Dehn Sommerville Manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912545183367168 |
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| author | Knill, Oliver |
| author_facet | Knill, Oliver |
| contents | Dehn-Sommerville manifolds are a class of finite abstract simplicial complexes that generalize discrete manifolds. Despite a simpler definition in comparison to manifolds, they still share most properties of manifolds. They especially satisfy all Dehn-Sommerville symmetries telling that half of the f-vector entries are redundant. They also share other properties with q-manifolds: for every Dehn-Sommerville q-manifold G and any function g: V(G) to A={0, ..., k} with positive k, the set of x such that g(x) contains A is a Dehn-Sommerville (q-k)-manifold if not empty. We also see that for Dehn-Sommerville q-manifolds, all higher characteristics w_m(G) agree with Euler characteristic that the chromatic number is bounded above by 2q+2 and that odd-dimensional Dehn-Sommerville manifolds are flat and form a monoid under the join operation. In general, Dehn-Sommerville manifolds are invariant under edge refinement, Barycentric refinement and Cartesian products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14372 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dehn Sommerville Manifolds Knill, Oliver Combinatorics 05Exx 68Rxx Dehn-Sommerville manifolds are a class of finite abstract simplicial complexes that generalize discrete manifolds. Despite a simpler definition in comparison to manifolds, they still share most properties of manifolds. They especially satisfy all Dehn-Sommerville symmetries telling that half of the f-vector entries are redundant. They also share other properties with q-manifolds: for every Dehn-Sommerville q-manifold G and any function g: V(G) to A={0, ..., k} with positive k, the set of x such that g(x) contains A is a Dehn-Sommerville (q-k)-manifold if not empty. We also see that for Dehn-Sommerville q-manifolds, all higher characteristics w_m(G) agree with Euler characteristic that the chromatic number is bounded above by 2q+2 and that odd-dimensional Dehn-Sommerville manifolds are flat and form a monoid under the join operation. In general, Dehn-Sommerville manifolds are invariant under edge refinement, Barycentric refinement and Cartesian products. |
| title | Dehn Sommerville Manifolds |
| topic | Combinatorics 05Exx 68Rxx |
| url | https://arxiv.org/abs/2508.14372 |