6533b82efe1ef96bd12929ba

RESEARCH PRODUCT

Geometry and quasisymmetric parametrization of Semmes spaces

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description

We consider decomposition spaces R/G that are manifold factors and admit defining sequences consisting of cubes-with-handles. Metrics on R/G constructed via modular embeddings of R/G into Euclidean spaces promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space R/G×R by R for any m ≥ 0 imposes quantitative topological constraints, in terms of the circulation and the growth of the cubes-with-handles, to the defining sequences for R/G. We give a necessary condition and a sufficient condition for the existence of 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.