2025/05/20 by Joshua G. Hinman, Hinman, Joshua
Arts and Humanities · Computer Science · Mathematics · #05E45 (Secondary) #52B05 (Primary) #Art, Technology, and Culture #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2505.13789
openalex publication_date 2025/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that every 4-polytope is determined by its edge-polygon incidences, solving an open problem of Grünbaum. For each d ≥ 3, we show that not every d-polytope is determined by its (d-3)-skeleton and dual (d-3)-skeleton together, answering a question of Samper. In the simplicial realm, we prove that for d ≥ 4 and \lceil (d)/(2) \rceil ≤ k ≤ d-2, every homology (d-1)-manifold is determined by the incidences of its k- and (k-1)-faces. For d ≥ 5 and \lceil (d+1)/(2) \rceil ≤ k ≤ d-2, we extend our proof to normal (d-1)-pseudomanifolds whose (2d-2k-1)-dimensional links are homology manifolds. Finally, we prove that not every normal (d-1)-pseudomanifold is determined by its (d-2)-skeleton.