Missing faces and topological embeddings of sphere skeleta

Abstract

Let S be a finite simplicial triangulation of the topological sphere S^(2d), and let T be a missing d-face. We prove that adjoining T to the d-skeleton of S prevents a topological embedding in S^(2d). For d ≥ 3, a graded Koszul calculation combines with established ambient-completion, anisotropy and metastable embedding theorems. For d = 2, we apply the Nevo–Wagner bistellar transfer theorem in the PL class supplied by the given triangulation: a contractible complementary manifold supplies the required disk map, and a flag subdivision supplies the base. This proves the statement of their Conjecture 1.3 without a standard PL structure assumption on the source. No claim of first priority or a new general method is made.

The idea in plain English

A triangulation builds a shape from vertices, edges, triangles and their higher-dimensional counterparts. Its skeleton retains only pieces up to a chosen dimension. A missing face is a piece that is absent even though its entire boundary is already present.

For an ordinary triangulated sphere, the edge network cannot be drawn on the sphere without crossings after adding any missing edge. This paper proves the corresponding statement in every even dimension: add a missing middle-dimensional face to the skeleton, and the enlarged space cannot fit into the original sphere without identifying distinct points.

Nevo and Wagner’s earlier theorem already rules out arbitrary continuous embeddings when the source has the standard piecewise-linear sphere structure. This result allows any finite triangulation of a topological sphere.

The precise result

Let d ≥ 0, let S be a finite simplicial complex whose realization is homeomorphic to S^(2d), and let T be a missing d-face on its actual vertex set. Then the realization of S^(d) ∪ {T} has no topological embedding into S^(2d).

The theorem establishes the statement of Nevo–Wagner Conjecture 1.3. For d = 2, the proof also gives a nonzero mod-2 van Kampen obstruction. The d = 0 statement is vacuous under the actual-vertex convention.

Why the result holds

Adding the missing face creates one extra class in a graded face-ring quotient. An embedding into a suitable PL sphere would force a surjective multiplication map between graded pieces of incompatible dimensions.

For d ≥ 3, a topological embedding supplies deleted-product data. Weber’s metastable theorem would then produce the PL embedding already ruled out by the algebraic argument.

Dimension four needs a separate argument. Nevo–Wagner transfer carries the obstruction through bistellar moves within the source’s own PL class, starting from a flag subdivision. A contractible complement supplies a disk map without requiring a standard PL complementary ball.

Scope and limitations

The source must be a finite triangulation of a topological even-dimensional sphere; the statement is not a theorem for every complex with the same homology.

The added simplex must be a missing face: every proper face must already be present, and its vertices must belong to the original complex.

The distinction from the earlier theorem concerns the source triangulation. Nevo and Wagner’s result for PL spheres already excludes arbitrary topological embeddings.

The proof combines established algebraic and topological theorems, including Nevo–Wagner transfer in dimension four and the ambient Lefschetz mechanism in higher dimensions.

Status and AI use

This is an unrefereed preprint. AI tools were used extensively in the research, proof development, computational checks, literature review, and writing. The supporting package contains the published source and verification materials.

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Collaborate on this result

I welcome substantive collaboration on an independent mathematical review of the face-ring and dimension-four arguments, or on developing a journal submission.

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

Cite this paper

Alec Kriebel. (2026). Missing faces and topological embeddings of sphere skeleta (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23310484

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Background and prior work

• Eran Nevo and Uli Wagner, On the embeddability of skeleta of spheres (2009): Proves the PL-source case, states the general topological-sphere case as Conjecture 1.3, and supplies the bistellar transfer used in dimension four.
• Karim Adiprasito, Combinatorial Lefschetz theorems beyond positivity (2019 version): Develops the ambient Lefschetz quotient mechanism that underlies the algebraic obstruction.
• Karim Adiprasito and Zuzana Patáková, A higher-dimensional version of Fáry’s theorem (2025): Supplies an ambient triangulation containing the original abstract complex after a PL embedding; the embedding map may change.
• Kalle Karu and Elizabeth Xiao, On the anisotropy theorem of Papadakis and Petrotou (2023): Provides the generic anisotropy input used to obtain the relevant Lefschetz isomorphism.
• Claude Weber, Plongements de polyèdres dans le domaine métastable (1967): Connects deleted-product data to the existence of a PL embedding in the dimensional range used for d ≥ 3.