Catalogues of PL-manifolds and complexity estimations via crystallization theory

Crystallization theory is a graph-theoretical representation method for compact PL-manifolds of arbitrary dimension, with or without boundary, which makes use of a particular class of edge-coloured graphs, which are dual to coloured (pseudo-) triangulations. These graphs are usually called gems, i.e. Graphs Encoding Manifolds, or crystallizations if the associated triangulation has the minimal number of vertices.One of the principal features of crystallization theory relies on the purely combinatorial nature of the representing objects, which makes them particularly suitable for computer manipulation.The present talk focuses on up-to-date results about:- generation of catalogues of PL-manifolds for increasing values of the vertex number of the representing graphs;- definition and/or computation of invariants for PL-manifolds, directly from the representing graphs.