Endpoint-homogeneous fans
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917633853489152 |
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| author | Brian, Will Rogina, Rene Gril |
| author_facet | Brian, Will Rogina, Rene Gril |
| contents | A fan $F$ is \emph{endpoint-homogeneous} if for any two endpoints $e,e'$ of $F$, there is a homeomorphism $h: F \rightarrow F$ such that $h(e) = e'$.
We prove there are uncountably many distinct homeomorphism types of endpoint-homogeneous smooth fans. To do this, we associate to each such fan $F$ a topological invariant, in the form of a characteristic subset $EPG(F) \subseteq [0,1]$ describing how the endpoints of $F$ limit onto any given blade of $F$. We describe precisely all the uncountably many different $X \subseteq [0,1]$ that can arise as $EPG(F)$ for some endpoint-homogeneous smooth fan $F$. We also prove the existence of $\frac{1}{n}$-homogeneous smooth fans for all $n \geq 5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05709 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Endpoint-homogeneous fans Brian, Will Rogina, Rene Gril General Topology A fan $F$ is \emph{endpoint-homogeneous} if for any two endpoints $e,e'$ of $F$, there is a homeomorphism $h: F \rightarrow F$ such that $h(e) = e'$. We prove there are uncountably many distinct homeomorphism types of endpoint-homogeneous smooth fans. To do this, we associate to each such fan $F$ a topological invariant, in the form of a characteristic subset $EPG(F) \subseteq [0,1]$ describing how the endpoints of $F$ limit onto any given blade of $F$. We describe precisely all the uncountably many different $X \subseteq [0,1]$ that can arise as $EPG(F)$ for some endpoint-homogeneous smooth fan $F$. We also prove the existence of $\frac{1}{n}$-homogeneous smooth fans for all $n \geq 5$. |
| title | Endpoint-homogeneous fans |
| topic | General Topology |
| url | https://arxiv.org/abs/2404.05709 |