Simplicial cell decompositions of $\mathbb{CP}^{\hspace{.3mm}n}$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909876760870912 |
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| author | Datta, Basudeb Spreer, Jonathan |
| author_facet | Datta, Basudeb Spreer, Jonathan |
| contents | According to a well-known result in geometric topology, we have \linebreak $\left (\mathbb{S}^2 \right)^{n}\!\!/\operatorname{Sym}(n) = \mathbb{CP}^{n}$, where $\operatorname{Sym}(n)$ acts on $\left (\mathbb{S}^2 \right)^{n}$ by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of $\mathbb{CP}^{n}$ for each $n \geq 2$. In more detail, we start with the standard two triangle crystallisation $S^2_3$ of the $2$-sphere $\mathbb{S}^2$, in its $n$-fold Cartesian product. We then construct a simplicial subdivision of this product and prove that the $\operatorname{Sym}(n)$ quotient of this subdivision yields a simplicial cell decomposition of $\mathbb{CP}^n$. The first derived subdivision of this cell complex is a simplicial triangulation of $\mathbb{CP}^n$. To the best of our knowledge, this is the first explicit description of triangulations of $\mathbb{CP}^n$ for $n \geq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simplicial cell decompositions of $\mathbb{CP}^{\hspace{.3mm}n}$ Datta, Basudeb Spreer, Jonathan Combinatorics Geometric Topology 57Q15, 57Q05, 05C15, 06A06 According to a well-known result in geometric topology, we have \linebreak $\left (\mathbb{S}^2 \right)^{n}\!\!/\operatorname{Sym}(n) = \mathbb{CP}^{n}$, where $\operatorname{Sym}(n)$ acts on $\left (\mathbb{S}^2 \right)^{n}$ by coordinate permutation. We use this fact to explicitly construct a regular simplicial cell decomposition of $\mathbb{CP}^{n}$ for each $n \geq 2$. In more detail, we start with the standard two triangle crystallisation $S^2_3$ of the $2$-sphere $\mathbb{S}^2$, in its $n$-fold Cartesian product. We then construct a simplicial subdivision of this product and prove that the $\operatorname{Sym}(n)$ quotient of this subdivision yields a simplicial cell decomposition of $\mathbb{CP}^n$. The first derived subdivision of this cell complex is a simplicial triangulation of $\mathbb{CP}^n$. To the best of our knowledge, this is the first explicit description of triangulations of $\mathbb{CP}^n$ for $n \geq 4$. |
| title | Simplicial cell decompositions of $\mathbb{CP}^{\hspace{.3mm}n}$ |
| topic | Combinatorics Geometric Topology 57Q15, 57Q05, 05C15, 06A06 |
| url | https://arxiv.org/abs/2401.09891 |