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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.02519 |
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| _version_ | 1866909223501168640 |
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| author | Bourque, Maxime Fortier |
| author_facet | Bourque, Maxime Fortier |
| contents | We use the Schwarz-Christoffel formula to show that for every $n\geq 3$, the space of labelled immersed $n$-gons in the plane up to similarity is homeomorphic to $\mathbb{R}^{2n-4}$. We then prove that all immersed triangles, quadrilaterals, and pentagons are embedded, from which it follows that the space of labelled simple $n$-gons up to similarity is homeomorphic to $\mathbb{R}^{2n-4}$ if $n\in \{3,4,5\}$. This was first shown by Gonzáles and López-López for $n=4$ and conjectured to be true for every $n\geq 5$ by González and Sedano-Mendoza. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02519 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The space of immersed polygons Bourque, Maxime Fortier Geometric Topology We use the Schwarz-Christoffel formula to show that for every $n\geq 3$, the space of labelled immersed $n$-gons in the plane up to similarity is homeomorphic to $\mathbb{R}^{2n-4}$. We then prove that all immersed triangles, quadrilaterals, and pentagons are embedded, from which it follows that the space of labelled simple $n$-gons up to similarity is homeomorphic to $\mathbb{R}^{2n-4}$ if $n\in \{3,4,5\}$. This was first shown by Gonzáles and López-López for $n=4$ and conjectured to be true for every $n\geq 5$ by González and Sedano-Mendoza. |
| title | The space of immersed polygons |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2406.02519 |