Physical Sciences and Mathematics Commons^™

A Natural Pseudometric On Homotopy Groups Of Metric Spaces, Jeremy Brazas, Paul Fabel

Mathematics Faculty Publications

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 …