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

TL;DR

This paper develops a Galois theory for branched coverings of (G,X)-manifolds, extends the framework to singular (G,X)-manifolds, and provides new solutions to open questions in the field.

Contribution

It introduces a natural notion of branched covering for (G,X)-manifolds, enriches the existing theory, and constructs developping maps for singular (G,X)-manifolds.

Findings

Solved two open questions of Montesinos.

Constructed an example related to another open question.

Extended (G,X)-manifold theory to include singular cases.

Abstract

Branched covering have a long history from ramification of Riemann surfaces to realization of 3-manifolds as covering ramified over a knots; from geometrical topology to algebraic geometry. The present work investigates a notion of branched covering "\`a 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 covering together with a description of the fiber above branching points. As a consequence, we solve two open questions of Montesinos and construct an example related to another open question. Second, we present a theory of singular (G,X)-manifolds and apply the theory of branched covering we developped to extend the usual framework of (G,X)-manifolds to singular (G,X)-manifolds; in particular, we construct a developping map for such singular manifolds. An application to singular locally Minkowski manifolds is given.