We consider decomposition spaces R3/G that are manifold factors and admit defining sequences consisting of cubes-with-handles of finite type. Metrics on R3/G constructed via modular embeddings of R3/G into a Euclidean space promote the controlled topology to a controlled geometry. The quasisymmetric parametrizability of the metric space R3/G×Rm by R3+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 R3/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.
- Decomposition space
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