Carlo Gagliardi

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.

Regular imbeddings of edge-coloured graphs

RiassuntoIn questo lavoro si introduce un particolare tipo di immersioni cellulari, dette regolari, per grafi colorati sugli spigoli. Facendo uso di tecniche sia geometriche che combinatorie si espongono alcuni teoremi generali di immersione per tali grafi, strettamente collegati si poliedri da essi rappresentati.AbstractA particular kind of 2-cell imbeddings, called regular, for edge-coloured graphs is introduced. By using both geometric and combinatorial techniques, some general imbedding theorems for such graphs, strictly related to the polyhedra they represent, are presented.

NON-ORIENTABLE 3-MANIFOLDS ADMITTING COLORED TRIANGULATIONS WITH AT MOST 30 TETRAHEDRA

We present the census of all non-orientable, closed, connected 3-manifolds admitting a rigid crystallization with at most 30 vertices. In order to obtain the above result, we generate, manipulate and compare, by suitable computer procedures, all rigid non-bipartite crystallizations up to 30 vertices.

Classifying PL 5-manifolds up to regular genus seven

In the present paper, we show that the only closed orientable 5-manifolds of regular genus less or equal than seven are the 5-sphere S 5 and the connected sums of m copies of S 1 × S 4 , with m ≤ 7. As a consequence, the genus of S 3 × S 2 is proved to be eight. This suggests a possible approach to the (3-dimensional) Poincare Conjecture, via the well-known classification of simply connected 5-manifolds, obtained by Smale and Barden

Regular genus: The boundary case

SuntoNel lavoro si introduce e si studia un nuovo tipo di immersione cellulare in superficie con bordo, per una particolare classe di grafi colorati sugli spigoli.Ciò permette, in analogia con quanto fatto nel caso chiuso in un precedente lavoro [13], di definite per ogni n-varietà con bordo una coppia di invarianti — il genere regolare ed il numero di buchi — che estendono gli omonimi concetti noti per le superficie con bordo. Di tali invarianti si studiano poi relazioni ed interpretazione geometrica, con particolare riguardo alla dimensione tre, dove essi sono collegati ad uno speciale spezzamento tipo-Heegaard.Una caratterizzazione di

On the genus of the complex projective plane

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Representing products of polyhedra by products of edge-colored graphs

n conclude il lavoro.SummaryA particular kind of 2-cell imbedding for a class of edge-coloured graphs into surfaces with boundary is introduced and studied. This allows to define, as in [13], where the closed case was traited, a pair of invariants — the regular genus and the hole-number — for every n-manifold with boundary. These invariants are proved to coincide with the classical ones in dimension two, and to be strictly related with a Heegaard-like handlebody decomposition in dimension three. A characterization of 261-1 concludes the work.

PL 4-manifolds admitting simple crystallizations: framed links and regular genus

RiassuntoIn questo lavoro si introduce e si studia una classe di poliedri 3-dimensionali, che generalizza quella delle varietàPL con bordo.SummaryIn this paper we introduce and study a class of 3-dimensional polyhedra generalizing that ofPL manifolds with boundary.