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Abstract. In this paper we consider completed coverings that are branched coverings in the sense of Fox. For completed coverings between PL manifolds we give a characterization of the existence of a monodromy representation and the existence of a locally compact monodromy representation. These results stem from a characterization for the discreteness of a completed normal covering. We also show that completed coverings admitting a monodromy representations are discrete and that the image of the branch set is closed.

Abstract. 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 f B f , then the local monodromy of f at y is perfect. In particular, for generalized branched covers between n-manifolds the dimension of f B 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 f B f has local dimension (n − 2).

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We show that every canonical Seifert surface is (up to isotopy) given by a knot diagram in which the (open) Seifert disks are pairwise disjoint.