Maria Rita Casali

A CATALOGUE OF ORIENTABLE 3-MANIFOLDS TRIANGULATED BY 30 COLORED TETRAHEDRA

The present paper follows the computational approach to 3-manifold classification via edge-colored graphs, already performed in [1] (with respect to orientable 3-manifolds up to 28 colored tetrahedra), in [2] (with respect to non-orientable 3-manifolds up to 26 colored tetrahedra), in [3] and [4] (with respect to genus two 3-manifolds up to 34 colored tetrahedra): in fact, by automatic generation and analysis of suitable edge-colored graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting colored triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decompositions and fibering structures. It is worth noting that, in the present work, a suitable use of elementary combinatorial moves yields an automatic partition of the elements of the generated crystallization catalogue into equivalence classes, which turn out to be in one-to-one correspondence with the homeomorphism classes of the represented manifolds.

Classification of Nonorientable 3-Manifolds Admitting Decompositions into ≤ 26 Coloured Tetrahedra

The present paper adopts a computational approach to the study of nonorientable 3-manifolds: in fact, we describe how to create an automaticcatalogue of all nonorientable 3-manifolds admitting coloured triangulationswith a fixed number of tetrahedra. In particular, the catalogue has been effectively produced and analysed for up to 26 tetrahedra, to reach the complete classification of all involved 3-manifolds. As a consequence, the following summarising result can be stated:THEOREM I. Exactly seven closed connected prime nonorientable3-manifolds exist, which admit a coloured triangulation consisting of atmost 26 tetrahedra.More precisely, they are the four Euclidean nonorientable 3-manifolds, the nontrivial S2 bundle overS1, the topological product between thereal projective plane RP2 andS1, and the torus bundle overS1, with monodromy induced by matrix(10 -11).

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.

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

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.

CLASSIFYING PL 5-MANIFOLDS BY REGULAR GENUS: THE BOUNDARY CASE

In the present paper, we face the problem of classifying clas ses of ori- entable PL 5-manifolds M 5 with h 1 boundary components, by making use of a combinatorial invariant called regular genusG(M5). In particular, a complete classifi- cation up to regular genus five is obtained: G(M 5 ) 5 M 5 # ∂ ( 4 1 )# (h) ∂ ,

A NOTE ABOUT COMPLEXITY OF LENS SPACES

AbstractWithin crystallization theory, (Matveev’s) complexity of a 3-manifold can be es-timated by means of the combinatorial notion of GM-complexity . In this paper, weprove that the GM-complexity of any lens space L(p,q), with p ≥ 3, is bounded byS(p,q)−3, where S(p,q) denotes the sum of all partial quotients in the expansion of qp as a regular continued fraction. The above upper bound had been already establishedwith regard to complexity; its sharpness was conjectured by Matveev himself and hasbeen recently proved for some infinite families of lens spaces by Jaco, Rubinstein andTillmann. As a consequence, infinite classes of 3-manifolds turn out to exist, wherecomplexity and GM-complexity coincide.Moreover, we present and briefly analyze results arising from crystallization cat-alogues up to order 32, which prompt us to conjecture, for any lens space L(p,q)with p ≥ 3, the following relation: k(L(p,q)) = 5 + 2c(L(p,q)), where c(M) denotesthe complexity of a 3-manifold M and k(M) + 1 is half the minimum order of acrystallization of M.Keywords: 3-manifold, complexity, crystallization, 2-bridge knot, 4-plat, lens spaceMathematics Subject Classification (2010): 57Q15, 57M27, 57M25, 57M15.

A note about bistellar operations on PL-manifolds with boundary

In 1990, U. Pachner proved that simplicial triangulations of the same PL-manifold (with boundary) are always connected by a finite sequence of transformations belonging to two different groups:shelling operations (and their inverses), which work mostly with the boundary triangulations, andbistellar operations, which affect only the interior of the triangulations.The purpose of this note is to prove that, in case of simplicial triangulations coinciding on the boundary, bistellar operations are sufficient to solve the homeomorphism problem.

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

Simple crystallizations are edge-colored graphs representing piecewise linear (PL) 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In this paper, we prove that any (simply-connected) PL 4-manifold M admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, M may be represented by a framed link yielding 𝕊3, with exactly β2(M) components (β2(M) being the second Betti number of M). As a consequence, the regular genus of M is proved to be the double of β2(M). Moreover, the characterization of any such PL 4-manifold by k(M) = 3β2(M), where k(M) is the gem-complexity of M (i.e. the non-negative number p − 1, 2p being the minimum order of a crystallization of M), implies that both PL invariants gem-complexity and regular genus turn out to be additive within the class of all PL 4-manifolds admitting simple crystallizations (in particular, within the class of all “standard” simply-connected PL 4-manifolds).

Estimating Matveev's complexity via crystallization theory

In [M.R. Casali, Computing Matveevs complexity of non-orientable 3-manifolds via crystallization theory, Topology Appl. 144(1-3) (2004) 201-209], a graph-theoretical approach to Matveevs complexity computation is introduced, yielding the complete classification of closed non-orientable 3-manifolds up to complexity six. The present paper follows the same point-of view, making use of crystallization theory and related results (see [M. Ferri, Crystallisations of 2-fold branched coverings of S^3, Proc. Amer. Math. Soc. 73 (1979) 271-276; M.R. Casali, Coloured knots and coloured graphs representing 3-fold simple coverings of S^3, Discrete Math. 137 (1995) 87-98; M.R. Casali, From framed links to crystallizations of bounded 4-manifolds, J. Knot Theory Ramifications 9(4) (2000) 443-458]) in order to significantly improve existing estimations for complexity of both 2-fold and three-fold simple branched coverings (see [O.M. Davydov, The complexity of 2-fold branched coverings of a 3-sphere, Acta Appl. Math. 75 (2003) 51-54] and [O.M. Davydov, Estimating complexity of 3-manifolds as of branched coverings, talk-abstract, Second Russian-German Geometry Meeting dedicated to 90-anniversary of A.D.Alexandrov, Saint-Petersburg, Russia, June 2002]) and 3-manifolds seen as Dehn surgery (see [G. Amendola, An algorithm producing a standard spine of a 3-manifold presented by surgery along a link, Rend. Circ. Mat. Palermo 51 (2002) 179-198]).