By a branched cover we refer to a continuous, open and discrete mapping, and the set of points where it fails to be locally injective its branch set. By the classical Stoilow Theorem, a branched cover between planar domains is locally equivalent to a complex power map and the equivalence is even quasiconformal when the original mapping is quasiregular. In higher dimensions the claim is not true, except for some special cases. Indeed, by the classical theorems of Church-Hemmingsen amd Martio-Rickman-Väisälä, a branched cover between euclidean n-domains is locally equivalent to a winding map when the image of the branch set is an n-2 -dimensional hyperplane. In 1979 Martio and Srebro showed that in dimension three a branched cover is topologically equivalent to a piecewise linear map when the image of the branch set is topologically a simplicial complex. In this talk we discuss a recent result, joint with Eden Prywes, that generalizes this Martio-Srebro result to all dimensions by utilizing some modern tools in topology; namely the Perelman Theorem / Poincare Conjecture.
427 Thackeray Hall