Counting triangulated d-manifolds, asymptotically

Abstract. In dimension d >= 3, take n simplices, and glue their facets in an arbitrary way. You obtain a topological space that is a pseudo-manifold, but not always a manifold. In how many ways, asymptotically, can you do it in order to obtain a manifold? We give partial answers to this question, in particular we determine the superexponential growth in dimension 3, in the special case of colored manifolds. This is work in progress with Guillem Perarnau.