On branched coverings of singular $(G,X)$-manifolds

Abstract

Branched coverings have a long history from ramification of Riemann surfaces to realization of 3 manifolds as coverings ramified over a knots; from geometrical topology to algebraic geometry. The present work investigates a notion of branched covering "à la Fox" which is particularly natural for (G, X)-manifolds. The work is two fold. First, we recall and enrich the current state of the art (based upon Montesinos) and develop a Galois theory for such branched coverings together with a description of the fiber above branching points. Second, we present a theory of singular (G, X)-manifolds and apply the theory of branched coverings we develop to extend the usual framework of (G, X)-manifolds to singular (G, X)-manifolds; in particular, we construct a developing map for such singular manifolds. An application to a singular locally Minkowski manifolds is given.