We develop a theory of geometrically controlled branched covering maps between metric spaces that are generalized cohomology manifolds. Our notion extends that of maps of bounded length distortion, or BLD-maps, from Euclidean spaces. We give a construction that generalizes an extension theorem for branched covers by I. Berstein and A. Edmonds. We apply the theory and the construction to show that certain reasonable metric spaces that were shown by S. Semmes not to admit bi-Lipschitz parametrizations by a Euclidean space nevertheless admit BLD-maps into Euclidean space of same dimension.
"Geometric branched covers between generalized manifolds." Duke Math. J. 113 (3) 465 - 529, 15 June 2002. https://doi.org/10.1215/S0012-7094-02-11333-7