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A natural pseudometric on homotopy groups of metric spaces
- Part of
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- Journal:
- Glasgow Mathematical Journal / Volume 66 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 08 November 2023, pp. 162-174
- Print publication:
- January 2024
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- Article
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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.
