Revista Matemática Iberoamericana

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Volume 30, Issue 3, 2014, pp. 893–960
DOI: 10.4171/RMI/802

Published online: 2014-08-27

Geometry and quasisymmetric parametrization of Semmes spaces

Pekka Pankka[1] and Jang-Mei Wu[2]

(1) University of Jyväskylä, Finland
(2) University of Illinois at Urbana-Champaign, USA

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 $\mathbb{S}^4$.

Keywords: Quasisymmetry, parametrization, quasisphere, decomposition space

Pankka Pekka, Wu Jang-Mei: Geometry and quasisymmetric parametrization of Semmes spaces. Rev. Mat. Iberoamericana 30 (2014), 893-960. doi: 10.4171/RMI/802