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 R3/G\mathbb{R}^3/G that are manifold factors and admit defining sequences consisting of cubes-with-handles of finite type. Metrics on R3/G\mathbb{R}^3/G constructed via modular embeddings of R3/G\mathbb{R}^3/G into a Euclidean space promote the controlled topology to a controlled geometry.

The quasisymmetric parametrizability of the metric space R3/G×Rm\mathbb{R}^3/G\times \mathbb{R}^m by R3+m\mathbb{R}^{3+m} for any m0m\ge 0 imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, on the defining sequences for R3/G\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 S4\mathbb{S}^4.

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