A non-face characterization of spheres on few vertices
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915421930651648 |
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| author | Huang, Shuai Miller, Jasper Rose-Levine, Daniel Simon, Steven |
| author_facet | Huang, Shuai Miller, Jasper Rose-Levine, Daniel Simon, Steven |
| contents | We prove a relatively simple combinatorial characterization of simplicial $d$-spheres on $d+4$ vertices. Our criteria are given in terms of the intersection patterns of a simplicial complex's family of minimal non-faces. Namely, let $Σ$ be a simplicial complex on $d+4$ vertices and let $\mathcal{F}$ be its family of minimal non-faces. Then $Σ$ is a $d$-sphere if and only if $|\mathcal{F}|=n\geq 3$ is odd and there is an ordering $A_0,\ldots, A_{n-1}$ of the minimal non-faces, indices taken modulo $n$, such that successive $A_i$ are disjoint and the alternating $\frac{(n-1)}{2}$-fold intersections $A_i\cap A_{i+2} \cap A_{i+4} \cap \cdots \cap A_{i+n-3}$ partition the vertex set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06120 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A non-face characterization of spheres on few vertices Huang, Shuai Miller, Jasper Rose-Levine, Daniel Simon, Steven Combinatorics 05E45, 52B35, 52B12 We prove a relatively simple combinatorial characterization of simplicial $d$-spheres on $d+4$ vertices. Our criteria are given in terms of the intersection patterns of a simplicial complex's family of minimal non-faces. Namely, let $Σ$ be a simplicial complex on $d+4$ vertices and let $\mathcal{F}$ be its family of minimal non-faces. Then $Σ$ is a $d$-sphere if and only if $|\mathcal{F}|=n\geq 3$ is odd and there is an ordering $A_0,\ldots, A_{n-1}$ of the minimal non-faces, indices taken modulo $n$, such that successive $A_i$ are disjoint and the alternating $\frac{(n-1)}{2}$-fold intersections $A_i\cap A_{i+2} \cap A_{i+4} \cap \cdots \cap A_{i+n-3}$ partition the vertex set. |
| title | A non-face characterization of spheres on few vertices |
| topic | Combinatorics 05E45, 52B35, 52B12 |
| url | https://arxiv.org/abs/2507.06120 |