\gamma$-entry and the statement coincides with the Charney–Davis conjecture in its odd- and even-dimensional forms respectively (the even-dimensional form is due to Gal–Januszkiewicz); in dimension 3 it is a theorem of Davis–Okun, proved by vanishing theorems for the $\ell^2$-homology (the homology theory for infinite group actions built on square-summable chains) of right-angled Coxeter groups (groups generated by involutions in which certain designated pairs commute and no other relations hold). In dimensions $\ge 5$, $\gamma_2 \ge 0$ is a lower-bound-theorem statement about face numbers of flag spheres, strictly between the classical $\gamma_1$ bound and the top-entry Charney–Davis quantities, and no direct geometric interpretation of it is known. In terms of the $g$-numbers, direct expansion gives $\gamma_2 = g_2 - (d-2)\bigl(f_0 - \tfrac{3}{2}(d+1)\bigr)$ for a $d$-sphere, so the conjecture demands roughly a $(d-2)$-fold multiple of the vertex count as a lower bound on $g_2$ — far beyond current methods, which combine vertex-link counting (each link of a vertex is itself a flag sphere, giving $f_1 \ge d\,f_0$) with stable-set estimates in the graph of the sphere.

Problem Statement

Prove or disprove the $\gamma_2$ conjecture for flag spheres (Conjecture 5.2 of Basak–Ghosh–Gupta 2026, the first nontrivial case of Gal's 2005 conjecture):

For every integer $d \ge 3$ and every flag simplicial $d$-sphere $\Delta$ — that is, a flag simplicial complex (every clique of its 1-skeleton is a face) whose geometric realization is homeomorphic to the $d$-sphere —

$$\gamma_2(\Delta) = f_1(\Delta) - (2d-1)f_0(\Delta) + (d-1)(2d+2) \ge 0,$$

equivalently, $f_1(\Delta) \ge (2d-1)f_0(\Delta) - (d-1)(2d+2)$, where $f_0(\Delta)$ and $f_1(\Delta)$ are the numbers of vertices and edges of $\Delta$, and $\gamma_2$ is the second entry of the $\gamma$-vector defined by the expansion $h_\Delta(t) = \sum_{0 \le i \le (d+1)/2} \gamma_i(\Delta)\, t^i (1+t)^{d+1-2i}$ of the $h$-polynomial $h_\Delta(t) = f_\Delta(t-1)$, where $f_\Delta(x) = \sum_{i=0}^{d+1} f_{i-1}\,x^{d+1-i}$ and $f_{-1} = 1$ counts the empty face.

Scope of an answer. An affirmative answer must cover all dimensions $d \ge 3$ and all vertex counts; proving the stronger statement for all flag homology $d$-spheres (Gal's original setting) is acceptable, and the already-settled dimension-3 case may be cited (Davis–Okun) rather than reproved. A negative answer must exhibit one explicit flag simplicial $d$-sphere with $d \ge 3$ and $\gamma_2(\Delta) < 0$. Results restricted to bounded dimension, a bounded number of vertices, or special subclasses of flag spheres (for instance barycentric subdivisions, Coxeter complexes, or nestohedra) do not settle the problem as stated.

References

1. Ś. R. Gal, "Real root conjecture fails for five- and higher-dimensional spheres", Discrete & Computational Geometry 34 (2005), no. 2, 269–284. DOI: 10.1007/s00454-005-1171-5. https://doi.org/10.1007/s00454-005-1171-5 (zbMATH review: https://zbmath.org/?q=an:1085.52005)
2. R. Charney, M. W. Davis, "The Euler characteristic of a nonpositively curved, piecewise Euclidean manifold", Pacific Journal of Mathematics 171 (1995), no. 1, 117–137. DOI: 10.2140/pjm.1995.171.117. https://doi.org/10.2140/pjm.1995.171.117
3. M. W. Davis, B. Okun, "Vanishing theorems and conjectures for the ℓ²-homology of right-angled Coxeter groups", Geometry & Topology 5 (2001), 7–74. DOI: 10.2140/gt.2001.5.7. https://doi.org/10.2140/gt.2001.5.7
4. Ś. R. Gal, T. Januszkiewicz, "Even- vs. Odd-dimensional Charney–Davis Conjecture", Discrete & Computational Geometry 44 (2010), no. 4, 802–804. DOI: 10.1007/s00454-009-9210-2. https://doi.org/10.1007/s00454-009-9210-2 (preprint: "Odd-dimensional Charney–Davis conjecture", arXiv:0905.1961, https://arxiv.org/abs/0905.1961)
5. E. Nevo, T. K. Petersen, "On γ-vectors satisfying the Kruskal–Katona inequalities", Discrete & Computational Geometry 45 (2011), no. 3, 503–521. DOI: 10.1007/s00454-010-9243-6. https://doi.org/10.1007/s00454-010-9243-6 (preprint: arXiv:0909.0694, https://arxiv.org/abs/0909.0694)
6. J.-P. Labbé, E. Nevo, "Bounds for entries of γ-vectors of flag homology spheres", SIAM Journal on Discrete Mathematics 31 (2017), no. 3, 2064–2078. DOI: 10.1137/17M1110298. https://doi.org/10.1137/17M1110298
7. H. Zheng, "Face enumeration on flag complexes and flag spheres" (survey), arXiv:1809.06835 (2018). https://arxiv.org/abs/1809.06835
8. M. Chudnovsky, E. Nevo, "Stable sets in flag spheres", European Journal of Combinatorics 110 (2023), Paper No. 103699. DOI: 10.1016/j.ejc.2023.103699. https://doi.org/10.1016/j.ejc.2023.103699
9. B. Basak, D. Ghosh, R. Kumar Gupta, "A complete classification of discrete d-pseudomanifolds with at most 2d+7 vertices", arXiv:2606.29753 (2026). https://arxiv.org/abs/2606.29753 (stated there as Conjecture 5.2)
10. I. Novik, H. Zheng, "Lower bounds on the g-numbers of spheres without large missing faces", arXiv:2604.16905 (2026). https://arxiv.org/abs/2604.16905
11. B. Coron, L. Ferroni, S. Li, "Matroid analogues of Gal's conjecture", arXiv:2604.04550 (2026). https://arxiv.org/abs/2604.04550