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A natural pseudometric on homotopy groups of metric spaces
Published online by Cambridge University Press: 08 November 2023
Abstract
For a path-connected metric space $(X,d)$, the
$n$-th homotopy group
$\pi _n(X)$ inherits a natural pseudometric from the
$n$-th iterated loop space with the uniform metric. This pseudometric gives
$\pi _n(X)$ 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
$\pi _n(X)$. Our main result is that the pseudometric topology agrees with the shape topology on
$\pi _n(X)$ if
$X$ is compact and
$LC^{n-1}$ or if
$X$ is an inverse limit of finite polyhedra with retraction bonding maps.
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust
References
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