A negative answer to Nevo's homology-sphere decomposition question

Abstract

We construct a finite two-dimensional abstract simplicial complex that is doubly Cohen–Macaulay over every field of characteristic zero, but contains no two-dimensional homology-sphere subcomplex over any field. This gives a negative answer to Nevo's question asking whether every doubly Cohen–Macaulay complex admits a decomposition into homology spheres with specified overlap properties. We use a finite quotient of an affine building, with all links equal to the incidence graph of the projective plane of order eleven. A direct spectral estimate controls the links after any vertex deletion; a weighted cocycle argument verifies the required homology vanishing. The sphere obstruction is an Euler-characteristic count. The construction and the spectral and curvature mechanisms build on established work, credited below.

The idea in plain English

A simplicial complex is built from vertices, edges, triangles and their higher-dimensional counterparts. Doubly Cohen–Macaulay complexes satisfy strong homological conditions that survive removal of any single vertex.

Nevo asked whether every such complex could be assembled from homology-sphere pieces with specified overlap properties. This paper gives a two-dimensional counterexample with no permitted sphere piece at all. The obstruction therefore appears before the overlap requirements can even be considered.

The construction uses a finite quotient of a well-studied geometric object called an affine building. Earlier building theory, spectral estimates and a curvature argument supply the main tools. The contribution is their application to the precise decomposition question, including the conditions after every vertex deletion.

The precise result

There exists a finite pure two-dimensional abstract simplicial complex K that is doubly Cohen–Macaulay over every field of characteristic zero and contains no two-dimensional homology-sphere subcomplex over any field.

Here a homology sphere must have the homology of a sphere both globally and in every face link. The excluded sphere subcomplex need not be induced. Having sphere homology only globally is a weaker condition and is not the definition used in the theorem.

The example gives a negative answer to Nevo’s homology-sphere replacement question, recorded as Problem 8 in the 2005 Oberwolfach report and Problem 3.7 in the 2008 paper. It does not contradict Nevo’s earlier decomposition into minimal cycle complexes, which are a broader class of pieces.

Every vertex link of K is the incidence graph of the projective plane of order eleven. After a vertex deletion, the relevant links are either that graph or the graph with one vertex removed. Both have normalized Laplacian gap strictly greater than one half.

Why the result holds

The construction starts with the vertex-regular affine-building lattice and faithful polynomial matrix representation of Lubotzky, Samuels and Vishne. Reducing those matrices modulo t⁹ gives a finite quotient. A degree bound excludes short identifications, so the quotient retains its projective-plane links.

The projective-plane incidence matrix satisfies an exact Gram identity. This controls the spectral gap even after one link vertex is removed. A finite weighted-cocycle argument then gives the required first-homology vanishing for K and every vertex deletion; connectivity and the remaining face-link conditions complete the doubly Cohen–Macaulay property.

Any homology-sphere subcomplex would have exactly two triangles on each edge and a cycle around each vertex. Those cycles have length at least six because they lie in the projective-plane links. Counting vertices, edges and triangles forces Euler characteristic at most zero, whereas a homology two-sphere has Euler characteristic two.

Scope and limitations

The doubly Cohen–Macaulay assertion is proved for characteristic-zero coefficient fields. It is not a claim over every field of positive characteristic. The field of order eleven in the geometric construction is distinct from the field used to compute homology.

The sphere exclusion uses the local-link definition of a homology sphere. It concerns actual subcomplexes of this finite complex, without subdivision or an enlarged ambient complex; it does not exclude all possible immersed or topologically embedded spheres.

The ambient complex is two-dimensional but is not a surface: each of its edges belongs to twelve triangles. The construction is an existence argument using a finite quotient; the accompanying finite checks do not enumerate that quotient or replace the proof.

The building, congruence method, spectral vanishing mechanism and curvature obstruction have substantial earlier sources. The paper applies them to the stated decomposition question. Its bounded literature search is not a certificate of worldwide priority.

Status and AI use

AI tools were used extensively in the argument, writing, literature research, computation and internal review. These internal reviews do not constitute independent human peer review or formal proof verification. The author is responsible for the work.

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Collaborate on this result

I welcome substantive collaboration on independent review of the building quotient, the vertex-deletion argument, or the comparison with the decomposition literature, as well as preparation for journal submission. Expertise in algebraic combinatorics, simplicial complexes or affine buildings would be especially useful.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). A negative answer to Nevo's homology-sphere decomposition question (Version 1.0.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23444839

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Background and prior work

• Nevo, Rigidity and the lower bound theorem for doubly Cohen–Macaulay complexes (2008): Theorem 3.4 gives the minimal-cycle decomposition. Problem 3.7 asks whether homology spheres can replace those pieces; this is the question addressed here.
• Nevo, Discrete Geometry, Oberwolfach Report 17/2005: Problem 8 on printed page 961 is the earlier workshop formulation of the homology-sphere replacement question.
• Swartz, g-elements, finite buildings and higher Cohen–Macaulay connectivity, v1: Section 4 states the local-link convention for homology spheres used in the decomposition literature. Global sphere homology alone is insufficient.
• Lubotzky, Samuels and Vishne, Explicit constructions of Ramanujan complexes of type Ãd, v2: Sections 2–4 supply the affine building, vertex-regular lattice and faithful polynomial representation. The present paper does not need the later global Ramanujan spectrum results.
• Lubotzky and Meshulam, A Moore bound for simplicial complexes (2007): The polynomial-degree versus congruence-divisibility method is an antecedent. The present paper gives its own short-displacement argument for the particular nonreduced quotient modulo t⁹.
• Ballmann and Świątkowski, On L²-cohomology and property (T) for automorphism groups of polyhedral cell complexes: The strict link-gap threshold is part of the established Garland-type spectral method. The present manuscript includes a finite two-dimensional weighted-cocycle proof.
• Januszkiewicz and Świątkowski, Simplicial nonpositive curvature (2006): Example 1.8(5) records the curvature obstruction to triangulated two-spheres with sufficiently long links. The manuscript uses a direct Euler count for its arbitrary, possibly non-induced sphere subcomplexes.