The smoothness of the real projective deformation spaces of orderable Coxeter 3-polytopes
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915219083624448 |
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| author | Choi, Suhyoung Park, Seungyeol |
| author_facet | Choi, Suhyoung Park, Seungyeol |
| contents | A Coxeter polytope is a convex polytope in a real projective space equipped with linear reflections in its facets, such that the orbits of the polytope under the action of the group generated by the linear reflections tessellate a convex domain in the real projective space. Vinberg proved that the group generated by these reflections acts properly discontinuously on the interior of the convex domain, thus inducing a natural orbifold structure on the polytope. In this paper, we consider labeled combinatorial polytopes $\mathcal{G}$ associated to such orbifolds, and study the deformation space $\mathcal{C} (\mathcal{G})$ of Coxeter polytopes realizing $\mathcal{G}$. We prove that if $\mathcal{G}$ is orderable and of normal type then the deformation space $\mathcal{C}(\mathcal{G})$ of real projective Coxeter 3-polytopes realizing $\mathcal{G}$ is a smooth manifold. This result is achieved by analyzing a natural map of $\mathcal{C} (\mathcal{G})$ into a smooth manifold called the realization space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03770 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The smoothness of the real projective deformation spaces of orderable Coxeter 3-polytopes Choi, Suhyoung Park, Seungyeol Geometric Topology Primary 57M50, Secondary 53A20, 53C15 A Coxeter polytope is a convex polytope in a real projective space equipped with linear reflections in its facets, such that the orbits of the polytope under the action of the group generated by the linear reflections tessellate a convex domain in the real projective space. Vinberg proved that the group generated by these reflections acts properly discontinuously on the interior of the convex domain, thus inducing a natural orbifold structure on the polytope. In this paper, we consider labeled combinatorial polytopes $\mathcal{G}$ associated to such orbifolds, and study the deformation space $\mathcal{C} (\mathcal{G})$ of Coxeter polytopes realizing $\mathcal{G}$. We prove that if $\mathcal{G}$ is orderable and of normal type then the deformation space $\mathcal{C}(\mathcal{G})$ of real projective Coxeter 3-polytopes realizing $\mathcal{G}$ is a smooth manifold. This result is achieved by analyzing a natural map of $\mathcal{C} (\mathcal{G})$ into a smooth manifold called the realization space. |
| title | The smoothness of the real projective deformation spaces of orderable Coxeter 3-polytopes |
| topic | Geometric Topology Primary 57M50, Secondary 53A20, 53C15 |
| url | https://arxiv.org/abs/2502.03770 |