Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

Abstract

The decomposition space $R^3∕Wh$ associated with the Whitehead continuum $Wh$ is not a manifold, but the product $\left(R^3∕Wh\right)\times R^m$ is homeomorphic to $R^{3+m}$ for any $m\ge 1$ (known since the 1960’s). We study the quasisymmetric structure on $\left(R^3∕Wh\right)\times R^m$ and show that the space $\left(R^3∕Wh\right)\times R^m$ 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 $R^{3+m}$.