On hyperspaces of knots and planar simple closed curves
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918140780216320 |
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| author | Krupski, Paweł Omiljanowski, Krzysztof |
| author_facet | Krupski, Paweł Omiljanowski, Krzysztof |
| contents | We consider the Vietoris hyperspaces $\mathcal S(\mathbb R^n)$ of simple closed curves in $\mathbb R^n$, $n=2,3$, and their subspaces $\mathcal S_P(\mathbb R^2)$ of planar simple closed polygons, $\mathcal K_P$ of polygonal knots, and $\mathcal K_T$ of tame knots. We prove that all the hyperspaces are strongly locally contractible, arcwise connected, infinite-dimensional Cantor manifolds, and $\mathcal S(\mathbb R^2)$ and $\mathcal K_T$ are strongly infinite-dimensional Cantor manifolds. Moreover, $\mathcal S_P(\mathbb R^2)$ and $\mathcal K_P$ are $σ$-compact, strongly countable-dimensional absolute neighborhood retracts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On hyperspaces of knots and planar simple closed curves Krupski, Paweł Omiljanowski, Krzysztof General Topology 57K10 (Primary) 54B20, 54F45, 54H05 (Secondary) We consider the Vietoris hyperspaces $\mathcal S(\mathbb R^n)$ of simple closed curves in $\mathbb R^n$, $n=2,3$, and their subspaces $\mathcal S_P(\mathbb R^2)$ of planar simple closed polygons, $\mathcal K_P$ of polygonal knots, and $\mathcal K_T$ of tame knots. We prove that all the hyperspaces are strongly locally contractible, arcwise connected, infinite-dimensional Cantor manifolds, and $\mathcal S(\mathbb R^2)$ and $\mathcal K_T$ are strongly infinite-dimensional Cantor manifolds. Moreover, $\mathcal S_P(\mathbb R^2)$ and $\mathcal K_P$ are $σ$-compact, strongly countable-dimensional absolute neighborhood retracts. |
| title | On hyperspaces of knots and planar simple closed curves |
| topic | General Topology 57K10 (Primary) 54B20, 54F45, 54H05 (Secondary) |
| url | https://arxiv.org/abs/2401.13084 |