Nevo's conjecture for 4-spheres without large missing faces: is $g_2 \geq g_1$?

Nevo (2009) conjectured that a homology sphere whose missing faces all have dimension at most $i$ satisfies componentwise nonnegativity of a generalized $g$-vector; the first open case is the class $S(2,4)$ of ($\mathbb{Z}/2\mathbb{Z}$-homology) 4-spheres with no missing faces of dimension greater than 2, where the conjecture asserts $g_2 \geq g_1$, equivalently $f_1 \geq 6f_0 - 21$ — a sharp bound attained by the 8-vertex join $K(2,4)=\partial\sigma^2 * \partial\sigma^2 * \partial\sigma^1$. Despite steady progress — Nevo's $f_1 \geq (5+6/91)f_0 - 15$ (2009) and the recent $g_2 \geq \frac{2}{5}f_0 - \frac{6}{5}$ of Novik–Zheng (2026), obtained via affine stress spaces, independence numbers, and level rings — the conjectured coefficient remains unreached and no counterexample is known. The problem remains open as of August 2026.

Background

A simplicial complex $\Delta$ on a vertex set $V$ is a collection of subsets of $V$ closed under taking subsets; its elements are faces, maximal faces are facets, and the dimension of a face is its cardinality minus one. The complex is pure if all its facets have the same dimension. A subset $\tau \subseteq V$ is a missing face of $\Delta$ (a minimal non-face) if $\tau$ is not a face while every proper subset of $\tau$ is; $\Delta$ is flag if every missing face has dimension at most 1. Write $f_i(\Delta)$ for the number of $i$-dimensional faces, so $f_0$ counts vertices and $f_1$ counts edges. A pure 4-dimensional complex $\Delta$ is a ($\mathbb{Z}/2\mathbb{Z}$-homology) 4-sphere if for every face $\sigma \in \Delta$ — including the empty face — the link $\mathrm{lk}(\sigma) = \{\tau \in \Delta : \tau \cap \sigma = \varnothing,\ \tau \cup \sigma \in \Delta\}$ has the same reduced homology over the field $\mathbb{F}_2$ as a sphere of dimension $4 - |\sigma|$; this class contains the boundary complexes of simplicial 5-polytopes (convex polytopes all of whose facets are simplices). Following Novik–Zheng (2026), $S(2,4)$ denotes the class of ($\mathbb{Z}/2\mathbb{Z}$-homology) 4-spheres all of whose missing faces have dimension at most 2 — spheres without large missing faces, a class strictly between flag 4-spheres and arbitrary 4-spheres.

Face numbers are conventionally packaged in the $h$- and $g$-vectors. For a $(d-1)$-dimensional complex the $h$-vector $(h_0,\dots,h_d)$ is defined by $\sum_{i=0}^{d} f_{i-1}(t-1)^{d-i} = \sum_{i=0}^{d} h_i t^{d-i}$, and the $g$-numbers are $g_0 = 1$, $g_i = h_i - h_{i-1}$. Only two entries are needed here: $g_1 = f_0 - (d+1)$ and $g_2 = f_1 - d f_0 + \binom{d+1}{2}$. For a 4-sphere ($d = 5$) these read $g_1 = f_0 - 6$ and $g_2 = f_1 - 5f_0 + 15$.

The classical Lower Bound Theorem asserts $g_2 \geq 0$ for all normal pseudomanifolds of dimension at least three (a normal pseudomanifold is a pure complex in which every face of codimension one lies in exactly two facets and in which the links of all positive-dimensional faces are connected), with equality exactly for stacked spheres — spheres obtained from the boundary of a simplex by repeatedly adding a new vertex beyond a single facet — and the Generalized Lower Bound Theorem (the linear inequalities of the $g$-theorem, with the equality case settled by Murai–Nevo in 2013) asserts $g_i \geq 0$ for all spheres, with $g_i = 0$ if and only if the sphere is $(i-1)$-stacked. Stacked spheres always have missing faces of high dimension, which suggests that spheres whose missing faces are all small should satisfy far stronger inequalities. At the flag extreme, the Charney–Davis conjecture (1995, motivated by the Hopf conjecture on Euler characteristics of nonpositively curved manifolds) predicts $g_k - g_{k-1} + \cdots + (-1)^k g_0 \geq 0$ for flag $(2k-1)$-spheres — known for $k=2$ by Davis–Okun — and Gal's conjecture (2005) predicts nonnegativity of all $\gamma$-numbers, the coefficients of the $h$-polynomial rewritten in the basis $t^k(1+t)^{d-2k}$; both are wide open beyond low dimensions.

Nevo (2009) proposed a unified conjecture interpolating between these endpoints. For $0 < i \leq d+1$, let $HS(i,d)$ denote the class of integer-homology $d$-spheres without missing faces of dimension greater than $i$ (here $d$ is the sphere's dimension, to be distinguished from the $d$ of the $h$- and $g$-number formulas above, which is the dimension plus one); expanding the $h$-polynomial in the basis of symmetric polynomials generated by $P_{d,i}(t) = (1+t+\cdots+t^i)^q(1+t+\cdots+t^r)$, where $d+1 = qi + r$, defines a generalized $g$-vector $g^{(i)}$. Nevo's Conjecture 1.5 asserts $g^{(i)}(\Delta) \geq 0$ componentwise for every $\Delta \in HS(i,d)$: for $i = d$ this is the Generalized Lower Bound Theorem (a theorem), for $i = 1$ it reduces to Gal's conjecture, and Nevo identified $HS(2,4)$ — whose $\mathbb{Z}/2\mathbb{Z}$-homology analogue is the class $S(2,4)$ — as the first open case. There the conjecture takes a concrete form: every $\Delta \in S(2,4)$ should satisfy $g_2(\Delta) \geq g_1(\Delta)$, i.e. $f_1(\Delta) \geq 6f_0(\Delta) - 21$. The bound is the strongest possible in the one-parameter family $g_2 \geq \lambda f_0 - (8\lambda - 2)$ of linear bounds: all of them are tight at $K(2,4) = \partial\sigma^2 * \partial\sigma^2 * \partial\sigma^1$. Here $\partial\sigma^k$ denotes the boundary complex of the $k$-dimensional simplex, and the join $\Delta * \Gamma$ of two complexes on disjoint vertex sets is the complex whose faces are all unions of a face of $\Delta$ with a face of $\Gamma$; thus $K(2,4)$ is the join of the boundaries of two triangles and of a segment (a pair of vertices). By Goff–Klee–Novik (2011), $f_0 \geq 8$ on $S(2,4)$, $K(2,4)$ is the unique 8-vertex member of the class, and it simultaneously minimizes every face number in the class, so the conjectured inequality holds with equality there. Nevo highlighted even the polytopal subcase — boundaries of simplicial 5-polytopes with no missing faces of dimension greater than 2 — as an open problem (his Problem 4.3).

Progress has been slow. The main tool is the theory of affine stresses: for a $(d-1)$-sphere $\Delta$ with a generic embedding of its vertices into $\mathbb{R}^d$, the vector space of affine $k$-stresses — homogeneous degree-$k$ polynomials in the vertex variables, supported on faces of $\Delta$, and annihilated by the partial derivatives along the $d$ coordinate functions and by the total derivative $\partial_c = \sum_v \partial_{x_v}$ — has dimension exactly $g_k(\Delta)$, a reformulation of the Hard Lefschetz theorem for spheres (a fundamental duality theorem in the algebraic study of face numbers). Lower bounds on $g_k$ follow from exhibiting large independent families of stresses, for instance from independent sets in the graph of $\Delta$ (the route opened by Chudnovsky–Nevo for flag spheres) or from the Nevo–Novinsky characterization of spheres with $g_2 = 1$. Along these lines Novik–Zheng (April 2026) proved $g_2 \geq \frac{2}{5}f_0 - \frac{6}{5}$ for all of $S(2,4)$, after observing the easier $g_2 \geq f_0/4$ from McMullen's integral formula relating the $g$-numbers of a complex to those of its vertex links; they also proved that an initial segment of the $g$-vector of any sphere without large missing faces is a level sequence — the Hilbert function (the sequence of dimensions of the graded components) of an Artinian level ring, i.e., a standard graded algebra of finite vector-space dimension whose socle, the subspace annihilated by all elements of positive degree, is concentrated in a single degree — yielding further inequalities. Nevertheless, the conjectured coefficient $\lambda = 1$ remains out of reach; the same paper states plainly that "we are still very far from proving Gal's or Nevo's conjectures".

Problem Statement

Prove or refute Nevo's conjecture for the class $S(2,4)$: every ($\mathbb{Z}/2\mathbb{Z}$-homology) 4-dimensional simplicial sphere $\Delta$ all of whose missing faces (minimal non-faces) have dimension at most 2 satisfies

$$g_2(\Delta) \geq g_1(\Delta),$$

where, for a 4-sphere, $g_1(\Delta) = f_0(\Delta) - 6$ and $g_2(\Delta) = f_1(\Delta) - 5f_0(\Delta) + 15$. Equivalently:

$$f_1(\Delta) \geq 6f_0(\Delta) - 21.$$

This is the first open case $(i,d) = (2,4)$ of Nevo's Conjecture 1.5 on generalized $g$-vectors of spheres without large missing faces, restated as Conjecture 2.8 in Novik–Zheng (2026). A proof must cover the full class of $\mathbb{Z}/2\mathbb{Z}$-homology 4-spheres (in particular not only polytopal spheres — boundaries of simplicial 5-polytopes — or PL, that is, piecewise-linear, spheres; even the polytopal subcase, Nevo's Problem 4.3, is open); a single explicit complex in $S(2,4)$ with $g_2 < g_1$ refutes the conjecture. In the parametrization $g_2 \geq \lambda f_0 - (8\lambda - 2)$ of Novik–Zheng, the largest admissible constant $\lambda$ is currently known only to lie in $[2/5, 1]$, and the conjecture asserts that $\lambda = 1$ is attained, which would be optimal since equality holds for $K(2,4) = \partial\sigma^2 * \partial\sigma^2 * \partial\sigma^1$ (the 8-vertex join defined in the background).

References

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