The Combinatorics of Piecewise Linear Manifolds by Colored Graphs

Crystallization theory is a combinatorial representation of piecewiselinear (closed connected) manifolds of arbitrary dimension. This theory differs from the most important representation methods for triangulated manifolds as for example Heegaard splittings, standard spines, surgery along framed links and branched coverings, which work well in dimension less than or equal four. Crystallizations form a particular class of edge-colored multigraphs arising from combinatorial triangulations of manifolds which are minimal with respect to the number of vertices. Classical results and techniques on crystallizations are reviewed from a graph-theoretical point of view, especially to pay attention to certain new combinatorial invariants as regular genus, complexity and average order. These invariants are shown to be related with the topology of manifolds. Several open problems and conjectures concerning them complete the survey paper.Mathematics Subject Classification: 57M15, 57Q15, 05C10