Luigi Grasselli

A graph-theoretical representation of PL-manifolds — A survey on crystallizations

This is a survey of the techniques and results developed by M. Pezzana and his group, which includes, besides the authors, A. Cavicchioli, P. Bandieri and A Donati.The original concept is that of “contracted triangulation”, which was introduced with the main goal of finding a “minimal” atlas for topological manifolds ([P1 1968], [P2 1974], [P3 1974], [FG2 1979]). Only later did the possibility of deducing a graph-theoretical tool — the crystallization — for representing P.L. manifolds occur as a major aspect of the theory ([P4 1975], [F1 1976]). This leads to an application of graph theory to P.L. topology, which seems not to have been explored before. Recently, other authors outside Italy have independently become interested in this subject.For the sake of conciseness, definitions and statements often appear in a form other than that of the quoted references.

2-symmetric crystallizations and 2-fold branched coverings of S 3

For each integer g > 1, a class Mg of ‘2-symmetric’ crystallizations, depending on a 2(g + 1)-tuple of positive integers satisfying simple conditions is introduced; the ‘2-symmetry’ implies that the represented closed, orientable 3-manifolds are 2-fold covering spaces of S3 branched over a link. Since every closed, orientable 3-manifold M of Heegaard genus g ⩽ 2 admits a crystallization belonging to Mg, we obtain an easy proof of the fact that M is a 2-fold covering space of S3 branched over a link. Further, the class contains all Lins–Mandel crystallizations S(b, l, t, c), with l odd, which are thus proved to represent 2-fold branched coverings of S3.

Topology in colored tensor models via crystallization theory

Abstract The aim of this paper is twofold. On the one hand, it provides a review of the links between random tensor models, seen as quantum gravity theories, and the PL-manifolds representation by means of edge-colored graphs ( crystallization theory ). On the other hand, the core of the paper is to establish results about the topological and geometrical properties of the Gurau-degree (or G-degree ) of the represented manifolds, in relation with the motivations coming from physics. In fact, the G-degree appears naturally in higher dimensional tensor models as the quantity driving their 1 ∕ N expansion, exactly as it happens for the genus of surfaces in the two-dimensional matrix model setting. In particular, the G-degree of PL-manifolds is proved to be finite-to-one in any dimension, while in dimension 3 and 4 a series of classification theorems are obtained for PL-manifolds represented by graphs with a fixed G-degree. All these properties have specific relevance in the tensor models framework, showing a direct fruitful interaction between tensor models and discrete geometry, via crystallization theory.

Representing products of polyhedra by products of edge-colored graphs

Given an (m + 1)-colored graph (Γ γ′) and an (n + 1)-colored graph (, γ″), representing two polyhedra P′, P″ respectively, we present a direct construction of an (m + n+1)-colored graph (Γ′⊠ Γ″, γ′ ⊠ γ″), which represents the product P′ × P″. Some examples, applications, and conjectures about the genus of manifold products are also presented.

Representing branched coverings by Edge-Coloured Graphs

Given a link L ⊂ S 3 , we describe a standard method for constructing a class Γ L,d of 4-coloured graphs representing all closed orientable 3-manifolds which are d -fold coverings of S 3 branched over the link L .

2-Symmetric Transformations for 3-Manifolds of Genus 2

As previously known, all 3-manifolds of genus 2 can be represented by edge-coloured graphs uniquely defined by 6-tuples of integers satisfying simple conditions. The present paper describes an elementary transformation on these 6-tuples which changes the associated graph but does not change the represented manifold. This operation is a useful tool in the classification problem for 3-manifolds of genus 2; in fact, it allows an equivalence relation to be defined on admissible 6-tuples so that equivalent 6-tuples represent the same manifold. Different equivalence classes can represent the same manifold; however, equivalence classes “almost always” contain infinitely many 6-tuples. Finally, minimal representatives of the equivalence classes are described.

Crystallizations of generalized Neuwirth manifolds

Given a group presentation φ, let M(

Tetrahedron manifolds via coloured graphs

) denote the class of all 3-manifolds which admit a Heegaard diagram inducing φ. We describe an algorithm which produces crystallizations of all 3-manifolds of M(

Simple contracted complexes are nonshellable

). As an application of this technique, we obtain Crystallizations of a wide class of 3-manifolds which generalizes, in a natural way, Neuwirth manifolds introduced in [N]. 1991 Mathematics Subject Classification: 57N10, 57Q15, 57M05.

A geometric description of ‘normal’ crystallizations

Abstract In his paper ‘Tetrahedron manifolds and space forms’, Molnar describes an infinite class of 3-manifolds (depending on two natural integers m , n ) by means of suitable face identifications on a tetrahedron. These manifolds can be represented by edge-coloured graphs. By making use of these combinatorial techniques, it is easy to show that they are 2-fold coverings of the 3-sphere, branched over suitable links. This immediately leads to the classification of these manifolds in terms of Seifert fibered spaces.