Local monodromy of branched covers and dimension of the branch set

We show that, if the local dimension of the image of the branch set of a discrete and open mapping f: M -> N between n-manifolds is less than (n - 2) at a point y of the image of the branch set fB(f), then the local monodromy of f at y is perfect. In particular, for generalized branched covers between n-manifolds the dimension of fB(f) is exactly (n-2) at the points of abelian local monodromy. As an application, we show that a generalized branched covering f : M -> N of local multiplicity at most three between n-manifolds is either a covering or fB(f) has local dimension (n - 2).