  We consider decomposition spaces $\R^3/G$ that are manifold factors and admit
defining sequences consisting of cubes-with-handles. Metrics on $\R^3/G$
constructed via modular embeddings into Euclidean spaces promote the controlled
topology to a controlled geometry. The quasisymmetric parametrizability of the
metric space $\R^3/G\times \R^m$ imposes quantitative topological constraints,
in terms of the circulation and growth, to the defining sequences for $\R^3/G$.
We give a necessary condition and a sufficient condition for the existence of
parametrization. The necessary condition answers 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 $\bS^4$.
