Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum

The decomposition space R³/Wh associated with the Whitehead continuum Wh is not a manifold, but the product (R³/Wh) × R^m is homeomorphic to R^3+m for any m≥1 (known since the 1960's). We study the quasisymmetric structure on.(R³/Wh) × R^m and show that the space (R³/Wh) × R^m may be equipped with a metric resembling R^3+m geometrically and measure theoretically-it is linearly locally contractible and Ahlfors (3+m)-regular-nevertheless the resulting space does not admit a quasisymmetric parametrization by R^3+m.