#### Volume 14, issue 2 (2010)

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Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

### Juha Heinonen and Jang-Mei Wu

Geometry & Topology 14 (2010) 773–798
##### Abstract

The decomposition space ${R}^{3}∕Wh$ associated with the Whitehead continuum $Wh$ is not a manifold, but the product $\left({R}^{3}∕Wh\right)×{R}^{m}$ is homeomorphic to ${R}^{3+m}$ for any $m\ge 1$ (known since the 1960’s). We study the quasisymmetric structure on $\left({R}^{3}∕Wh\right)×{R}^{m}$ and show that the space $\left({R}^{3}∕Wh\right)×{R}^{m}$ may be equipped with a metric resembling ${R}^{3+m}$ geometrically and measure theoretically—it is linearly locally contractible and Ahlfors $\left(3+m\right)$–regular—nevertheless the resulting space does not admit a quasisymmetric parametrization by ${R}^{3+m}$.

 In memory of Juha Heinonen, an extraordinary mathematician and friend, who passed away after the mathematical work in this paper was completed. – J-M W
##### Keywords
Whitehead continuum, decomposition space, quasisymmetric parametrization, quasisymmetric map
##### Mathematical Subject Classification 2000
Primary: 30C65
Secondary: 57N16, 57N45