Reconstructing Polytopes and Pseudomanifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915293543006208 |
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| author | Hinman, Joshua |
| author_facet | Hinman, Joshua |
| contents | We prove that every 4-polytope is determined by its edge-polygon incidences, solving an open problem of Grünbaum. For each $d \geq 3$, 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 $d \geq 4$ and $\lceil \frac{d}{2} \rceil \leq k \leq d-2$, every homology $(d-1)$-manifold is determined by the incidences of its $k$- and $(k-1)$-faces. For $d \geq 5$ and $\lceil \frac{d+1}{2} \rceil \leq k \leq d-2$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13789 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reconstructing Polytopes and Pseudomanifolds Hinman, Joshua Combinatorics 52B05 (Primary), 05E45 (Secondary) We prove that every 4-polytope is determined by its edge-polygon incidences, solving an open problem of Grünbaum. For each $d \geq 3$, 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 $d \geq 4$ and $\lceil \frac{d}{2} \rceil \leq k \leq d-2$, every homology $(d-1)$-manifold is determined by the incidences of its $k$- and $(k-1)$-faces. For $d \geq 5$ and $\lceil \frac{d+1}{2} \rceil \leq k \leq d-2$, 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. |
| title | Reconstructing Polytopes and Pseudomanifolds |
| topic | Combinatorics 52B05 (Primary), 05E45 (Secondary) |
| url | https://arxiv.org/abs/2505.13789 |