Good metric spaces without good parameterizations
type: Item
instance of: scholarly article
title: Good metric spaces without good parameterizations
zbMATH Open document ID: 0854.57018
DOI: 10.4171/RMI/198
author: Stephen Semmes
published in: Revista Matemática Iberoamericana
publication date: 1997-01-12T00:00:00Z
full work available at URL: https://eudml.org/doc/39510
review text: This excellently written paper deals with the question of when a metric space \(M\) (in fact, a subspace of a Euclidean space), that is topologically \(\mathbb{R}^n\), admits a parametrization \(f : \mathbb{R}^n \to M\) with good metric properties, more specifically, a parametrization \(f\) which is quasisymmetric. (A mapping \(f : (X,d) \to (Y,\rho)\) between metric spaces is called quasisymmetric, provided there exists a homeomorphism \(\eta: [0,\infty) \to [0,\infty)\) such that \(d(x,x') \leq td(x,x'')\) implies \(\rho(f(x), f(x')) \leq \eta(t) \rho(f(x), f(x''))\) for all \(t > 0\), \(x, x', x'' \in X).\) The author provides counterexamples to certain optimistic conjectures, by providing examples of topological copies of \(\mathbb{R}^3\) in \(\mathbb{R}^4\) admitting no quasisymmetric parametrization, but possessing many other nice Euclidean-like properties. (These sets enjoy good bounds on their geodesic functions and good mass bounds (Ahlfors regularity). They are smooth except for reasonably tame degenerations near small sets, they are uniformly rectifiable, and they have good properties in terms of analysis). The constructions of these examples repeat (with some subtle extra analysis) the well-known examples of the Whitehead continuum, Bing's dogbone space, and Bing doubling.
reviewed by: Taras Banakh
Mathematics Subject Classification ID: 57N12
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 885338
zbMATH Keywords: topological manifold
zbMATH Keywords: parametrization
zbMATH Keywords: quasisymmetric
zbMATH Keywords: Whitehead continuum
zbMATH Keywords: Bing's dogbone space
zbMATH Keywords: Bing doubling
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 885338