JAVIER BRACHO AND LUIS MONTEJANO THE COMBINATORICS OF COLORED TRIANGULATIONS OF MANIFOLDS ABSTRACT.Foundations for the topic of crystallizations are proposed through the more general concept of colored triangulations. Classic results and techniques of crystallizations are reviewed from this point of view. A new set of combinatorial invariants of manifolds is defined, and related to the fundamental group and other known invariants. A universal group theoretic approach for this theory is introduced. A new combinatorial approach to the topology of PL-manifolds has been developing in recent years. It is based on the facts that a graph with colored edges provides precise instructions to construct a space, and that any manifold (i.e. PL-manifold) is obtained in this way. Thus, manifolds may be studied through graph theory. The idea of the construction, due to Pezzana ([18], [-19]) and further developed by Gagliardi, Ferri and their group (see the survey [-7], and the references), is to take for each vertex of the graph one copy of a standard geometric simplex (whose faces correspond to the colors), and then, each (colored) edge tells us to glue two simplexes along one of their faces (the color says which). Clearly, the space so constructed comes with a rich simplicial structure, which we call a colored complex. It is not a classic simplicial complex for two simplexes may meet in more than a single subsimplex (in this sense, it has the flexibility of a pseudocomplex [15] or of a semisimplicial complex [,17]). But, on the other hand, it is quite rigid, for every simplex is canonically isomorphic to a standard one (another resemblance to semisimplicial complexes). Thus, the study of manifolds through colored graphs, via colored complexes, is qualitatively different from the classic PL-combinatorial approach, and leads to results of a different nature. In Section 1 we start from a new general point of view, something like a 'Thinker's Toy' (as in [2]). A colored graph G (Subsection 1.1) is thought of as an 'instructions manual' to build spaces: if we supply a colored space X (a space with a fixed subspace of each color (Subsection 1.2) to act as a 'building block', we obtain a new space [G;X[ (Subsection 1.3) by glueing copies as G says. Of course, the main interest lies on [G[ (obtained, as above, when X is the standard simplex, (Example 1.4(i)), of which all manifolds are examples (Example 1.4(ii)). But the spaces [G; Xr have a lot to do with [G[. To talk about links, regular neighborhoods, canonical decompositions and other interesting subspaces of rG[, one only has to analyze the basic 'building blocks' and the obvious functoriality of the construction (Section 1-2) takes care of the rest. Geometriae Dedieata 22 (1987), 303-328. © 1987 by D. Reidel Publishing Company.