  We extend the proof of Reifenberg's Topological Disk Theorem to allow the
case of sets with holes, and give sufficient conditions on a set $E$ for the
existence of a bi-Lipschitz parameterization of $E$ by a $d$-dimensional plane
or smooth manifold. Such a condition is expressed in terms of square
summability for the P. Jones numbers $\beta_1(x,r)$. In particular, it applies
in the locally Ahlfors-regular case to provide very big pieces of bi-Lipschitz
images of $\R^d$.
