Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum
type: Item
instance of: scholarly article
title: Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum
zbMATH Open document ID: 1198.30024
author: Juha Heinonen
author: Jang-Mei Wu
published in: Geometry \& Topology
publication date: 2010-02-23T00:00:00Z
review text: Let \(Wh\) denote the Whitehead continuum in \(\mathbb{R}^3\) [\textit{J. H. C. Whitehead}, Q. J. Math., Oxf. Ser. 6, 268--279 (1935; Zbl 0013.08103 and JFM 61.0607.01)] and \(\mathbb{R}^3/Wh\) the associated decomposition space. Now \(\mathbb{R}^3/Wh\) is not a manifold and hence not homeomorphic to \(\mathbb{R}^3\). However, \(\mathbb{R}^3/Wh \times \mathbb{R}\) is homeomorphic to \(\mathbb{R}^4\). The late first author and \textit{S. Semmes} [Conform. Geom. Dyn. 1, No. 1, 1--12 (1997; Zbl 0885.00006)] asked if \(\mathbb{R}^3/Wh \times \mathbb{R}^m\), \(m \geq 1\), could admit a quasisymmetric parametrization. The metric in the resulting space is required to be such that the space is linearly locally contractible and Ahlfors \((3+m)\)-regular. In this paper the authors show that the answer to this question is no. The proof makes use of the fact that the meridians of the solid tori used in the construction of \(Wh\) are trivial in homology of the complement of \(Wh\) but nontrivial in homotopy. Also Hausdorff-measure estimates are used extensively.
Mathematics Subject Classification ID: 30C65
Mathematics Subject Classification ID: 57N16
Mathematics Subject Classification ID: 57N45
zbMATH DE Number: 5673943
zbMATH Keywords: Whitehead continuum
zbMATH Keywords: quasisymmetry
reviewed by: O. Martio
MaRDI profile type: Publication
full work available at URL: gt.2010.14.773
OpenAlex ID: W2067209129
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DOI: 10.2140/GT.2010.14.773
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description: scientific article; zbMATH DE number 5673943