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.