Geometry and quasisymmetric parametrization of Semmes spaces

Abstract

We consider decomposition spaces ${\mathbb{R}}^3/G$ that are manifold factors and admit defining sequences consisting of cubes-with-handles of finite type. Metrics on ${\mathbb{R}}^3/G$ constructed via modular embeddings of ${\mathbb{R}}^3/G$ into a Euclidean space promote the controlled topology to a controlled geometry.

The quasisymmetric parametrizability of the metric space ${\mathbb{R}}^3/G\times {\mathbb{R}}^m$ by ${\mathbb{R}}^{3+m}$ for any $m\ge 0$ imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, on the defining sequences for ${\mathbb{R}}^3/G$. We give a necessary condition and a sufficient condition for the existence of such a parametrization.

The necessary condition answers negatively a question of Heinonen and Semmes on quasisymmetric parametrizability of spaces associated to the Bing double. The sufficient condition gives new examples of quasispheres in ${\mathbb{S}}^4$.