Abstract We introduce a novel and constructive definition of gluing data, and give the first rigorous proof that a universal manifold satisfying the Hausdorff condition can always be constructed from any set of gluing data. We also present a class of spaces called parametric pseudo-manifolds, which under certain conditions, are manifolds embedded in Rn and defined from sets of gluing data. We give a construction for building a set of gluing data from any simplicial surface in R3. This construction is an improvement of the construction given in Siqueira et al. (2009) , where the results were stated without proof. We also give a complete proof of the correctness of this construction making use of the crucial “property A.” The above results enable us to develop a methodology that explicitly yields manifolds in Rn arising in several graphics and engineering applications.