Classifying PL 4-manifolds via crystallizations: results and open problems

Crystallization theory is a graph-theoretical representation method for compact PL-manifolds of arbitrary dimension, which makes use of a particular class of edge-coloured graphs, which are dual to coloured (pseudo-)triangulations. The purely combinatorial nature of crystallizations makes them particularly suitable for automatic generation and classication, as well as for the introduction and study of graph-defined invariants for PL-manifolds. The present survey paper focuses on the 4-dimensional case, presenting up-to-date results about the PL classication of closed 4-manifolds, by means of two such PL invariants: regular genus and gem-complexity. Open problems are also presented, mainly concerning different classication of 4-manifolds in TOP and DIFF=PL categories, and a possible approach to the 4-dimensional Smooth Poincare Conjecture