Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum
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.
scientific article; zbMATH DE number 5673943
scientific article