We give a sufficient condition for a metric (homology) manifold to be locally
bi-Lipschitz equivalent to an open subset in $\rn$. The condition is a Sobolev
condition for a measurable coframe of flat 1-forms. In combination with an
earlier work of D. Sullivan, our methods also yield an analytic
characterization for smoothability of a Lipschitz manifold in terms of a
Sobolev regularity for frames in a cotangent structure. In the proofs, we
exploit the duality between flat chains and flat forms, and recently
established differential analysis on metric measure spaces. When specialized to
$\rn$, our result gives a kind of asymptotic and Lipschitz version of the
measurable Riemann mapping theorem as suggested by Sullivan.