Geometry and quasisymmetric parametrization of Semmes spaces
Pankka, P., & Wu, J.-M. (2014). Geometry and quasisymmetric parametrization of Semmes spaces. Revista matematica Iberoamericana, 30 (3), 893-960. doi:10.4171/rmi/802
Published inRevista matematica Iberoamericana
© 2014 European Mathematical Society. This is a final draft version of an article whose final and definitive form has been published by EMS. Published in this repository with the kind permission of the publisher.
We consider decomposition spaces R 3 /G that are manifold factors and admit defining sequences consisting of cubes-with-handles of finite type. Metrics on R 3 /G constructed via modular embeddings of R 3 /G into a Euclidean space promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space R 3 /G×R m by R 3+m for any m ≥ 0 imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, on the defining sequences for 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 S 4 .