Geometry and quasisymmetric parametrization of Semmes spaces

  • Pekka Pankka

    University of Jyväskylä, Finland
  • Jang-Mei Wu

    University of Illinois at Urbana-Champaign, USA


We consider decomposition spaces that are manifold factors and admit defining sequences consisting of cubes-with-handles of finite type. Metrics on constructed via modular embeddings of into a Euclidean space promote the controlled topology to a controlled geometry.

The quasisymmetric parametrizability of the metric space by for any imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, on the defining sequences for . 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 .

Cite this article

Pekka Pankka, Jang-Mei Wu, Geometry and quasisymmetric parametrization of Semmes spaces. Rev. Mat. Iberoam. 30 (2014), no. 3, pp. 893–960

DOI 10.4171/RMI/802