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| Natura: | Preprint |
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2022
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| Accesso online: | https://arxiv.org/abs/2210.12567 |
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| _version_ | 1866913930538909696 |
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| author | Brazas, Jeremy Fischer, Hanspeter |
| author_facet | Brazas, Jeremy Fischer, Hanspeter |
| contents | We present a 2-dimensional Peano continuum $\mathbb{T}\subseteq \mathbb{R}^3$ with the following properties: (1) There is a universal covering projection $q:\overline{\mathbb{T}}\rightarrow \mathbb{T}$ with uncountable fundamental group $π_1(\overline{\mathbb{T}})$; (2) For every $1\not=[\overlineα]\in π_1(\overline{\mathbb{T}},\ast)$, there is a covering projection $r:(E,e)\rightarrow (\overline{\mathbb{T}},\ast)$ such that $[\overlineα]\not\in r_\#π_1(E,e)$; (3) There is no universal covering projection $r:E\rightarrow \overline{\mathbb{T}}$; (4) The universal object $p:\widetilde{\mathbb{T}}\rightarrow \mathbb{T}$ in the category of fibrations with unique path lifting (and path-connected total space) over $\mathbb{T}$ has trivial fundamental group $π_1(\widetilde{\mathbb{T}})=1$; (5) $p:\widetilde{\mathbb{T}}\rightarrow \mathbb{T}$ is not a path component of an inverse limit of covering projections over $\mathbb{T}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_12567 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space Brazas, Jeremy Fischer, Hanspeter Algebraic Topology 55R05, 57M10 We present a 2-dimensional Peano continuum $\mathbb{T}\subseteq \mathbb{R}^3$ with the following properties: (1) There is a universal covering projection $q:\overline{\mathbb{T}}\rightarrow \mathbb{T}$ with uncountable fundamental group $π_1(\overline{\mathbb{T}})$; (2) For every $1\not=[\overlineα]\in π_1(\overline{\mathbb{T}},\ast)$, there is a covering projection $r:(E,e)\rightarrow (\overline{\mathbb{T}},\ast)$ such that $[\overlineα]\not\in r_\#π_1(E,e)$; (3) There is no universal covering projection $r:E\rightarrow \overline{\mathbb{T}}$; (4) The universal object $p:\widetilde{\mathbb{T}}\rightarrow \mathbb{T}$ in the category of fibrations with unique path lifting (and path-connected total space) over $\mathbb{T}$ has trivial fundamental group $π_1(\widetilde{\mathbb{T}})=1$; (5) $p:\widetilde{\mathbb{T}}\rightarrow \mathbb{T}$ is not a path component of an inverse limit of covering projections over $\mathbb{T}$. |
| title | A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space |
| topic | Algebraic Topology 55R05, 57M10 |
| url | https://arxiv.org/abs/2210.12567 |