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.
Glasgow Mathematical Journal
Cambridge University Press
Brazas, J., & Fabel, P. (2023). A natural pseudometric on homotopy groups of metric spaces. Glasgow Mathematical Journal, 1-17. http://dx.doi.org/10.1017/S0017089523000393