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 $\mathbb{S}^4$.