Geometry and quasisymmetric parametrization of Semmes spaces

TL;DR

This paper explores the geometric and topological conditions under which certain 3-dimensional decomposition spaces can be parametrized quasisymmetrically, providing new insights and examples in geometric topology.

Contribution

It establishes necessary and sufficient conditions for quasisymmetric parametrization of Semmes spaces, linking topology with controlled geometric structures.

Findings

Provides a necessary condition related to circulation and growth for quasisymmetric parametrization.

Offers a sufficient condition that yields new examples of quasispheres in four-dimensional spheres.

Answers a question of Heinonen and Semmes regarding Bing double spaces.

Abstract

We consider decomposition spaces $\R^3/G$ that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on $\R^3/G$ constructed via modular embeddings into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space $\R^3/G\times \R^m$ imposes quantitative topological constraints, in terms of the circulation and growth, to the defining sequences for $\R^3/G$. We give a necessary condition and a sufficient condition for the existence of parametrization. The necessary condition answers 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 $\bS^4$.