Sparse higher-dimensional spaces with no hidden parity obstructions

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Family 177 · Combinatorics
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

An expander graph is sparse but difficult to separate: a substantial set of vertices has many edges leading out. Can an analogous phenomenon hold for spaces made of edges, triangles, tetrahedra and higher-dimensional pieces, while every vertex still meets only a bounded number of pieces?

In higher dimensions, there is an additional difficulty. Local parity tests may all pass even when the tested data do not arise from the expected global construction. Such hidden obstructions are measured by cohomology. A full coboundary expander must both exclude these obstructions and quantitatively detect distance from the valid global data.

The manuscript claims arbitrarily large examples in every fixed dimension at least three, with uniformly bounded vertex incidence and expansion in every lower degree. Combined with known graph and two-dimensional results, this supplies bounded-degree coboundary expanders over the two-element field in every positive dimension.

2. Background and essential definitions

A simplicial complex is a collection of finite vertex sets closed under taking subsets. Its vertices are one-element faces, edges are two-element faces, triangles are three-element faces, and so on. An $i$-face has $i+1$ vertices. A pure $d$-dimensional complex is one in which every face belongs to some $d$-face.

The field $\mathbb F_2=\{0,1\}$ uses addition modulo two. An $i$-cochain $f$ assigns a bit to each $i$-face. Equivalently, it selects the faces assigned one.

Its coboundary $\delta_i f$ assigns a bit to each $(i+1)$-face by adding the bits on its $i$-dimensional subfaces:

$$(\delta_i f)(\tau)=\sum_{\substack{\sigma\subset\tau\\|\sigma|=i+1}}f(\sigma) \quad\text{in }\mathbb F_2.$$

For $i=0$, a selected vertex set has coboundary exactly the edges crossing to its complement. For $i=1$, an edge labeling fails the triangle test when a triangle contains an odd number of labeled edges.

Applying two successive coboundaries gives zero: every smaller face is counted twice. Therefore the space of coboundaries

$$B^i=\operatorname{im}\delta_{i-1}$$

lies inside the space of cocycles $Z^i=\ker\delta_i$. A cocycle passes every local parity test. A coboundary comes from data one dimension lower. Their quotient $H^i=Z^i/B^i$ is cohomology; it records the possible gap between those notions. In degree zero, the paper takes $B^0$ to be the constant functions.

The norm $\|f\|$ is a weighted fraction of faces labeled one. Sample a top-dimensional face uniformly and then one of its $i$-faces uniformly; the norm is the probability of seeing a one. The distance $\operatorname{dist}(f,B^i)$ is the smallest such fraction of changes needed to turn $f$ into a coboundary.

3. Development history and previous partial results

The higher-dimensional expansion viewpoint grew through work on random complexes by Linial and Meshulam, then Meshulam and Wallach. These models keep all lower-dimensional faces and add top-dimensional faces randomly, studying when cohomological obstructions disappear.

Gromov connected filling inequalities with topological overlap and asked for large complexes combining bounded local incidence with uniform expansion properties. The resulting theory has several related notions, and their differences matter for this family.

Lubotzky and Meshulam constructed two-dimensional expanders from random Latin squares. Lubotzky, Luria and Rosenthal used random Steiner systems in every dimension. These constructions bound how many top faces meet a codimension-one face. That is weaker than bounding incidence at vertices: their lower-dimensional skeletons are complete, so a vertex meets increasingly many edges as the complex grows.

For bounded vertex degree, Kaufman, Kazhdan and Lubotzky obtained cosystolic and topological expansion from Ramanujan complexes. Evra and Kaufman established bounded-degree cosystolic expanders in every dimension. Among other properties, cosystolic expansion controls distance to the space $Z^i$ of cocycles. It can coexist with nonzero cohomology.

Chapman and Lubotzky proved bounded-degree two-dimensional coboundary expansion over $\mathbb F_2$. Other recent work obtained degree-one coboundary expansion with suitable non-Abelian coefficients. These results did not supply all lower cochain degrees simultaneously in every higher dimension.

The new manuscript uses an established algebraic construction program and a cosystolic expansion theorem of Oppenheim and Valentiner-Branth. Its additional task is to arrange cohomology vanishing in the same sequence of bounded-degree complexes, converting the available distance-to-cocycles inequality into full coboundary expansion.

4. What the new manuscripts claim

For every fixed integer $d\geq3$, the manuscript states that there are constants $D<\infty$ and $\varepsilon>0$, and finite connected pure $d$-dimensional complexes $X_m$, such that their number of vertices tends to infinity and every vertex belongs to at most $D$ top-dimensional faces.

For every degree $0\leq i<d$ and every $i$-cochain $f$, the claimed inequality is

$$\|\delta_i f\|_{X_m,i+1} \geq\varepsilon\operatorname{dist}_{X_m,i}(f,B^i(X_m)).$$

The constants may depend on $d$, but not on the size parameter $m$, the degree $i$ within that range, or the cochain $f$.

The statement has two inseparable parts. First, if $\delta_i f=0$, the inequality forces $f\in B^i$. Thus every cocycle is a coboundary in the stated degrees: there are no nonzero cohomology classes there. Second, a cochain far from $B^i$ must fail a proportionate number of local tests.

In degree zero, distance to the constant functions measures the smaller weighted side of a vertex cut. The inequality becomes an ordinary expansion statement. In degree one, it says that an edge labeling far from every labeling induced by vertex bits must violate many triangle parity tests.

The degree bound is genuinely at vertices. Since the complex is pure and $d$ is fixed, bounding the number of incident top faces also bounds the number of incident lower faces. This avoids the growing complete skeletons of earlier bounded-codimension-one-degree constructions.

5. Why the advance matters

Importance: 5/5. The claim resolves a central existence problem in higher-dimensional expansion by combining three requirements: arbitrarily large size, bounded incidence at vertices, and full coboundary expansion in all degrees below the dimension.

The distinction from expansion toward cocycles is structural. If a nontrivial cocycle exists, it passes every local test while remaining a positive distance from the coboundaries. No quantitative improvement in the local test constant can remove that obstruction. It must be eliminated by topology or algebra.

The result also sharpens the analogy with graph expanders. Sparse graphs can globally connect a large vertex set using bounded local degree. Here a similarly sparse collection of higher-dimensional faces is claimed to control global consistency of face labelings at every lower level. Such complexes are relevant to topological combinatorics and local testing, although a particular coding or algorithmic application requires additional specifications beyond this theorem.

6. The main idea

Build large complexes with fixed local pieces

The construction starts with matrix groups over a polynomial ring $\mathbb F_q[t]$, where $q$ is a sufficiently large fixed odd prime depending on the desired dimension. A suitable group is reduced modulo successive powers $t^m$, producing larger finite groups.

A coset complex is built from a group and chosen subgroups. Its vertices are tagged cosets, and a collection of vertices forms a face when the corresponding cosets have a common group element. Keeping the local subgroups fixed bounds the number of faces meeting a vertex, while the growing quotient groups increase the total size.

The manuscript first constructs complexes in a higher ambient dimension, then keeps only faces of dimension at most $d$. This retained subcomplex is called the $d$-skeleton.

Remove the global obstructions

An infinite version of the coset complex is placed inside a geometric object called an affine building. Buildings organize many simplex-shaped pieces according to matrix-group symmetries. The paper uses this geometry to prove homological vanishing in an appropriate range.

Passing that conclusion to each finite quotient is the delicate algebraic stage. Existing stability results initially control an inverse system of congruence subgroups, rather than immediately giving vanishing at every individual level. The manuscript combines them with transfer maps associated with finite quotients of odd order.

Working over $\mathbb F_2$ is important here: odd integers act invertibly on these coefficients. The resulting averaging and surjectivity statements turn the system-level information into the required finite-level vanishing. This is how the paper eliminates the hidden cocycles in each constructed complex.

Combine vanishing with quantitative expansion

The existing Kac–Moody–Steinberg expansion theorem supplies a uniform estimate of the form

$$\|\delta_i f\|\geq\eta\operatorname{dist}(f,Z^i)$$

on the relevant complexes. Once the new argument establishes $Z^i=B^i$, the target subspace becomes the space of coboundaries.

One further step is needed because the final theorem uses the $d$-skeleton and its own face-sampling weights. The manuscript compares how many top faces contain each retained face. Uniform bounds control the change in norms, and a common normalization factor cancels between successive dimensions. Thus both the expansion constant and the vertex-incidence bound remain independent of the growing quotient size.

7. Scope, limitations, and open questions

The result is over $\mathbb F_2$. It does not assert the same construction and constants for every coefficient field or for arbitrary non-Abelian coefficients. The use of odd-order group quotients is specifically coordinated with characteristic two.

The dimension is fixed before choosing the constants. There is no uniform bound on $D$ or $\varepsilon$ as $d$ itself tends to infinity. The theorem also controls cochain degrees below $d$, not the top degree, where there are no higher faces available for the same coboundary test.

The sequence has unbounded size; it is not a promise of a complex on every prescribed number of vertices. Nor does the headline optimize the local degree or the expansion constant.

Vanishing of lower cohomology should not be confused with contractibility of the finite complexes. Contractibility is a stronger topological condition. The claimed conclusion is exactly the vanishing and quantitative inequality needed for coboundary expansion in the specified range.

8. Sources and further reading

The manuscript introduces its face weights and distinguishes cocycles from coboundaries before stating the construction. Its final section explains how the existing expansion theorem and the new vanishing result combine.

Primary background includes Gromov’s topological expansion framework, random Steiner-system expanders, Evra–Kaufman’s bounded-degree cosystolic expanders, Chapman–Lubotzky’s two-dimensional result, and Oppenheim–Valentiner-Branth’s KMS expansion theorem. These distinguish the earlier local expansion inputs from the all-degree coboundary conclusion claimed here.

Source manuscripts

• M1. OpenAI, Bounded-degree coboundary expanders in every dimension (September 24, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.