Contracted triangulations as branched coverings

We prove that every closed connected orientable n-dimensional pseudomanifold, which is n-colored on its vertex set, is a covering of the n-sphere branched over the (n-2)-skeleton of an n-simplex. This extends a theorem of Alexander (Bull. Amer. Math. Soc. 26 (1920)) and a theorem of Ramirez (An. Inst. Mat. Univ. Nac. Autonoma Mexico 15 (1975)). Using the concept of contracted triangulation, every closed connected orientable 3-manifold M is represented as a covering of the 3-sphere branched over an universal graph G, so that the cardinality of the fiber of each point of G depends only on the number of the 3-simplexes of a contracted triangulation of M.