A natural pseudometric on homotopy groups of metric spaces

TL;DR

This paper introduces a natural pseudometric on homotopy groups of metric spaces, exploring its properties and showing its topology aligns with the shape topology under certain conditions.

Contribution

It defines a new pseudometric on homotopy groups and establishes conditions under which its topology matches the shape topology, linking metric and shape theoretic structures.

Findings

Pseudometric induces a topological group structure on homotopy groups.

Topology from the pseudometric coincides with the shape topology for certain compact spaces.

The pseudometric topology is independent of the metric for compact spaces.

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.