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.

Regular Genus and Products of Spheres

“…10b, 10c, 10d with 14b, 14c, 14d) by 5-colored edges. For the remaining vertices of G (4,2) we consider the edges as they are in the graphs G (3,2) and G (4,1) , representing S 3 × S 2 and S 4 × S 1 , respectively, according to step (5). Now we simplify the graph G (4,2) by cancelling the dipole of type 1 formed by the vertices 15a and 14a joined by one edge labelled by color 4, as these vertices belong to different connected components of G b 4 .…”

“…15b, 15c, 15d with 20b, 20c, 20d) by 6-colored edges. For the remaining vertices of G (5,2) we consider the edges as they are in the graphs G (4,2) and G (5,1) , representing S 4 × S 2 and S 5 × S 1 , respectively, according to step (5). Since the graph G (5,2) has a double symmetry we can draw only a part of it (see Figure 3).…”

“…Colored graphs represent a useful combinatorial approach to the topology of piecewise linear (PL) manifolds. For more details see for example the survey papers [1], [5], [7], [10], and [18]. An n-pseudocomplex K(G) can be associated to (G, c) by the following rules.…”

Dotted Links, Heegaard Diagrams, and Colored Graphs for PL 4-manifolds

“…Since (G, g) does not contain ρ 3 -pairs, Proposition 4.2 can not be applied. Notwithstanding this, it is easy to check that a finite sequence of dipole eliminations (more precisely, the subsequent eliminations of 1-dipole {v 1 , v 2 } and 2-dipoles {v 3 , v 4 }, {v 5 , v 6 }, {v 7 , v 8 }, {v 9 , v 10 }, according to the captions of Fig. 13(b)) transforms (G, g) into a 4-colored graph containing a ρ 3 -pair of color 2 (which corresponds to the pair of edges {e, f } of (G, g), according to the captions of by MIUR of Italy (project "Strutture geometriche delle varietà reali e complesse") and by the Università degli Studi di Modena e Reggio Emilia (project "Strutture finite e modelli discreti di strutture geometriche continue").…”

“…The aim of the present paper is to establish a connection between the 4-manifold representation method by dotted framed links (or equivalently-in the closed caseby Heegaard diagrams) and the so called crystallization theory, which visualizes general PL-manifolds by means of edge-colored graphs (see [11], [1], [5], [10], [14], [16], [22],. .…”