SIAM Journal on Discrete Mathematics· 2026Q1
Reconstructing Polytopes and Pseudomanifolds
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- Q1SCImago
- 2026year
Short summary
Every 4-polytope is determined by its edge-polygon incidences, resolving a long-standing open problem.
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Key points
- Every 4-polytope is uniquely determined by its edge-polygon incidences.
- Not all k-polytopes are determined by their k-skeleton and dual k-skeleton together.
- Homology k-manifolds are determined by incidences of their (k-1)- and 1-faces for specific dimensions.
- Normal k-pseudomanifolds are not always determined by their (k-1)-skeleton.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract. We prove that every 4-polytope is determined by its edge-polygon incidences, solving an open problem of Grünbaum. For each [Formula: see text], we show that not every [Formula: see text]-polytope is determined by its [Formula: see text]-skeleton and dual [Formula: see text]-skeleton together, answering a question of Samper. In the simplicial realm, we prove that for [Formula: see text] and [Formula: see text], every homology [Formula: see text]-manifold is determined by the incidences of its [Formula: see text]- and [Formula: see text]-faces. For [Formula: see text] and [Formula: see text], we extend our proof to normal [Formula: see text]-pseudomanifolds whose [Formula: see text]-dimensional links are homology manifolds. Finally, we prove that not every normal [Formula: see text]-pseudomanifold is determined by its [Formula: see text]-skeleton.
The authors' abstract, as published at the source. SIAM Journal on Discrete Mathematics, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics