Fonctionne avec Indico
v3.2.9
The notion of (G,X)-structure on a manifold introduced by Ehresman and popularized by Thurston is an interpretation of "geometric structure" allowing via the developing map Theorem to efficiently express and prove uniformization results such as: "every compact locally Euclidean manifold is isomorphic to a quotient of the Euclidean space by a discrete group of isometry". Singularities, such as conical singularities, are common place in constructions of manifolds endowed with such structures via gluings, quotients, suspensions or compactifications. On the one hand, the notion of singular (G,X)-manifold is not so well-defined and is usually understood as "there is a (G,X)-structure on the complement of the (n-2)-facet of a simplicial decomposition". On the other hand, we may want to use developing maps and holonomies to express and prove uniformization results for singular (G,X)-manifolds. Furthermore, leaving the realm of metric structures, interesting singularities become more sophisticated (and interesting). In order to help manipulations of singular (G,X)-manifolds, we give a topological caracterization of reasonable singular locii and provide tools allowing to build developing maps in non-trivial situations. Doing so, we prove some new results on branched covering à la Fox. To illustrate these methods we apply them to a mess-like uniformization result of flat Lorentzian 3-manifolds endowed with a type of cuspidal ends.