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A natural pseudometric on homotopy groups of metric spaces

Jeremy Brazas, Paul Fabel

TL;DR

The paper defines a natural pseudometric $\rho$ on $\pi_n(X,x_0)$ derived from the uniform metric on the $n$-loops $\Omega^n(X,x_0)$, yielding a topological group $\pi_n^{\text{met}}(X,x_0)$ whose topology is independent of the base metric when $X$ is compact. It then situates this metric topology among classical topologies on homotopy groups, notably the quotient, $\tau$, and shape topologies, and shows that $\rho$-topology is always at least as fine as the shape topology. Under two natural hypotheses—$X$ being $LC^{n-1}$ or $X$ an inverse limit of finite polyhedra with retraction bonding maps—the $\rho$-topology coincides with the shape topology, linking metric geometry to shape theory. These results enable applying shape-theoretic methods to metric-enriched homotopy groups in broad classes of spaces and clarify when the geometric pseudometric preserves shape information.

Abstract

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.

A natural pseudometric on homotopy groups of metric spaces

TL;DR

The paper defines a natural pseudometric on derived from the uniform metric on the -loops , yielding a topological group whose topology is independent of the base metric when is compact. It then situates this metric topology among classical topologies on homotopy groups, notably the quotient, , and shape topologies, and shows that -topology is always at least as fine as the shape topology. Under two natural hypotheses— being or an inverse limit of finite polyhedra with retraction bonding maps—the -topology coincides with the shape topology, linking metric geometry to shape theory. These results enable applying shape-theoretic methods to metric-enriched homotopy groups in broad classes of spaces and clarify when the geometric pseudometric preserves shape information.

Abstract

For a path-connected metric space , the -th homotopy group inherits a natural pseudometric from the -th iterated loop space with the uniform metric. This pseudometric gives the structure of a topological group and when is compact, the induced pseudometric topology is independent of the metric . In this paper, we study the properties of this pseudometric and how it relates to previously studied structures on . Our main result is that the pseudometric topology agrees with the shape topology on if is compact and or if is an inverse limit of finite polyhedra with retraction bonding maps.
Paper Structure (8 sections, 22 theorems, 14 equations, 2 figures)

This paper contains 8 sections, 22 theorems, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topologies on homotopy groups
  4. The quotient topology
  5. The "tau topology"
  6. The shape topology
  7. A pseudometric on homotopy groups
  8. Comparison with the Shape Topology

Key Result

Theorem 1.1

Let $(X,d)$ be a path-connected compact metric space and $n\geq 1$. The topology induced by the natural pseudometric $\rho$ on $\pi_n(X,x_0)$ is at least as fine as the shape topology. Moreover, these two topologies agree in the following two cases:

Figures (2)

  • Figure 1: The mapping torus of the shift map $f:\mathbb{E}_2\to\mathbb{E}_2$ of the $2$-dimensional earring space.
  • Figure 2: The space $B$ in the case $n=1$ is the union of a cylinder over $S^1$ and a non-compact surface that limits on the cylinder. The space $X$ is constructed by connecting the two path components with an arc.

Theorems & Definitions (52)

  • Theorem 1.1
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3
  • Remark 4.1
  • Lemma 4.2: Isometric Inversion
  • proof
  • Lemma 4.3: Isometric translations
  • proof
  • Lemma 4.4
  • ...and 42 more