A natural pseudometric on homotopy groups of metric spaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917901041139712 |
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| author | Brazas, Jeremy Fabel, Paul |
| author_facet | Brazas, Jeremy Fabel, Paul |
| contents | For a path-connected metric space $(X,d)$, the $n$-th homotopy group $π_n(X)$ inherits a natural pseudometric from the $n$-th iterated loop space with the uniform metric. This pseudometric gives $π_n(X)$ the structure of a topological group and when $X$ is compact, the induced pseudometric topology is independent of the metric $d$. In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on $π_n(X)$. Our main result is that the pseudometric topology agrees with the shape topology on $π_n(X)$ if $X$ is compact and $LC^{n-1}$ or if $X$ is an inverse limit of finite polyhedra with retraction bonding maps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_14141 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A natural pseudometric on homotopy groups of metric spaces Brazas, Jeremy Fabel, Paul Algebraic Topology Metric Geometry 55Q52, 55P55, 54E35, 54C56 For a path-connected metric space $(X,d)$, the $n$-th homotopy group $π_n(X)$ inherits a natural pseudometric from the $n$-th iterated loop space with the uniform metric. This pseudometric gives $π_n(X)$ the structure of a topological group and when $X$ is compact, the induced pseudometric topology is independent of the metric $d$. In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on $π_n(X)$. Our main result is that the pseudometric topology agrees with the shape topology on $π_n(X)$ if $X$ is compact and $LC^{n-1}$ or if $X$ is an inverse limit of finite polyhedra with retraction bonding maps. |
| title | A natural pseudometric on homotopy groups of metric spaces |
| topic | Algebraic Topology Metric Geometry 55Q52, 55P55, 54E35, 54C56 |
| url | https://arxiv.org/abs/2211.14141 |