
Maria Rita CASALI
Professore Ordinario Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede exMatematica

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2024
 Geminduced trisections of compact PL 4manifolds
[Articolo su rivista]
Casali, M. R.; Cristofori, P.
abstract
The idea of studying trisections of closed smooth 4manifolds via (singular) triangulations, endowed with a suitable vertexlabelling by three colors, is due to Bell, Hass, Rubinstein and Tillmann, and has been applied by Spreer and Tillmann to standard simplyconnected 4manifolds, via the socalled simple crystallizations. In the present paper we propose generalizations of these ideas by taking into consideration a possible extension of trisections to compact PL 4manifolds with connected boundary, which is related to Birman's special Heegaard sewing, and by analyzing geminduced trisections, i.e. trisections that can be induced not only by simple crystallizations, but also by any 5colored graph encoding a PL 4manifold with empty or connected boundary. This last notion gives rise to that of Gtrisection genus, as an analogue, in this context, of the wellknown trisection genus. We give conditions on a 5colored graph ensuring one of its geminduced trisections  if any  to realize the Gtrisection genus, and prove how to determine it directly from the graph. As a consequence, we detect a class of closed simplyconnected 4manifolds, comprehending all standard ones, for which both Gtrisection genus and trisection genus coincide with the second Betti number and also with half the value of the graphdefined PL invariant regular genus. Moreover, the existence of geminduced trisections and an estimation of the Gtrisection genus via surgery description is obtained, for each compact PL 4manifold admitting a handle decomposition lacking in 3handles.
2023
 Classifying compact 4manifolds via generalized regular genus and Gdegree
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
$(d+1)$colored graphs, i.e. edgecolored graphs that are $(d+1)$regular, have already been proved to be a useful representation tool for compact PL $d$manifolds, thus extending the theory (known as crystallization theory) originally developed for the closed case. In this context, combinatorially defined PL invariants play a relevant role.
The present paper focuses in particular on generalized regular genus and Gdegree: the first one extending to higher dimension the classical notion of Heegaard genus for 3manifolds, the second one arising, within theoretical physics, from the theory of random tensors as an approach to quantum gravity in dimension greater than two.
We establish several general results concerning the two invariants, in relation with invariants of the boundary and with the rank of the fundamental group, as well as their behaviour with respect to connected sums.
We also compute both generalized regular genus and Gdegree for interesting classes of compact $d$manifolds, such as handlebodies, products of closed manifolds by the interval and $mathbb D^2$bundles over $mathbb S^2.$
The main results of the paper concern dimension 4, where it is obtained the classification of all compact PL manifolds with generalized regular genus at most one, and of all compact PL manifolds with Gdegree at most 18; moreover, in case of empty or connected boundary, the classifications are extended to generalized regular genus two and to Gdegree 24.
2023
 Kirby diagrams and 5colored graphs representing compact 4manifolds
[Articolo su rivista]
Casali, M. R.; Cristofori, P.
abstract
It is wellknown that in dimension 4 any framed link (L, c) uniquely represents the PL 4manifold M^4 (L, c) obtained from D^4 by adding 2handles along (L, c). Moreover, if trivial dotted components are also allowed (i.e. in case of a Kirby diagram (L^(*), d)), the associated PL 4manifold M^4(L^(*), d) is obtained from D^4 by adding 1handles along the dotted components and 2handles along the framed components. In this paper we study the relationships between framed links and/or Kirby diagrams and the representation theory of compact PL manifolds by edgecolored graphs: in particular, we describe how to construct algorithmically a (regular) 5colored graph representing M^4(L^(*), d), directly "drawn over" a planar diagram of (L^(*), d), or equivalently how to algorithmically obtain a triangulation of M^4(L^(*),d). As a consequence, the procedure yields triangulations for any closed (simplyconnected) PL 4manifold admitting handle decompositions without 3handles. Furthermore, upper bounds for both the invariants gemcomplexity and regular genus of M^4(L^(*), d) are obtained, in terms of the combinatorial properties of the Kirby diagram.
2023
 TOPOLOGY IN COLORED TENSOR MODELS
[Poster]
Casali, Maria Rita; Cristofori, Paola; Grasselli, Luigi
abstract
From a “geometric topology” point of view, the theory of manifold representation by means of edgecolored graphs has been deeply studied since 1975 and many results have been achieved: its great advantage is the possibility of encoding, in any dimension, every PL dmanifold by means of a totally combinatorial tool.
Edgecolored graphs also play an important rôle within colored tensor models theory, considered as a possible approach to the study of Quantum Gravity: the key tool is the Gdegree of the involved graphs, which drives the 1/N expansion in the higher dimensional tensor models context, exactly as it happens for the genus of surfaces in the twodimensional matrix model setting.
Therefore, topological and geometrical properties of the represented PL manifolds, with respect to the Gdegree, have speciﬁc relevance in the tensor models framework, showing a direct fruitful interaction between tensor models and discrete geometry, via edgecolored graphs.
In colored tensor models, manifolds and pseudomanifolds are (almost) on the same footing, since they constitute the class of polyhedra represented by edgecolored Feynman graphs arising in this context; thus, a promising research trend is to look for classiﬁcation results concerning all pseudomanifolds represented by graphs of a given Gdegree. In dimension 4, the goal has already been achieved  via singular 4manifolds  for all compact PL 4manifolds with connected boundary up to Gdegree 24.
In the same dimension, the existence of colored graphs encoding different PL manifolds with the same underlying TOP manifold, suggests also to investigate the ability of tensor models to accurately reﬂect geometric degrees of freedom of Quantum Gravity.
2022
 GEOMETRIA
[Monografia/Trattato scientifico]
Casali, Maria Rita; Gagliardi, Carlo; Grasselli, Luigi
abstract
l presente testo sviluppa argomenti tradizionalmente trattati negli insegnamenti di ``Geometria''
(ovvero di ``Algebra lineare e Geometria'') nell'ambito delle lauree di primo livello, ed è particolarmente rivolto agli studenti dei vari Corsi di Laurea in Ingegneria, e di quelli in Matematica, Fisica e Informatica.
2021
 Compact 4manifolds admitting special handle decompositions
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
In this paper we study colored triangulations of compact PL $4$manifolds with empty or connected boundary which induce handle decompositions lacking in 1handles or in 1 and 3handles, thus facing also the problem, posed by Kirby, of the existence of {em special handlebody decompositions} for any simplyconnected closed PL $4$manifold. In particular, we detect a class of compact simplyconnected PL $4$manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links.
Moreover, this class includes any compact simplyconnected PL $4$manifold with empty or connected boundary having colored triangulations that minimize the combinatorially defined PL invariants {em regular genus, gemcomplexity} or {em Gdegree} among all such manifolds with the same second Betti number.
2020
 Crystallizations of compact 4manifolds minimizing combinatorially defined PLinvariants
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo
abstract
The present paper is devoted to present a unifying survey about some special classes of crystallizations of compact PL $4$manifolds with empty or connected boundary, called semisimple and weak semisimple crystallizations, with a particular attention to their properties of minimizing combinatorially defined PLinvariants, such as the regular genus, the Gurau degree, the gemcomplexity and the (geminduced) trisection genus.
The main theorem, yielding a summarizing result on the topic, is an original contribution.
Moreover, in the present paper the additivity of regular genus with respect to connected sum is proved to hold for all compact $4$manifolds with empty or connected boundary which admit weak semisimple crystallizations.
2019
 Combinatorial properties of the Gdegree
[Articolo su rivista]
Casali, M. R.; Grasselli, L.
abstract
A strong interaction is known to exist between edgecolored graphs (which encode PL pseudomanifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the "Gdegree" of the involved graphs, which drives the 1/N expansion in the tensor models context. In the present paper  by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph  we prove that, in any even dimension d greater or equal to 4, the Gdegree of all bipartite graphs, as well as of all (bipartite or nonbipartite) graphs representing singular manifolds, is an integer multiple of (d1)!. As a consequence, in even dimension, the terms of the 1/N expansion corresponding to odd powers of 1/N are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context. In particular, in the 4dimensional case, where the Gdegree is shown to depend only on the regular genera with respect to an arbitrary pair of "associated" cyclic permutations, several results are obtained, relating the Gdegree or the regular genus of 5colored graphs and the Euler characteristic of the associated PL 4manifolds.
2018
 Gdegree for singular manifolds
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; Grasselli, Luigi
abstract
The Gdegree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models.
The present paper studies the properties of the Gdegree for the large class of graphs representing singular manifolds (including closed PL manifolds).
In particular, the complete topological classification up to Gdegree 6 is obtained in dimension 3, where all 4colored graphs represent singular manifolds.
2018
 TOPOLOGY IN COLORED TENSOR MODELS
[Poster]
Casali, M. R.; Cristofori, P.; Grasselli, L.
abstract
From a “geometric topology” point of view, the theory of manifold representation by means of edgecolored graphs has been deeply studied since 1975 and many results have been achieved: its great advantage is the possibility of encoding, in any dimension, every PL dmanifold by means of a totally combinatorial tool.
Edgecolored graphs also play an important rôle within colored tensor models theory, considered as a possible approach to the study of Quantum Gravity: the key tool is the Gdegree of the involved graphs, which drives the 1/N expansion in the higher dimensional tensor models context, exactly as it happens for the genus of surfaces in the twodimensional matrix model setting.
Therefore, topological and geometrical properties of the represented PL manifolds, with respect to the Gdegree, have specific relevance in the tensor models framework, show ing a direct fruitful interaction between tensor models and discrete geometry, via edgecolored graphs.
In colored tensor models, manifolds and pseudomanifolds are (almost) on the same footing, since they constitute the class of polyhedra represented by edgecolored Feynman graphs arising in this context; thus, a promising research trend is to look for classification results concerning all pseudomanifolds  or, at least, singular dmanifolds, if d ≥ 4  represented by graphs of a given Gdegree.
In dimension 4, the existence of colored graphs encoding different PL manifolds with the same underlying TOP manifold, suggests also to investigate the ability of ten sor models to accurately reflect geometric degrees of freedom of Quantum Gravity.
2018
 Topology in colored tensor models via crystallization theory
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; Dartois, Stèphane; Grasselli, Luigi
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 PLmanifolds representation by means of edgecolored graphs (crystallization theory). On the other hand, the core of the paper is to establish results about the topological and geometrical properties of the Guraudegree (or Gdegree) of the represented manifolds, in relation with the motivations coming from physics.
In fact, the Gdegree 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 twodimensional matrix model setting.
In particular, the Gdegree of PLmanifolds is proved to be finitetoone in any dimension, while in dimension 3 and 4 a series of classification theorems are obtained for PLmanifolds represented by graphs with a fixed Gdegree.
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.
2017
 Lower bounds for regular genus and gemcomplexity of PL 4manifolds
[Articolo su rivista]
Basak, B.; Casali, Maria Rita
abstract
Within crystallization theory, two interesting PL invariants for dmanifolds have been introduced and studied, namely, gemcomplexity and regular genus. In the present paper we prove that for any closed connected PL 4manifold M, its gemcomplexity k(M) and its regular genus G(M) satisfy
k(M)≥3χ(M)+10m−6 and G(M)≥2χ(M)+5m−4,
where rk(π1(M))=m. These lower bounds enable to strictly improve previously known estimations for regular genus and gemcomplexity of product 4manifolds. Moreover, the class of semisimple crystallizations is introduced, so that the represented PL 4manifolds attain the above lower bounds. The additivity of both gemcomplexity and regular genus with respect to connected sum is also proved for such a class of PL 4manifolds, which comprehends all ones of “standard type”, involved in existing crystallization catalogs, and their connected sums.
2016
 Classifying PL 4manifolds via crystallizations: results and open problems
[Capitolo/Saggio]
Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo
abstract
Crystallization theory is a graphtheoretical representation method for compact PLmanifolds of arbitrary dimension, which makes use of a particular class of edgecoloured graphs, which are dual to coloured (pseudo)triangulations. The purely combinatorial nature of crystallizations makes them particularly suitable for automatic generation and classication, as well as for the introduction and study of graphdefined invariants for PLmanifolds.
The present survey paper focuses on the 4dimensional case, presenting uptodate results about the PL classication of closed 4manifolds, by means of two such PL invariants: regular genus and gemcomplexity.
Open problems are also presented, mainly concerning different classication of 4manifolds in TOP and DIFF=PL categories, and a possible approach to the 4dimensional Smooth Poincare Conjecture.
2016
 GEOMETRIA
[Monografia/Trattato scientifico]
Casali, Maria Rita; Gagliardi, Carlo; Grasselli, Luigi
abstract
Il presente testo sviluppa argomenti tradizionalmente trattati nei corsi di “Geometria” (ovvero di “Algebra e Geometria”) nell'ambito delle lauree di primo livello, ed è particolarmente rivolto agli studenti dei vari Corsi di Laurea in Ingegneria, e di quelli in Matematica, Fisica ed Informatica. Il testo è suddiviso in due parti:  la prima parte contiene gli elementi fondamentali di Algebra lineare;  la seconda, di carattere più propriamente geometrico, riguarda le principali proprietà degli spazi euclidei, sviluppando in tale ambito la teoria delle coniche e delle quadriche.
L'esposizione risulta articolata, come ovvio per ogni teoria matematica, in Definizioni e Proposizioni (o Teoremi, nel caso in cui gli enunciati rivestano particolare importanza). Un ruolo significativo viene attribuito a Osservazioni ed Esempi atti a:  chiarire concetti, risultati, dimostrazioni;  stimolare i necessari collegamenti tra i vari argomenti;  motivare la genesi dei concetti e dei problemi;  evidenziare i casi notevoli di particolare rilievo nell'ambito di una teoria generale;  indicare possibili generalizzazioni o descrizioni alternative di una teoria. Ciò può consentire inoltre al Docente di “dosare” con maggiore libertà, secondo le proprie convinzioni ed esperienze didattiche, il peso da attribuire, durante le lezioni, ai vari argomenti del corso.
Definizioni, Proposizioni e Osservazioni sono dotati di una numerazione progressiva all'interno di ogni Capitolo; l'esposizione della teoria è arricchita inoltre da esempi notevoli, con numerazione autonoma all'interno di ogni Capitolo.
Con l'eccezione delle principali proprietà degli insiemi numerici fondamentali e dell'utilizzo di una teoria “ingenua”, non rigorosamente assiomatica, degli insiemi (peraltro, brevemente richiamata nel primo Capitolo), il testo appare essenzialmente autocontenuto. In particolare, non risulta necessario alcun prerequisito di Geometria euclidea così come viene sviluppata, a partire da un sistema di assiomi, nelle Scuole secondarie. Seguendo l'impostazione algebrica ormai dominante nelle varie teorie matematiche e quindi in una ottica di “algebrizzazione della Geometria”, i concetti e i risultati di natura geometrica, compresi quelli relativi alla Geometria euclidea, sono infatti ricavati da conoscenze di tipo algebrico precedentemente introdotte. Abbiamo cercato tuttavia di non fare perdere contenuto geometrico a tali concetti, sia mediante il metodo con cui questi vengono presentati, sia facendo spesso ricorso ad Osservazioni ed Esempi atti ad aiutare il lettore a ritrovare, pure in ambiti più generali, le proprietà geometriche già note.
La scelta privilegiata è stata quella di sviluppare la teoria, sia dal punto di vista algebrico che da quello geometrico, per spazi di dimensione finita n; le dimensioni due e tre sono tuttavia sempre illustrate in modo dettagliato, come casi particolari e nelle loro specificità, sfruttandone le caratteristiche di rappresentatività. Tale scelta di generalità nella dimensione è dovuta essenzialmente a due considerazioni: da un lato riteniamo opportuno evitare inutili ripetizioni nella enunciazione della teoria per le varie dimensioni particolari, dall'altro siamo convinti che lo sviluppo della teoria in ambito ragionevolmente generale sia un ottimo stimolo allo sviluppo della capacità di astrazione e generalizzazione che è obiettivo fondamentale di ogni corso di matematica, anche nell'ambito dei nuovi ordinamenti degli studi universitari.
La presente III edizione risulta integrata, rispetto a quelle precedenti, in primo luogo con l'introduzione di test di valutazione al termine di ciascuna delle due parti (Algebra lineare e Geometria euclidea) in cui il testo è suddiviso, rendendo così possibile al lettore una verifica del proprio livello di comprensione delle tematiche trattate.
È stata inoltre realizzata una rivisitazione sostanziale di alcuni argomenti (in pa
2016
 PL 4manifolds admitting simple crystallizations: framed links and regular genus
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo
abstract
Simple crystallizations are edgecolored graphs representing piecewise linear (PL) 4manifolds with the property that the 1skeleton of the associated triangulation equals the 1skeleton of a 4simplex. In this paper, we prove that any (simplyconnected) PL 4manifold M admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, M may be represented by a framed link yielding S^3, with exactly β_2(M) components (β_2(M) being the second Betti number of M). As a consequence, the regular genus of M is proved to be the double of β_2(M). Moreover, the characterization of any such PL 4manifold by k(M)=3β_2(M), where k(M) is the gemcomplexity of M (i.e. the nonnegative number p−1, 2p being the minimum order of a crystallization of M), implies that both PL invariants gemcomplexity and regular genus turn out to be additive within the class of all PL 4manifolds admitting simple crystallizations (in particular, within the class of all “standard” simplyconnected PL 4manifolds).
2015
 A note about complexity of lens spaces
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
Within crystallization theory, (Matveev's) complexity of a 3manifold can be estimated by means of the combinatorial notion of GMcomplexity. In this paper, we prove that the GMcomplexity of any lens space L(p,q), with p greater than 2, is bounded by S(p,q)3, where S(p,q) denotes the sum of all partial quotients in the expansion of q/p as a regular continued fraction. The above upper bound had been already established with regard to complexity; its sharpness was conjectured by Matveev himself and has been recently proved for some infinite families of lens spaces by Jaco, Rubinstein and Tillmann. As a consequence, infinite classes of 3manifolds turn out to exist, where complexity and GMcomplexity coincide.
Moreover, we present and briefly analyze results arising from crystallization catalogues up to order 32, which prompt us to conjecture, for any lens space L(p,q) with p greater than 2, the following relation: k(L(p,q)) = 5 + 2 c(L(p,q)), where c(M) denotes the complexity of a 3manifold M and k(M)+1 is half the minimum order of a crystallization of M
2015
 Cataloguing PL 4manifolds by gemcomplexity
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
We describe an algorithm to subdivide automatically a given set of PL nmanifolds (via coloured triangulations or, equivalently, via crystallizations) into classes whose elements are PLhomeomorphic. The algorithm, implemented in the case n = 4, succeeds to solve completely the PLhomeomorphism problem among the catalogue of all closed connected PL 4manifolds up to gemcomplexity 8 (i.e.,
which admit a coloured triangulation with at most 18 4simplices).
Possible interactions with the (not completely known) relationship among the different classications in the TOP and DIFF=PL categories are also investigated.
As a first consequence of the above PL classification, the nonexistence of exotic PL 4manifolds up to gemcomplexity 8 is proved. Further applications of the tool are described, related to possible PLrecognition of different triangulations of the K3surface.
2014
 REVIEW OF: "Benedetti Riccardo  Petronio Carlo, Spin structures on 3manifolds via arbitrary triangulations, Algebr. Geom. Topol. 14, No. 2, 10051054 (2014)". [DE062726187]
[Recensione in Rivista]
Casali, Maria Rita
abstract
In the present paper a combinatorial encoding of spin structures based on arbitrary triangulations of oriented
compact 3manifolds is introduced. The goal is achieved by means of the notion of weak branching, which
turns out to be related to the notion of Z/2Ztaut structure on triangulations, introduced by Luo (see [Proc.
Am. Math. Soc. 140, No. 3, 10531068 (2012; Zbl 1250.57028)] and [Proc. Am. Math. Soc. 141, No. 1,
335350 (2013; Zbl 1272.57004)]).
In particular, by taking into account the set of all pairs (M, s) (M being a compact oriented 3manifold and
s being a spin structure on M), the authors claim:
 “given any (loose) triangulation T of M, with ideal vertices at the components of @M and possibly internal
vertices, and any s, we encode s by decorating T with certain extra combinatorial structures;
 we exibit combinatorial moves on decorated triangulations relating to each other any two that encode the
same (M, s).”
A dual version of the above encoding is also presented, in terms of special spines dual to triangulations (see
[Acta Appl. Math. 19, No.2, 101130 (1990; Zbl 0724.57012)]).
A first application of the described techniques is contained in [BaseilhacBenedetti, Analytic families of
quantum hyperbolic invariants and their asymptotical behaviour, I, arXiv:1212.4261]. Further possible applications
are also pointed out, concerning “an effective construction of the Roberts spinrefined TuraevViro
invariants and of the related Blanchet spinrefined ReshetikhinTuraev invariants of the double of a manifold”.
2014
 REVIEW OF: "Fominykh Evgeny  Wiest Bert, Upper bounds for the complexity of torus knot complements, J. Knot Theory Ramifications 22, No. 10, Article ID 1350053, 19 p. (2013)". [DE062240723]
[Recensione in Rivista]
Casali, Maria Rita
abstract
The definition of Matveev complexity c(M) of a compact 3manifold with nonempty boundary M is based on
the existence of an almost simple spine for M: see [Acta Appl. Math. 19 (2), 101130 (1990; Zbl 0724.57012)].
The complexity of a given 3manifold is generally hard to compute from the theoretical point of view, leaving
aside the concrete enumeration of its spines: see, for example, [ACM monogr. 9 (2003; Zbl 1048.57001)] and
[Algebr. Geom. Topol. 11 (3), 12571265 (2011; Zbl. 1229.57010)], together with their references.
In the paper under discussion the authors establish an upper bound for the Matveev complexity of any
Seifert fibered 3manifold with nonempty boundary M, by realizing M as an assembling of several copies
of five particular building blocks, whose skeleta contain a known number of true vertices. As a consequence,
they obtain potentially sharp bounds on the Matveev complexity of torus knot complements.
2014
 REVIEW OF: "Kawauchi Akio, Splitting a 4manifold with infinite cyclic fundamental group, revised, J. Knot Theory Ramifications 22, No. 14, Article ID 1350081, 9 p. (2013)". [DE062730205]
[Recensione in Rivista]
Casali, Maria Rita
abstract
In [Osaka J. Math. 31(3), 489495 (1994; Zbl 0849.57018 )], the author stated that every closed connected
orientable 4manifold M with infinite cyclic fundamental group is TOPsplit, i.e. it is homeomorphic to
the connected sum (S1 × S3)#M1, M1 being a closed simply connected 4manifold. However, in [Manuscr.
Math. 93(4), 435442 (1997; Zbl 0890.57034)], Hambleton and Teichner obtained a counterexample to the
above general statement.
In the paper under review, the author makes a revision and proves that TOPsplittability holds under the
additional hypothesis that a finite covering of M is TOPsplit. In particular, the original statement turns
out to be true in the case of indefinite intersection form, as well as for any smooth spin 4manifold (with
infinite cyclic fundamental group).
The proof of the revised statement makes use of notions developed in [Knots in Hellas 98, Ser. Knots
Everything. 24 (World Scientific Publishing), 208228 (2000; Zbl 0969.57020)] and [Atti Semin. Mat. Fis.
Univ. Modena 48(2), 405424 (2000; Zbl 1028.57019)], together with the key result  proved in [Osaka J.
Math. 31(3), 489495 (1994; Zbl 0849.57018 )]  that every closed connected orientable 4manifold M with
infinite cyclic fundamental group is homology cobordant to (S1 × S3)#M1.
Consequences about surfaceknots in S4 are also considered (see [J. Knot Theory Ramifications 4(2), 213224
(1995; Zbl 0844.57020)]).
2014
 REVIEW OF: "Taylor Scott A., Bandtaut sutured manifolds, Algebr. Geom. Topol. 14, No. 1, 157215 (2014)". [DE062342080]
[Recensione in Rivista]
Casali, Maria Rita
abstract
A well known theorem of Lackenby ([Math. Ann. 308, No.4, 615632 (1997; Zbl 0876.57015)]) relates Dehn
surgery properties of a knot to the intersection between the knot and essential surfaces in the 3manifold.
In the paper under review, the author extends Lackenby’s Theorem to the case of 2handles attached to a
sutured 3manifold along a suture, by determining the relationship between an essential surface in a sutured
3manifold, the number of intersections between the boundary of the surface and one of the sutures, and the
cocore of the 2handle in the manifold after attaching a 2handle along the suture.
The author makes use of Scharlemann’s combinatorial version of sutured manifold theory ([J. Differ. Geom.
29, No.3, 557614 (1989; Zbl 0673.57015)]) and takes inspiration from Gabai’s proof that, under suitable
hypotheses, there is at most one way to fill a torus boundary component of a 3manifold so that the Thurston
norm decreases ([J. Differ. Geom. 26, 461478 (1987; Zbl 0627.57012)]). On the other hand, in order to
prove the theorem, bandtaut sutured manifolds are introduced and bandtaut sutured manifold hierarchies
are proved to exist.
As an application, the paper shows that tunnels for tunnel number one knots or links in any 3manifold can
be isotoped to lie on a branched surface corresponding to a certain taut sutured manifold hierarchy of the
knot or link exterior.
Other interesting applications are contained in [Trans. Am. Math. Soc. 366, No. 7, 37473769 (2014; Zbl
06303179)], where band sums are proved to satisfy the cabling conjecture, and new proofs that unknotting
number one knots are prime and that genus is superadditive under band sum are obtained.
2013
 Coloured graphs representing PL 4manifolds
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
Crystallization theory is a representation method for compact PL manifolds by means of a particular class of edgecoloured graphs. The combinatorial nature of this representation allows to elaborate and implement algorithmic procedures for the generation and analysis of catalogues of closed PL nmanifolds. In this paper we discuss the concepts which are involved in these procedures for n = 4 and present classification results arising from the study of the initial segment of the catalogue.
2013
 Computing Matveev's complexity via crystallization theory: The boundary case
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
The notion of GemMatveev complexity (GMcomplexity) has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper, we extend the definition to the case of nonempty boundary and prove that for each compact irreducible and boundaryirreducible 3manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via GMcomplexity, we obtain an estimation of Matveev's complexity for all Seifert 3manifolds with base D2 and two exceptional fibers and, therefore, for all torus knot complements.
2013
 Gammaclass_4dim: A program to subdivide a set of rigid crystallizations of closed 4manifolds into equivalence classes, whose elements represent PLhomeomorphic manifolds.
[Software]
Casali, Maria Rita; Cristofori, Paola
abstract
Gammaclass_4dim is a program yielding, from any given list X of crystallizations of 4dimensional PLmanifolds, the automatic partition of the elements of X into equivalence classes, such that each class contains only crystallizations representing the same PLmanifold. Moreover, the program attempts the identification of the represented 4manifolds by means of comparison of the representatives of each class with known catalogues of crystallizations and/or by means of splitting into connected sums.
Gammaclass_4dim is based on the existence of elementary combinatorial moves available for crystallizations of PLmanifolds of any dimension (i.e. the wellknown "dipole moves", together with the so called "blobs" and "flips", introduced in [S. Lins  M. Mulazzani, Blobs and flips on gems, Journal of Knot Theory and its Ramifications 15 (2006), 10011035].
The program has already been tested for known catalogues of crystallizations of 4manifolds, by making use of a fixed admissible sequence of the above moves; further applications are in progress.
2013
 REVIEW OF: "Birman Joan  Brinkmann Peter  Kawamuro Keiko, Polynomial invariants of pseudoAnosov maps, J. Topol. Anal. 4, No. 1, 1347 (2012)". [DE060376660]
[Recensione in Rivista]
Casali, Maria Rita
abstract
In [Topology 34, No.1, 109140 (1995; Zbl 0837.57010)], Bestvina and Handel gave an algorithmic proof of
Thurston’s classification theorem for mapping classes (see e.g., [Astrisque, 6667 (1979; Zbl 0446.5700523)]).
If [F] is a pseudoAnisov map acting on an orientable surface S, their algorithm allows to construct a graph
G (homotopic to S when S is punctured), a suitable map f : G ! G (called train track map) and the
associated transition matrix T (whose PerronFrobenius eigenvalue is the dilatation of [F]: see [The theory
of matrices, Vol. 2, AMS Chelsea Publishing (1959; Zbl 0927.15002)]).
The dilatation (F) is an invariant of the conjugacy class [F] in the modular group of S, studied in [Ann.
Sci. c. Norm. Supr. (4) 33, No. 4, 519560 (2000; Zbl 1013.57010)] and in several subsequent papers.
The present paper introduces a new approach to the study of invariants of [F], when [F] is pseudoAnisov:
starting from BestvinaHandel algorithm, the authors investigate the structure of the characteristic polynomial
of the transition matrix T and obtain two new integer polynomials (both containing (F) as their
largest real root), which turn out to be invariants of the given pseudoAnisov mapping class.
The degrees of these new polynomials, as well as of their product, are invariants of [F], too; simple formulas
are given for computing them by a counting argument from an invariant train track.
The paper gives also examples of genus 2 pseudoAnisov maps having the same dilatation, which are distinguished
by the new invariants.
2013
 REVIEW OF: "Jaco William  Rubinstein J.Hyam  Tillmann Stephan, Z2Thurston norm and complexity of 3manifolds, Math. Ann. 356, No. 1, 122 (2013)". [DE061684636]
[Recensione in Rivista]
Casali, Maria Rita
abstract
Given a closed, irreducible 3manifold M (different from S3, RP2 and L(3, 1)), the complexity of M is known
to be the minimum number of tetrahedra in a (pseudosimplicial) triangulation of M: see [Acta Appl. Math.
19, No.2, 101130 (1990; Zbl 0724.57012)] and [Algorithms and Computation in Mathematics 9, Springer
(2007; Zbl 1128.57001)].
In [Algebr. Geom. Topol. 11(3), 12571265 (2011; Zbl 1229.57010)] the authors found a lower bound for the
complexity of M, in case M having a connected double cover (or, equivalently, a nontrivial Z2cohomology
class).
The present paper makes use of the notion of Z2Thurston norm (an analogue of Thurston’s norm, defined
in [Mem. Am. Math. Soc. 339, 99130 (1986; Zbl 0585.57006)]) in order to obtain a new lower bound on
the complexity of M, if M admits multiple Z2cohomology classes. Moreover, the minimal triangulations
realizing this bound are characterized, in terms of normal surfaces consisting entirely of quadrilateral discs.
It is worthwhile to note that the combinatorial structure of a minimal triangulation turns out to be governed
by 0efficiency ([J. Differ. Geom. 65(1), 61168 (2003; Zbl 1068.57023)]) and low degree edges ([J. Topol. 2(1),
157180 (2009; Zbl 1227.57026)]), and that the unique minimal triangulation of the generalized quaternionic
space S3/Q8k (8k 2 Z+)  already obtained by the authors via the first lower bound  actually realizes this
new bound, too.
2013
 REVIEW OF: "Kotschick D.  Neofytidis, C., On threemanifolds dominated by circle bundles, Math. Z. 274, No. 12, 2132 (2013)". [DE06176500X]
[Recensione in Rivista]
Casali, Maria Rita
abstract
Given two closed oriented nmanifolds M and N, M is said to dominate N if a nonzero degree map from
M to N exists. From dimension n = 3 on, the domination relation fails to be an ordering.
By a result of [Math. Semin. Notes, Kobe Univ. 9, 159180 (1981; Zbl 0483.57003)], every 3manifold turns
out to be dominated by a surface bundle over the circle; on the other hand, in [J. Lond. Math. Soc. 79 (3),
545561 (2009; Zbl 1168.53024)] and [Groups Geom. Dyn. 7 (1), 181204 (2013; Zbl 06147449)] it is shown
that 3manifolds dominated by products cannot have hyperbolic or Sol3geometry, and must often be prime.
In the present paper, the authors give a complete characterization of 3manifolds dominated by products,
by proving that a closed oriented 3manifold is dominated by a product if and only if it is finitely covered
either by a product or by a connected sum of copies of S2 × S1. It is worthwhile to note that the same
characterization may also be formulated in terms of Thurston geometries, or in terms of purely algebraic
properties of the fundamental group.
Moreover, the authors determine which 3manifolds are dominated by nontrivial circle bundles, and which
3manifold groups are presentable by products (according to [J. Lond. Math. Soc. 79 (3), 545561 (2009;
Zbl 1168.53024)]).
2013
 REVIEW OF: "Li Tao, Rank and genus of 3manifolds, J. Am. Math. Soc. 26, No. 3, 777829 (2013)". [DE061686069]
[Recensione in Rivista]
Casali, Maria Rita
abstract
In the 1960’s,Waldhausen conjectured the equality, for each closed orientable 3manifoldM, between the rank
r(M) of the fundamental group 1(M) and the Heegaard genus g(M) of M ([Algebr. Geom. Topol. 32(2),
313322 (1978; Zbl 0397.57007)]). Various counterexamples may be found in the literature: in particular,
Boileau and Zieschang obtained a Seifert fibered space with r(M) = 2 and g(M) = 3 ([Invent. Math. 76,
455468 (1984; Zbl 0538.57004)]), while Schultens and Weidman proved the existence of graph manifolds
with discrepancy g(M) − r(M) arbitrarily large ([Pac. J. Math. 231 (2), 481510 (2007; Zbl 1171.57020)]).
In 2007, the above Conjecture has been reformulated, by restricting the attention to hyperbolic 3manifolds:
see [Geometry and Topology Monographs 12, 335349 (2007; Zbl 1140.57009)].
The present paper gives a negative answer to this “modern version” of Waldhausen Conjecture. In fact, a
closed orientable hyperbolic 3manifold M is proved to exist, so that g(M) − r(M) is arbitrarily large.
Actually, the author produces an (atoroidal) 3manifold with boundary ¯M with r( ¯M ) < g( ¯M), and the
closed counterexample is constructed starting from ¯M , via the so called annulus sum (see [Geom. Topol. 14
(4), 18711919 (2010; Zbl 1207.57031)]). Moreover, every 2bridge knot exterior is proved to be a JSJ piece
of a closed 3manifold M with r(M) < g(M).
2012
 Catalogues of PLmanifolds and complexity estimations via crystallization theory
[Articolo su rivista]
Casali, Maria Rita
abstract
Crystallization theory is a graphtheoretical representation method for compact PLmanifolds of arbitrary dimension, with or without boundary, which makes use of a particular class of edgecoloured graphs, which are dual to coloured (pseudo) triangulations. These graphs are usually called gems, i.e. Graphs Encoding Manifolds, or crystallizations if the associated triangulation has the minimal number of vertices.One of the principal features of crystallization theory relies on the purely combinatorial nature of the representing objects, which makes them particularly suitable for computer manipulation.The present talk focuses on uptodate results about: generation of catalogues of PLmanifolds for increasing values of the vertex number of the representing graphs; definition and/or computation of invariants for PLmanifolds, directly from the representing graphs.
2012
 Complexity computation for compact 3manifolds via crystallizations and Heegaard diagrams
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola; M., Mulazzani
abstract
The idea of computing Matveev complexity by using Heegaard decompositions has been recently developed by two different approaches: the first one for closed 3manifolds via crystallization theory, yielding the notion of GemMatveev complexity; the other one for compact orientable 3manifolds via generalized Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this paper we extend to the nonorientable case the definition of modified Heegaard complexity and prove that for closed 3manifolds GemMatveev complexity and modified Heegaard complexity coincide. Hence, they turn out to be useful different tools to compute the same upper bound for Matveev complexity.
2012
 REVIEW OF: "Burton Benjamin A., Maximal admissible faces and asymptotic bounds for the normal surface solution space, J. Comb. Theory, Ser. A 118, No. 4, 14101435 (2011)". [DE058788316]
[Recensione in Rivista]
Casali, Maria Rita
abstract
The present paper deals about the use of normal surface theory in 3dimensional computational topology:
see Kneser’s foundational paper ([Jahresbericht D. M. V. 38, 248260 (1929; JFM 55.0311.03)]), together
with the developments by Haken and JagoOertel ([Acta Math. 105, 245375 (1961; Zbl 0100.19402)], [Math.
Z. 80, 89120 (1962; Zbl 0106.16605)], [Topology 23, 195209 (1984; Zbl 0545.57003)]).
In particular, the author faces the crucial problem of enumerating normal surfaces in a given (triangulated)
3manifold, via the underlying procedure of enumerating admissible vertices of a highdimensional polytope
(admissibility being a powerful but nonlinear and nonconvex constraint).
The main results of the present paper are significant improvements upon the best known asymptotic bounds
on the number of admissible vertices (see [J. ACM 46, No. 2, 185211 (1999; Zbl 1065.68667)], [B.A.Burton,
The complexity of the normal surface solution space, in: SCG’10: Proceedings of the TwentySixth Annual
Symposium on Computational Geometry, ACM Press, 2010, pp.201209] and [Math. Comput. 79, No. 269,
453484 (2010; Zbl pre05776230)]).
To achieve these results, the author examines the layout of admissible points within polytopes in both the
standard normal surface coordinate system and the streamlined quadrilateral coordinate system. These
points are proved to correspond to wellbehaved substructures of the face lattice, and the properties of
the corresponding “admissible faces” are studied. Key lemmata include upper bounds on the number of
maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the
two coordinate systems mentioned above.
2012
 REVIEW OF: "Garoufalidis Stavros, Knots and tropical curves, Champagnerkar, Abhijit (ed.) et al., Interactions between hyperbolic geometry, quantum topology and number theory. Proceedings of a workshop, June 3–13, 2009 and a conference, June 15–19, 2009, Columbia University, New York, NY, USA. Providence, RI: American Mathematical Society (AMS) (ISBN 97808218 49606/pbk). Contemporary Mathematics 541, 83101 (2011)." [DE059544674]
[Recensione in Rivista]
Casali, Maria Rita
abstract
A sequence of rational functions in a variable q is qholonomic (see [J. Comput. Appl. Math. 32, No.3,
321368 (1990; Zbl 0738.33001)] and [Invent. Math. 103, No.3, 575634 (1991; Zbl 0739.05007)]) if it satisfies
a linear recursion with coefficient polynomials in q and qn.
In virtue of a fundamental result by WilfZeilberger, Quantum Topology turns out to provide us with a
plethora of qholonomic sequences of natural origin. In particular, the present paper takes into account the
qholonomic sequence of Jones polynomials of a knot and its parallels (see [Geom. Topol. 9, 12531293 (2005;
Zbl 1078.57012)]).
The author associates a tropical curve (see [Contemporary Mathematics 377, 289317 (2005; Zbl 1093.14080)]
and [Math. Mag. 82, No. 3, 163173 (2009; Zbl 1227.14051)]) to each qholonomic sequence; in particular, to
every knot K a tropical curve is associated, via the Jones polynomial of K and its parallels. As a consequence,
a relation is established between the AJ Conjecture ([Geometry and Topology Monographs 7, 291309 (2004;
Zbl 1080.57014)]) and the Slope Conjecture ([Quantum Topol. 2, No. 1, 4369 (2011; Zbl 1228.57004)]),
which relate the Jones polynomial of K and its parallels respectively to the SL(2;C) character variety and
to slopes of incompressible surfaces.
The paper gives also an explicit computation of the tropical curve for the 41, 52 and 61 knots, verifying
in these cases the duality between the tropical curve and a Newton subdivision of the Apolynomial of the
knot.
2012
 REVIEW OF: "Schleimer Saul, The end of the curve complex, Groups Geom. Dyn. 5, No. 1, 169176 (2011)". [DE059733321]
[Recensione in Rivista]
Casali, Maria Rita
abstract
If S is a genus g surface with b boundary components, so that 3g − 3 + b 2, then the curve complex C(S)
has a vertex for each isotopy class of essential nonperipheral simple closed curves in S and a ksimplex for
each collection of k + 1 disjoint vertices having disjoint representatives.
By regarding each simplex as a Euclidean simplex of sidelength one, C(S) turns out to be Gromov hyperbolic
([Invent. Math. 138, No.1, 103149 (1999; Zbl 0941.32012)]).
The present paper proves that, if the surface S has exactly one boundary component and genus two or more,
than for each vertex ! 2 C(S) and for any r 2 N, the subcomplex spanned by C0(S) − B(!, r) is connected
(where B(!, r) denotes the ball of radius r about the vertex !).
In order to prove the above result, the author makes use of the fact that the complex of curves has no dead
ends (Prop. 3.1 of this paper) and of the so called Birman short exact sequence (see [Annals of Mathematics
Studies 82, Princeton (1975; Zbl 0305.57013)] and [Acta Math. 146, 231270 (1981; Zbl 0477.32024)]).
Note that, for the considered surfaces, the above result directly answers a question of Masur’s, and answers
a question of G.Bell and K.Fujiara ([J. Lond. Math. Soc., II. Ser. 77, No. 1, 3350 (2008; Zbl 1135.57010)])
in the negative. It is also evidence for a positive answer to a question of P.Storm (already verified in an
independent way by Gabai in [Geom. Topol. 13, No. 2, 10171041 (2009; Zbl 1165.57015)]).
2012
 REVIEW of: "Friedl Stefan  Vidussi, Stefano, Twisted Alexander polynomials detect fibered 3manifolds, Ann. Math. (2) 173, No. 3, 15871643 (2011)." [DE05960690X]
[Recensione in Rivista]
Casali, Maria Rita
abstract
It is well known that, if a knot K S3 is fibered, then the Alexander polynomial is monic and the degree
equals twice the genus of K. Various generalizations of this result have been performed, showing that twisted
Alexander polynomials give necessary conditions for (N, ) (where N is a compact, connected, oriented 3
manifolds with empty or toroidal boundary and 2 H1(N;Z)) to fiber: see [Ann. Sci. Ecol. Norm. Super.
(4) 35, No. 2, 153171 (2002; Zbl 1009.57021)], [Trans. Am. Math. Soc. 355, No.10, 41874200 (2003;
Zbl 1028.57004)], [Comment. Math. Helv. 80, No. 1, 5161 (2005; Zbl 1066.57008)], [Topology 45, No. 6,
929953 (2006; Zbl 1105.57009)] and [T.Kitayama, Normalization of twisted Alexander invariants, preprint
2007 (arXiv 0705.2371)].
In general, the constraint of monicness and degree for the ordinary Alexander polynomial falls short from
characterizing fibered 3manifolds. The main result of present paper shows that that the collection of
all twisted Alexander polynomials does detect fiberedness; equivalently, it proves that twisted Alexander
polynomials detect whether (N, ) fibers under the assumption that the Thurston norm of is known.
Moreover, by making use of some of their previous works (see in particular [Am. J. Math. 130, No. 2, 455
484 (2008; Zbl 1154.57021)]), the authors show that, if a manifold of the form S1 × N3 admits a symplectic
structure, then N fibers over S1.
2012
 REVIEW of: "Wong Helen, Quantum invariants can provide sharp Heegaard genus bounds, Osaka J. Math. 48, No. 3 (2011), 709717". [DE059690467]
[Recensione in Rivista]
Casali, Maria Rita
abstract
It is wellknown that the Heegaard genus g(M) of a 3manifold M (i.e. the smallest integer so that M has
a Heegaard splitting of that genus) is generally very difficult to compute. The present paper investigates
the effectiveness of a lower bound on g(M) deriving from the ReshetikhinTuraev invariants (see [Commun.
Math. Phys. 121, No.3, 351399 (1989; Zbl 0667.57005)] for a basic reference about the invariants, and
[Topology 37, No.1, 219224 (1998; Zbl 0892.57005)] and [Quantum invariants of knots and 3manifolds,
de Gruyter Studies in Mathematics 18, Walter de Gruyter, Berlin (1994; Zbl 0812.57003)] for the quoted
bound).
Until advent of the quantum invariants, the best known lower bound on g(M) was the rank of the fundamental
group of M, r( 1M). In [Invent. Math. 76, 455468 (1984; Zbl 0538.57004)], Boileau and Zieschang
presented a particular set of Seifert fibered spaces M, with the relatively rare property that g(M) = 3, while
r( 1M) = 2.
By studying the examples of Boileau and Zieschang, the author proves that quantum invariants may be used
to provide a lower bound on g(M) which is both simpler to calculate and strictly larger than r( 1M).
2011
 Computational aspects of crystallization theory: complexity, catalogues and classifications of 3manifolds
[Articolo su rivista]
Bandieri, Paola; Casali, Maria Rita; Cristofori, Paola; Grasselli, Luigi; M., Mulazzani
abstract
The present paper is a survey of uptodate results in 3dimensional crystallization theory, in particular along the following directions: generation and analysis of catalogues of PLmanifolds for increasing values of the vertex number of the representing graphs; definition and/or computation of invariants for PLmanifolds, directly from the representing graphs.In particular, with regard to PLmanifold invariants, the authors focus on gems considered as an useful tool for computing Matveev complexity.
2011
 Gaifullin Alexander A., Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class [Recensione]
[Recensione in Rivista]
Casali, Maria Rita
abstract
The problem of finding explicit combinatorial formulae for the Pontryagin classes of triangulated manifolds was originally faced by Gabrielov, Gelfand and Losik in [Funct. Anal. Appl. 9, 103115 (1975; Zbl 0312.57016)], and successively taken into account by various authors: see Russ. Math. Surv. 60, No. 4, 615644 (2005; Zbl 1139.57026)] for a survey on the topic, with a comparison among different formulae. The present paper discusses the only two known combinatorial formulae for the first Pontryagin class that can be used for real computation, i.e. the classical GabrielovGelfandLosik formula and the local formula obtained by the author in [Izv. Math. 68, No. 5, 861910 (2004; Zbl 1068.57022)]. Note that the first formula is presented not according to the original approach, based on endowing a triangulated manifold with locally flat connections, but according to MacPherson’s approach, based on the construction of a homology Gaussian mapping for a combinatorial manifold: see [Semin. Bourbaki, Vol. 1976/77, Lect. Notes Math. 677, 105124 (1978; Zbl 0388.57013)]. A detailed exposition of the second formula is provided, together with the related notions and results concerning bistellar moves (see [Abh. Math. Semin. Univ. Hamb. 57, 6986 (1987; Zbl 0651.52007)] and [Eur. J. Comb. 12, No.2, 129145 (1991; Zbl 0729.52003)]) and the existence and uniqueness of universal local formulae for polynomials in rational Pontryagin classes.
In the present paper the author succeeds in considerably simplifying his own explicit formula for the first Pontryagin class, by giving a new simpler algorithm for decomposing a cycle in the graph of bistellar moves of twodimensional combinatorial spheres into a linear combination of elementary cycles.
2011
 REVIEW OF: "Howie James, Can Dehn surgery yield three connected summands?, Groups Geom. Dyn. 4, No. 4, 785797 (2010)".[DE05880964X]
[Recensione in Rivista]
Casali, Maria Rita
abstract
The {\it Cabling Conjecture} of GonzalesAcuna and Short ([Math. Proc. Camb. Philos. Soc. 99, 89102 (1986; Zbl 0591.57002)]) states that Dehn surgery on a knot in $\Bbb S^3$ can produce a reducible 3manifold only if the knot is a cable knot and
the surgery slope is that of a cabling annulus.
Since the case of cable knot yields exactly two connected summands (see [Math. Proc. Camb. Philos. Soc. 102, No. 1, 97101 (1987; Zbl 0655.57500)]), a weaker conjecture may be stated, asserting that a manifold obtained by Dehn surgery (i.e. $M^3=M(K, r)$, where $K$ is a knot in $\Bbb S^3$ and $r \in \Bbb Q \cup \{\infty\}$) cannot by expressed as a connected sum of three nontrivial manifolds.
Note that results by [Topology Appl. 87, No.1, 7378 (1998; Zbl 0926.57020)], [Topology Appl. 98, No.13, 355370 (1999; Zbl 0935.57024)] and [J. Pure Appl. Algebra 173, No.2, 167176 (2002; Zbl 1026.20019)] ensure that if $M(K, r)$ has three connected summands, than two of these must be lens spaces and the third must be a $\Bbb Z$homology sphere. Moreover, in the same hypothesis, $r$ must be an integer: see [Math. Proc. Camb. Philos. Soc. 102, No. 1, 97101 (1987; Zbl 0655.57500)].
\medskip
The present paper faces the problem of three connected summands by making use of standard techniques of intersection graphs by Scharlemann and GordonLuecke.
The main result proves that, if $K$ has bridenumber $b$ and $M(K, r)= M_1 \#M_2\#M_3,$ $M_1$ and $M_2$ being lens spaces and $M_3$ being a homology sphere (but not a homotopy sphere), then
$$\pi_1(M_1) + \pi_1(M_2) \le b+1.$$
As a consequence, the inequality
$$r = \pi_1(M_1) \cdot \pi_1(M_2) \le \frac{b(b+2)}{4}$$
is obtained, which is a sharpening of a similar inequality due to Sayari (see [J. Knot Theory Ramifications 18, No. 4, 493504 (2009; Zbl 1188.57004)]).
2011
 REVIEW OF: "Kirk Paul, The impact of QFT on lowdimensional topology, Ocampo Hern\'an (ed.) et al., Geometric and topological methods for quantum field theory. Papers based on the presentations at the summer school `Geometric and topological methods for quantum field theory', Villa de Leyva, Colombia, July 220, 2007. Cambridge: Cambridge University Press (ISBN 9780521764827/hbk; 9780511717970/ebook). 153 (2010)".[DE058315336]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2011
 REVIEW OF: "Mroczkowski Maciej, Kauffman Bracket Skein module of the connected sum of two projective spaces, J. Knot Theory Ramifications 20, No. 5, 651675 (2011)".[DE059180169]
[Recensione in Rivista]
Casali, Maria Rita
abstract
{\it Skein modules}, which are invariants of 3manifolds as well as of links in these manifolds, were introduced by Przytycki ([Bull. Pol. Acad. Sci., Math. 39, No.12, 91100 (1991; Zbl 0762.57013)]) and Turaev ([J. Sov. Math. 52, No.1, 27992805 (1990; Zbl 0706.57004)]).
In the present paper, diagrams and Reidemeister moves for links in a twisted $\mathbb S^1$bundles over a nonorientable surface are introduced, and the Kauffman bracket skein module (KBSM) of $\mathbb R P^3 \times \mathbb R P^3$ is computed.
Note that the notion of diagrams of links in $F \times \mathbb S^1$ ($F$ being an orientable surface)), was introduced in [Topology Appl. 156, No. 10, 18311849 (2009; Zbl 1168.57010)], together with Reidemeister moves for such diagrams.
After having extended the above notions to the case $N \hat \times \mathbb S^1$ ($N$ being a nonorientable surface), the author takes into account the particular case $N= \mathbb RP^2$, so that $N \hat \times \mathbb S^1 = \mathbb R P^3 \times \mathbb R P^3.$
The full computation of KBSM for $\mathbb R P^3 \times \mathbb R P^3$ shows that it has torsion (as it happens for $\mathbb S^1 \times \mathbb S^2$: see [Math. Z. 220, No.1, 6573 (1995; Zbl 0826.57007)]), but  unlike the KBSM of $\mathbb S^1 \times \mathbb S^2$  it does not split as a sum of cyclic modules.
A new computation of KBSM of both $\mathbb S^1 \times \mathbb S^2$ and the lens space $L(p,1)$ completes the paper.
2011
 REVIEW OF: "Rafi Kasra  Schleimer Saul, Curve complexes are rigid, Duke Math. J. 158, No. 2, 225246 (2011)".[DE059178142]
[Recensione in Rivista]
Casali, Maria Rita
abstract
The {\it curve complex} of a surface was introduced into the study of Teichmüller space by Harvey (see [Riemann surfaces and related topics: Proc. 1978 Stony Brook Conf., Ann. Math. Stud. 97, 245251 (1981; Zbl 0461.30036)]) as an analogous of the Tits building of a symmetric space.
The present paper deals about geometric structure of the curve complex (see [Invent. Math. 138, No.1, 103149 (1999; Zbl 0941.32012)] and [Geom. Funct. Anal. 10, No.4, 902974 (2000; Zbl 0972.32011)]) of an orientable connected compact surface $S$, in case the
{\it complexity} $\csi(S)$ of $S$ is greater or equal to two, where $\csi(S)=3g3+b$, $g$ (resp. $b$) being the genus (resp. the number of boundary components) of $S$.
By making use of the notions of {\it cobounded ending lamination} and of {\it marking complex}, together with some key results due to Gabai (see [Geom. Topol. 13, No. 2, 10171041 (2009; Zbl 1165.57015)]) and to BerhrstockKleinerMinskyMosher (see [{\it Geometry and rigidity of mapping class group}, arXiv:0801.2006v4]), the authors prove that any quasi isometry of the curve complex is bounded distance from a simplicial automorphism. As a consequence, the quasiisometry type of the curve complex determines the homeomorphism type of the surface.
2011
 REVIEW OF: "Spaggiari Fulvia, Regular genus and products of spheres, J. Korean Math. Soc. 47, No. 5, 925934 (2010)".[DE057900743]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2011
 REVIEW OF: "Takao Kazuto, A refinement of Johnson's bounding for the stable genera of Heegaard splittings, Osaka J. Math. 48, No. 1, 251268 (2011)".[DE058823344]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2010
 Characterization of minimal 3manifolds by edgecoloured graphs
[Articolo su rivista]
Casali, Maria Rita
abstract
We characterize combinatorial representations of minimal 3manifolds by means of edgecoloured graphs. This enables their recognition among existing crystallization catalogues, and contemporarily enables the automatic construction of efficient and exhaustive catalogues representing all minimal 3manifolds up to a fixed genus.
2010
 GEOMETRIA
[Monografia/Trattato scientifico]
Casali, Maria Rita; Gagliardi, Carlo; Grasselli, Luigi
abstract
Il presente testo sviluppa argomenti tradizionalmente trattati nei corsi di “Geometria” (ovvero di “Algebra e Geometria”) nell'ambito delle lauree di primo livello, ed è particolarmente rivolto agli studenti delle Facoltà di Ingegneria e dei Corsi di Laurea in Matematica, Fisica ed Informatica. Il testo è suddiviso logicamente in due parti:  la prima parte contiene gli elementi fondamentali di Algebra lineare;  la seconda parte, di carattere più propriamente geometrico, riguarda le principali proprietà degli spazi euclidei, sviluppando in tale ambito la teoria delle coniche e delle quadriche. La presente edizione risulta integrata, rispetto a quella precedente, da una rivisitazione sostanziale della teoria delle coniche e delle quadriche, dall'inserimento di nuovi argomenti e complementi (algebre di Boole, isometrie del piano euclideo...), oltre che dalla aggiunta di nuove osservazioni ed esempi lungo tutto lo sviluppo del testo.L'esposizione risulta articolata, come ovvio per ogni teoria matematica, in Definizioni e Proposizioni (o Teoremi, nel caso in cui gli enunciati rivestano particolare importanza). Particolare rilievo viene attribuito ad Osservazioni ed Esempi atti a:  chiarire concetti, risultati, dimostrazioni;  stimolare i necessari collegamenti tra i vari argomenti;  motivare la genesi dei concetti e dei problemi;  evidenziare i casi notevoli di particolare rilievo nell'ambito di una teoria generale;  indicare possibili generalizzazioni o descrizioni alternative di una teoria. Ciò può consentire inoltre al Docente di “dosare” con maggiore libertà, secondo le proprie convinzioni ed esperienze didattiche, il peso da attribuire, durante le lezioni, ai vari argomenti del corso. Con l'eccezione delle principali proprietà degli insiemi numerici fondamentali e dell'utilizzo di una teoria “ingenua”, non rigorosamente assiomatica, degli insiemi (peraltro, brevemente richiamata nel primo Capitolo), il testo appare essenzialmente autocontenuto. In particolare, non risulta necessario alcun prerequisito di Geometria euclidea così come viene sviluppata, in modo sintetico, a partire da un sistema di assiomi, nelle Scuole secondarie. Seguendo l'impostazione algebrica ormai dominante nelle varie teorie matematiche e quindi in una ottica di “algebrizzazione della Geometria”, i concetti ed i risultati di natura geometrica, compresi quelli relativi alla Geometria euclidea, sono infatti ricavati da conoscenze di tipo algebrico precedentemente introdotte. Abbiamo cercato tuttavia di non fare perdere contenuto geometrico a tali concetti, sia mediante il metodo con cui questi vengono presentati, sia facendo spesso ricorso ad Osservazioni ed Esempi atti ad aiutare il lettore a ritrovare, pure in ambiti più generali, le proprietà geometriche già note. La scelta privilegiata è stata quella di sviluppare la teoria, sia dal punto di vista algebrico che da quello geometrico, per spazi di dimensione finita n; le dimensioni due e tre sono tuttavia sempre illustrate in modo dettagliato, come casi particolari e nelle loro specificità, sfruttandone le caratteristiche di rappresentatività. Tale scelta di generalità nella dimensione è dovuta essenzialmente a due considerazioni: da un lato riteniamo opportuno evitare inutili ripetizioni nella enunciazione della teoria per le varie dimensioni particolari, dall'altro siamo convinti che lo sviluppo della teoria in ambito ragionevolmente generale sia un ottimo stimolo allo sviluppo della capacità di astrazione e generalizzazione che è obiettivo fondamentale di ogni corso di matematica, anche nell'ambito dei nuovi ordinamenti degli studi universitari.
2010
 REVIEW OF: "Burton, Benjamin A., Converting between quadrilateral and standard solution sets in normal surface theory, Algebr. Geom. Topol. 9, No. 4, 21212174 (2009)". [DE056242025]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2010
 REVIEW OF: "Hongler Cam Van Quach  Weber Claude, Link projections and flypes, Acta Math. Vietnam. 33, No. 3, 433457 (2008)". [DE055891064]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2010
 REVIEW OF: "Lee, Sang Youl; Seo, Myoungsoo, Formulas for the Casson invariant of certain integral homology 3spheres, J. Knot Theory Ramifications 18, No. 11, 15511576 (2009)". [DE056622866]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2010
 REVIEW of: "Todd Robert G., Khovanov homology and the twist number of alternating knots, J. Knot Theory Ramifications 18, No. 12, 16511662 (2009)". [DE056622768].
[Recensione in Rivista]
Casali, Maria Rita
abstract
2010
 “Computational and Geometric Topology”  A conference in honour of Massimo Ferri and Carlo Gagliardi on their 60th birthday.
[Altro]
Bandieri, Paola; Casali, Maria Rita; A., Cattabriga; Cristofori, Paola; P., Frosini; Grasselli, Luigi; Landi, Claudia; M., Mulazzani
abstract
La conferenza ha inteso mettere in contatto ricercatori provenienti sia dall'ambito matematico che da quello ingegneristico, accomunati dall'interesse per tecniche topologiche di carattere geometrico e computazionale. Questi strumenti di ricerca sono essenziali in vari settori scientifici e per molteplici classi di applicazioni. In topologia geometrica risultano di particolare importanza le ricerche in teoria dei nodi, connesse allo studio di strutture biologiche (p.e. il confronto di dati genetici) e in fisica (con particolare riferimento alla teoria delle stringhe). La topologia computazionale si è invece rivelata indispensabile per la descrizione di forme al calcolatore e per la loro comparazione, con conseguenti ricadute nelle applicazioni che richiedono manipolazione grafica, confronto di modelli e reperimento di informazioni visuali. Tutto ciò ha ovvie importanti ricadute nel trattamento di dati in Internet. Tutti questi ambiti applicativi richiedono lo sviluppo di nuovi approcci teorici e competenze fortemente e intrinsecamente interdisciplinari, che l'iniziativa ha favorito.Il convegno si è articolato in sei conferenze su invito, tenute da alcuni tra i massimi esperti internazionali, della durata di 50 minuti ciascuna e da numerose comunicazioni di 30 minuti. Ha vauto lo scopo di divulgare nuovi risultati in Topologia Geometrica e Computazionale, ed ha coinvolto sia docenti che giovani ricercatori, nonché studenti di dottorato di ricerca in Matematica e/o in Ingegneria.Conferenzieri principali:Herbert Edelsbrunner (Duke University, Durham, NC, USA) Tomasz Kaczynski (Université de Sherbrooke, Canada)Sóstenes Lins (Departamento de Matemática, UFPE, Brasile)Sergei Matveev (Chelyabinsk State University, Russia) José María Montesinos (Universidad Complutense, Madrid, Spagna)Marian Mrozek (Jagiellonian University, Kraków, Polonia)
2009
 REVIEW of: "Ayala R.  Fernández L.M.  Vilches J.A., Critical elements of proper discrete Morse functions, Mathematica Pannonica 19/2 (2008), 171185" [DEB09154]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2009
 REVIEW of: "Cooper Daryl  Tillmann Stephan, The Thurston norm via normal surfaces,Pac. J. Math. 239, No. 1, 115 (2009)". [Zbl 1165.57018]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2009
 REVIEW of: "Dasbach Oliver T.  Lin, XiaoSong, A volumish theorem for the Jones polynomial of alternating knots,Pac. J. Math. 231, No. 2, 279291 (2007)". [Zbl 1166.57002]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2009
 REVIEW of: "Friedl S., Reidemeister torsion, the Thurston norm and Harvey's invariants, Pac. J. Math. 230, No. 2, 271296 (2007)". [Zbl 1163.57008]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2009
 REVIEW of: "Perles Micha A  Martini Horst  Kupitz Yaakov S., A JordanBrouwer separation theorem for polyhedral pseudomanifolds, Discrete Comput. Geom. 42, No. 2, 277304 (2009)" [Zbl pre05586556]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2009
 REVIEW of: "Repovš D.  Rosicki W.  Zastrow A.  Željko M., Constructing nearembeddings of codimension one manifolds with countable dense singular sets, Glas. Mat. III. Ser. 44 (1), 255258 (2009)." [Zbl 1175.57016]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2009
 REVIEW of: "Thomassen Carsten  Vella Antoine, Graphlike continua, augmenting arcs, and Menger's theorem, Combinatorica 28, No. 5, 595623 (2008)". [Zbl pre05580969]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 A catalogue of orientable 3manifolds triangulated by 30 coloured tetrahedra
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
The present paper follows the computational approach to 3manifold classification via edgecoloured graphs, already performed in [1] (with respect to orientable 3manifolds up to 28 coloured tetrahedra), in [2] (with respect to nonorientable3manifolds up to 26 coloured tetrahedra), in [3] and [4] (with respect to genus two 3manifolds up to 34 coloured tetrahedra): in fact, by automatic generation and analysis of suitable edgecoloured graphs, called crystallizations, we obtain a catalogue of all orientable 3manifolds admitting coloured 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 oneto one correspondence with the homeomorphism classes of the represented manifolds.
2008
 CRYSTALLIZATION CATALOGUES AND ARCHIVES OF CLOSED 3MANIFOLDS WITH LOW GEMCOMPLEXITY
[Software]
Casali, Maria Rita; Cristofori, Paola
abstract
CRYSTALLIZATION CATALOGUES is a collection of algorithmic procedures, which can be used to construct essential catalogues of bipartite and/or non bipartite edgecoloured graphs representing all orientable and/or non orientable 3manifolds triangulated by a given number of coloured tetrahedra; the elements of the obtained catalogues may further be classified (i.e. subdivided into homeomorphism classes), as a first step toward the topological recognition of the involved manifolds. The output data of the C++ program (originally described in [M.R.Casali, Classification of nonorientable 3manifolds admitting decompositions into 26 coloured tetrahedra, Acta Appl. Math. 54 (1999), 7597]) generating catalogue C^2p of rigid bipartite crystallizations up to 2p vertices and/or catalogue ~C^2p of rigid non bipartite crystallizations up to 2p vertices are available, according to the number of vertices, at the Web page: http://cdm.unimo.it/home/matematica/casali.mariarita/CATALOGUES.htmThe Web page contains detailed results about existing catalogues ~C^26, C^28 and C^30 which are not included in the associated papers (for example: complete description of the involved manifolds, survey tables with related topological invariants, data about the reduced catalogues of clusterless crystallizations…). Further, a comparative analysis of both complexity and geometric properties of manifolds represented by the subsequent subsets C^2p, p compreso tra 1 e 15, of all crystallizations in C^30 with exactly 2p vertices is also presented.
2008
 Gammaclass: A program to subdivide a set of rigid crystallizations of closed 3manifolds into equivalence classes, whose elements represent homeomorphic manifolds
[Software]
Casali, Maria Rita; Cristofori, Paola
abstract
Gammaclass is a program which implements the algorithm described in in [Casali M.R., Cristofori P., A catalogue of orientable 3manifolds triangulated by 30 coloured tetrahedra, Journal of Knot Theory and its Ramifications 17 (2008), no.5, 579599], with respect to a fixed (finite) set S of admissible sequences of elementary combinatorial moves: it yields, from any given list X of crystallizations, the automatic partition of the elements of X into equivalence classes, such that each class contains only crystallizations representing the same manifold. Moreover, the program tries the identification of the represented manifolds by means of comparison of the representatives of each class with known catalogues of crystallizations and/or splitting into connected sums.Program Gammaclass has already allowed the recognition and cataloguing of all manifolds represented by rigid bipartite and non bipartite crystallizations up to 30 vertices.
2008
 REVIEW OF: "Balogh J.  Leanos J.  Pan S.  Richter R.B. Salazar G., The convex hull of every optimal pseudolinear drawing of $K_n$ is a triangle, Australas. J. Comb. 38 (2007), 155162".[Zbl 1138.05015]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REVIEW OF: "Burton Benjamin A., Enumeration of nonorientable 3manifolds using facepairing graphs and unionfind, Discrete Comput. Geom. 38 (3), 527571". [Zbl 1133.57001]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REVIEW OF: "Chen Yichao  Liu Yanpei, A note on lower bounds for maximum genus, Util. Math. 73 (2007), 2331".[Zbl 1138.05016]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REVIEW OF: "Falbel Elisha, A spherical CR structure on the complement of the figure eight knot with discrete holonomy, J. Differ. Geom. 79 (1) (2008), 69110".[Zbl 1148.57025]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REVIEW OF: "Flapan Erica  Naimi Ramin, The Ytriangle move does not preserve intrinsic knottedness, Osaka J. Math. 45 (1) (2008), 107111".[Zbl 1145.05019]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REVIEW OF: "Funar Louis, Surface cubications mod flips, Manuscr. Math. 125 (3), 285307".[Zbl 1144.05022]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REVIEW OF: "Hilden Mike  Montesinos Josè M.  Tejada Debora  Toro Margarita, Representing 3manifolds by triangulations of S^3. I: a constructive approach, Rev. Colomb. Mat. 39 (2) (2005), 6386". [Zbl 1130.57008]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REVIEW OF: "Li Tao, Saddle tangencies and the distance of Heegaard splittings, Algebr. Geom. Topol. 7 (2007), 11191134". [Zbl 1134.57005]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REVIEW OF: "Poenaru V.  Tanasi C., Some remarks on geometric simple connectivity in dimension four  part A, Rend. Semin. Mat. Torino 65 (3) (2007), 313344". [Zbl 1150.57005]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REVIEW OF: "Scharlemann Martin, Generalized property R and the Schoenflies conjecture, Comment. Math. Helv. 83 (1) (2008), 421449". [Zbl 1148.57032]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REVIEW OF: "Taylor Scott A., On noncompact Heegaard splittings, Algebr. Geom. Topol. 7 (2007), 603672". [Zbl 1134.57006]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 REWIEW OF: "Przytycki, Piotr, A The fixed point theorem for simplicial nonpositive curvature, Math. Proc. Camb. Philos. Soc. 144(3) (2008), 683695"[Zbl 1152.20038]
[Recensione in Rivista]
Casali, Maria Rita
abstract
2008
 Representing and recognizing 3manifolds obtained from Ibundles over the Klein bottle
[Articolo su rivista]
Casali, Maria Rita
abstract
As it is wellknown, the boundary of the orientable Ibundle $K X^sim I $ over the Klein bottle K is a torus; thus  in analogy with torus bundle construction (see [S])  any integer matrix A of order two with determinant 1 (resp. +1) uniquely defines an orientable (resp. nonorientable) 3manifold $(K X^sim I) cup (KX^sim I)/A$, which we denote by KB(A). In thepresent paper an algorithmic procedure is described, which allowsto construct, directly from any such matrix A, an edgecolouredgraph representing the manifold KB(A) associated to A. As a consequence, it is proved via regular genus (see [G]) thatthe Heegaard genus of any such manifold is less or equal to four;moreover, six elements of existing catalogues of orientable 3manifolds represented by edgecoloured graphs (see [L] and [CC_2]) are directly recognized as manifolds of type KB(A).
2007
 DUKE III: A program to handle edgecoloured graphs representing PL ndimensional manifolds
[Software]
Casali, Maria Rita; Cristofori, Paola
abstract
One of the main features of crystallization theory relies on the purely combinatorial nature of the representing objects, which makes them particularly suitable for computer manipulation. This fact allows a computational approach to the study of PL nmanifolds, which has been performed by means of several functions, collected in a unified program, called DUKE III. DUKE III allows automatic manipulation of edgecoloured graphs representing PL nmanifolds (code computation, checking possible isomorphism between edgecoloured graphs, construction of boundary graph, checking bipartition, connectedness, rigidity and planarity conditions, combinatorial moves, invariants computation...). Furthermore, DUKE III allows automatic recognition of orientable 3manifolds triangulated by at most 30 coloured tetrahedra and of nonorientable 3manifolds triangulated by at most 26 coloured tetrahedra (by making use of existing electronic archives of all rigid bipartite crystallizations up to 30 vertices and nonbipartite ones up to 26 vertices, due to the same research team).
2007
 Estimating Matveev's complexity via crystallization theory
[Articolo su rivista]
Casali, M. R.
abstract
In [M.R. Casali, Computing Matveev's complexity of nonorientable 3manifolds via crystallization theory, Topology Appl. 144(13) (2004) 201209], a graphtheoretical approach to Matveev's complexity computation is introduced, yielding the complete classification of closed nonorientable 3manifolds up to complexity six. The present paper follows the same pointof view, making use of crystallization theory and related results (see [M. Ferri, Crystallisations of 2fold branched coverings of S^3, Proc. Amer. Math. Soc. 73 (1979) 271276]; [M.R. Casali, Coloured knots and coloured graphs representing 3fold simple coverings of S^3, Discrete Math. 137 (1995) 8798]; [M.R. Casali, From framed links to crystallizations of bounded 4manifolds, J. Knot Theory Ramifications 9(4) (2000) 443458]) in order to significantly improve existing estimations for complexity of both 2fold and threefold simple branched coverings (see [O.M. Davydov, The complexity of 2fold branched coverings of a 3sphere, Acta Appl. Math. 75 (2003) 5154] and [O.M. Davydov, Estimating complexity of 3manifolds as of branched coverings, talkabstract, Second RussianGerman Geometry Meeting dedicated to 90anniversary of A.D.Alexandrov, SaintPetersburg, Russia, June 2002]) and 3manifolds seen as Dehn surgery (see [G. Amendola, An algorithm producing a standard spine of a 3manifold presented by surgery along a link, Rend. Circ. Mat. Palermo 51 (2002) 179198]).
2007
 TORUS BUNDLE: A program to construct edgecoloured graphs representing torus bundles over the circle.
[Software]
Casali, Maria Rita
abstract
TORUS BUNDLE is based on an algorithmic construction of edgecoloured graphs representing 3manifolds, which are torus bundles over S^1 (see [M.R.Casali, Representing and recognizing torus bundles over S^1, Boletín de la Sociedad Matemática Mexicana, 10(3) (2005), 89106]). The algorithm starts from a regular integer matrix A which describes the monodromy and contains at least one zero element. This program has allowed the topological recognition of all torus bundles among the 3manifolds represented by the existing crystallization catalogues C^28 and ~C^26. Note that TORUS BUNDLE is actually independent from DUKE III, but related to it: it supplies  as output  the code of an edgecoloured graph which can be successively inserted in DUKE III in order to be analyzed and simplified, for recognition of the represented manifold....
2006
 Computing Matveev's complexity via crystallization theory: the orientable case
[Articolo su rivista]
Casali, Maria Rita; Cristofori, Paola
abstract
By means of a slight modification of the notion of GMcomplexity introduced in [Casali, M.R., Topol. Its Appl., 144: 201209, 2004], the present paper performs a graphtheoretical approach to the computation of (Matveev's) complexity for closed orientable 3manifolds. In particular, the existing crystallization catalogue C28 available in [Lins, S., Knots and Everything 5, World Scientific, Singapore, 1995] is used to obtain upper bounds for the complexity of closed orientable 3manifolds triangulated by at most 28 tetrahedra. The experimental results actually coincide with the exact values of complexity, for all but three elements. Moreover, in the case of at most 26 tetrahedra, the exact value of the complexity is shown to be always directly computable via crystallization theory.
2006
 c_GM: A program to compute GMcomplexity of edgecoloured graphs representing closed 3manifolds
[Software]
Casali, Maria Rita; Cristofori, Paola
abstract
c_GM is a C++ program which implements the algorithmic procedure described in [M.R. Casali, Computing Matveev's complexity of nonorientable 3manifolds via crystallization theory, Topology and its Applications 144 (13) (2004), 201209], to estimate Matveev's complexity of a 3manifold starting from the code of an associated edgecoloured graph (GMcomplexity computation). This program has already allowed to compute GMcomplexity of all nonorientable 3manifolds represented by edgecoloured graphs up to 26 vertices (catalogue ~C26) and of all orientable 3manifolds represented by edgecoloured graphs up to 28 vertices (catalogue C28), giving a significant help to the classification of the involved manifolds; classes of manifolds for which the estimation is actually exact have been also detected. Furthermore, a comparison between different notions of complexity has been performed with the aid of this program: see [M.R. Casali, Computing Matveev's complexity of nonorientable 3manifolds via crystallization theory, Topology and its Applications 144 (13) (2004), 201209] and [M.R. Casali  P.Cristofori, Computing Matveev's complexity via crystallization theory: the orientable case, Acta Applicandae Mathematicae 92 (2006), 113123]. The program computes the GMcomplexity both of a single edgecoloured graph and of a list of edgecoloured graphs. It also computes the minimal GMcomplexity of a set of crystallizations representing the same manifold, thus providing upper bounds for the complexity of the manifold itself.c_GM interacts with Duke III program for handling edgecoloured graphs, since it recognizes Duke’s encoding of graphs and it can run on catalogues of crystallizations generated and classified through the procedures of CRYSTALLIZATION CATALOGUES and program Gamma_class.
2004
 Computing Matveev’s complexity of nonorientable 3manifolds via crystallization theory
[Articolo su rivista]
Casali, Maria Rita
abstract
The present paper looks at Matveev's complexity (introduced in 1990 and based on the existence of a simple spine for each compact 3manifold: see [Acta Appl. Math. 19 (1990), 101130]) through another combinatorial theory for representing 3manifolds, which makes use of particular edgecoloured graphs, called crystallizations. Crystallization catalogue $\tilde C^{26}$ for closed nonorientable 3manifolds (due to [Acta Appl. Math. 54 (1999), 7597]) is proved to yield upper bounds for Matveev's complexity of any involved 3manifold. As a consequence, an improvement of Amendola and Martelli classification of closed nonorientable irreducible and $P^2$irreducible 3manifolds up to complexity c=6 is obtained.
2004
 Dotted links, Heegaard diagrams and coloured graphs for PL 4manifolds
[Articolo su rivista]
Casali, Maria Rita
abstract
The present paper is devoted to establish a connection between the 4manifold representation method by dotted framed links (or  in the closed case  by Heegaard diagrams) and the so called crystallization theory, which visualizes general PLmanifolds by means of edgecoloured graphs.In particular, it is possible to obtain a crystallization of aclosed 4manifold $M^4$ starting from a Heegaard diagram$(\#_m(S^1 X S^2), \omega),$ and the algorithmicity of the whole process depends on the effective possibility of recognizing $(\#_m(S^1 X S^2), \omega)$ to be a Heegaard diagram by crystallization theory.
2004
 Representing and recognizing torus bundles over S1
[Articolo su rivista]
Casali, Maria Rita
abstract
As it is wellknown, torus bundles over $S^1$ areidentified by means of regular integer matrices of order two (see[S]); in the present paper an algorithmic procedure is described,which allows to construct, directly from any matrix $A \in GL (2;Z)$, an edgecoloured graph representing the torus bundle$TB(A)$ associated to $A.$ As a consequence, five topologically undetected elements of Lins's catalogue of orientable 3manifolds (see [L]) are finally recognized as torus bundles over $S^1$.
2003
 On the regular genus of 5manifolds with free fundamental group
[Articolo su rivista]
Casali, Maria Rita
abstract
In the present paper, we obtain the following classification of closed orientable PL 5manifolds M^5 with free fundamental group of rank m, so that the difference between the regular genus G(M^5) and m is less or equal to eight: (a) G(M^5) = m iff $M^5= \#_m (S^1 X S^4)$; (b) it is impossible m+1 less than or equal to G(M^5) less than or equal to m+7; (c) if G(M^5) = m+8, then either $M^5= \#_m (S^1 X S^4) # (S^2 X S^3)$ or $M^5= \#_m (S^1 X S^4) # (S^2 X_\sim S^3)$. As a consequence, we complete the classification of PL 5manifolds up to regular genus eight, and compute the regular genus of the 5dimensional real projective space RP5: if G(M^5) = 8, then either $M^5= S^2 X S^3$ or $M^5= S^2 X_\sim S^3$ or $M^5= \#_8 (S^1 X S^4)$; $G(S^2 X S^3) = 8$; $G(S^2 X_\sim S^3)$ greater than or equal to 8; $G(RP^5) = 9$.
2002
 An equivalence criterion for PLmanifolds
[Articolo su rivista]
Casali, Maria Rita
abstract
Within geometric topology of PL nmanifolds (with or withoutboundary), a representation theory exists, which makes use of (n+1)coloured graphs. Aim of this paper is to translate the homeomorphism problem for the represented manifolds into an equivalence problem for (n+1)coloured graphs, by means of a finite number of graphmoves, called dipole moves. Actually,the same problem was already faced  and solved  in [FG], but for closed nmanifolds, only; here, the whole class of PLmanifolds is considered, and the uptodate knowledge involved in the proof (i.e. shelling and bistellar operations, among all) throw a deeper light on the previous results, too. Moreover, the equivalence criterion for PLmanifolds via dipole moves is proved to be equivariant with respect to the boundary triangulation.
2001
 A code for mbipartite edgecoloured graphs
[Articolo su rivista]
Casali, Maria Rita; Gagliardi, Carlo
abstract
An (n+1)coloured graph $(\Gamma,\gamma)$ is said to be mbipartite if m is the maximum integer so that every mresidue of $(\Gamma,\gamma)$ (i.e. every connected subgraph whose edges are coloured by only m colours) is bipartite; obviously, every (n+1)coloured graph, with $n \ge 2$, results to be mbipartite for some m, with $\ 2 \le m \le n+1$. In this paper, a numerical code of length $(2nm+1) \times q$ is assigned to each mbipartite (n+1)coloured graph of order 2q.Then, it is proved that any two such graphs have the same code if and only if they are colourisomorphic, i.e. if a graph isomorphism exists, which transforms the graphs one into the other, up to permutation of the edgecolouring. More precisely, if H is a given group of permutations on the colour set, we face the problem of algorithmically recognizing Hisomorphic coloured graphs by means of a suitable definition of Hcode.
2001
 Geometria
[Monografia/Trattato scientifico]
Casali, Maria Rita; Grasselli, Luigi; Gagliardi, Carlo
abstract
Il presente testo sviluppa argomenti tradizionalmente trattati nei corsi di “Geometria” (ovvero di “Algebra e Geometria”) nell'ambito delle lauree di primo livello, ed è particolarmente rivolto agli studenti delle Facoltà di Ingegneria e dei Corsi di Laurea in Matematica, Fisica ed Informatica.Il testo è suddiviso logicamente in due parti: la prima parte contiene gli elementi fondamentali di Algebra lineare; la seconda parte, di carattere più propriamente geometrico, riguarda le principali proprietà degli spazi euclidei, sviluppando in tale ambito la teoria delle coniche e delle quadriche.L'esposizione risulta articolata, come ovvio per ogni teoria matematica, in Definizioni e Proposizioni (o Teoremi, nel caso in cui gli enunciati rivestano particolare importanza). Particolare rilievo viene attribuito ad Osservazioni ed Esempi atti a: chiarire concetti, risultati, dimostrazioni; stimolare i necessari collegamenti tra i vari argomenti; motivare la genesi dei concetti e dei problemi; evidenziare i casi notevoli di particolare rilievo nell'ambito di una teoria generale; indicare possibili generalizzazioni o descrizioni alternative di una teoria.Ciò può consentire inoltre al Docente di “dosare” con maggiore libertà, secondo le proprie convinzioni ed esperienze didattiche, il peso da attribuire, durante le lezioni, ai vari argomenti del corso.Con l'eccezione delle principali proprietà degli insiemi numerici fondamentali e dell'utilizzo di una teoria “ingenua”, non rigorosamente assiomatica, degli insiemi (peraltro, brevemente richiamata nel primo Capitolo), il testo appare essenzialmente autocontenuto. In particolare, non risulta necessario alcun prerequisito di Geometria euclidea così come viene sviluppata, in modo sintetico, a partire da un sistema di assiomi, nelle Scuole secondarie.Seguendo l'impostazione algebrica ormai dominante nelle varie teorie matematiche e quindi in una ottica di “algebrizzazione della Geometria”, i concetti ed i risultati di natura geometrica, compresi quelli relativi alla Geometria euclidea, sono infatti ricavati da conoscenze di tipo algebrico precedentemente introdotte. Abbiamo cercato tuttavia di non fare perdere contenuto geometrico a tali concetti, sia mediante il metodo con cui questi vengono presentati, sia facendo spesso ricorso ad Osservazioni ed Esempi atti ad aiutare il lettore a ritrovare, pure in ambiti più generali, le proprietà geometriche già note.La scelta privilegiata è stata quella di sviluppare la teoria, sia dal punto di vista algebrico che da quello geometrico, per spazi di dimensione finita n; le dimensioni due e tre sono tuttavia sempre illustrate in modo dettagliato, come casi particolari e nelle loro specificità, sfruttandone le caratteristiche di rappresentatività. Tale scelta di generalità nella dimensione è dovuta essenzialmente a due considerazioni: da un lato riteniamo opportuno evitare inutili ripetizioni nella enunciazione della teoria per le varie dimensioni particolari, dall'altro siamo convinti che lo sviluppo della teoria in ambito ragionevolmente generale sia un ottimo stimolo allo sviluppo della capacità di astrazione e generalizzazione che è obiettivo fondamentale di ogni corso di matematica, anche nell'ambito dei nuovi ordinamenti degli studi universitari.
2001
 Representing manifolds by crystallization theory: foundations, improvements and related results
[Articolo su rivista]
Bandieri, Paola; Casali, Maria Rita; Gagliardi, Carlo
abstract
Crystallization theory was born in Italy during the 70's, due to Mario Pezzana and his school, as a combinatorial representation tool for piecewiselinear (PL) manifolds of arbitrary dimension. The present paper performes a  not exaustive  survey of the main results of PLtopology achieved through crystallization theory, both by the italian school of M. Pezzana and by other researchers of different schools, which contributed to the development of the ideas. In many cases, research problems and open questions are reviewed, together with the most recent  sometimes unpublished  results.
2000
 From framed links to crystallizations of bounded 4manifolds
[Articolo su rivista]
Casali, Maria Rita
abstract
It is wellknown that every 3manifold $M^3$ may be represented by a framed link (L, c), which indicates the Dehnsurgery from $S^3$ to $M^3 = M^3(L, c)$; moreover, $M^3$ is the boundary of a PL 4manifold $M^4 = M^4(L, c)$, which is obtained from $D^4$ by adding 2handles along the framed link (L, c). In this paper we study the relationships between the above representations and the representation theory of general PLmanifolds by edgecoloured graphs: in particular, we describe how to construct a 5coloured graph representing $M^4 = M^4(L, c)$, directly from a planar diagram of (L, c). As a consequence, relations between the combinatorial properties of the link L and both the Heegaard genus of $M^3 = M^3(L, c)$ and the regular genus of $M^4 = M^4(L, c)$ are obtained.
1998
 Average order of coloured triangulations: The general case
[Articolo su rivista]
Casali, Maria Rita
abstract
In [Combinatorics of triangulations of 3manifolds, Trans. Amer. Math. Soc. 337 (2) (1993), 891906], Luo and Stong introduced the notion of "average edge order" $\mu_0(K) = \frac {3 F_0(K)}{E_0(K),$ K being a triangulation of a closed 3manifold M with $E_0(K)$ edges and $F_0(K)$ triangles. The present paper extends the above notion to the "average (n2)simplex order" of a coloured triangulation K of a compact PL nmanifold $M^n$ with $\alpha_i(K)$ isimplices: $\mu(K) = \frac {n \alpha_{n1}(K)}{\alpha_{n2}(K)$.Main properties of $\mu(K)$ and its relations with the topology of $M^n$, both in the closed and bounded case, are investigated; the obtained results show the existence of strong analogies with the 3dimensional simplicial case (see the quoted paper by Luo and Stong, together with [The average edge order of triangulations of 3manifolds, Osaka J. Math. 33(1986), 761773] by Tamura).
1998
 Classification of nonorientable 3manifolds admitting decompositions into <= 26 coloured tetrahedra
[Articolo su rivista]
Casali, Maria Rita
abstract
The present paper adopts a computational approach to the study of nonorientable 3manifolds: in fact, we describe how to create an automatic catalogue of all nonorientable 3manifolds admitting coloured triangulations with 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 3manifolds. As a consequence, the following summarising result can be stated: THEOREM I. Exactly seven closed connected prime nonorientable 3manifolds exist, which admit a coloured triangulation consisting of at most 26 tetrahedra. More precisely, they are the four Euclidean nonorientable 3manifolds, the nontrivial $S^2$bundle over $S^1$, the topological product between the real projective plane $RP^2$ and $S^1$, and the torus bundle over $S^1$, with monodromy induced by matrix (0,1;1,1).
1997
 A combinatorial proof of Rohlin Theorem
[Articolo su rivista]
Casali, Maria Rita; Gagliardi, Carlo
abstract
We present an algorithmic and combinatorial proof of the following wellknown theorem, originally proved by Rohlin: `Every closed orientable 3manifold $M^3$ bounds a simply connected orientable 4manifold $M^4$.' More precisely, an edgecoloured graph representing $M^4$ is obtained as the final result of a finite and welldetermined sequence of `admissible moves', starting from any given edgecoloured graph representing $M^3$.
1997
 An equivalence criterion for 3manifolds
[Articolo su rivista]
Casali, Maria Rita
abstract
Within geometric topology of 3manifolds (with or withoutboundary), a representation theory exists, which makes use of 4coloured graphs. Aim of this paper is to translate the homeomorphism problem for the represented manifolds into an equivalence problem for 4coloured graphs, by means of a finite number of graphmoves, called "dipole moves". Moreover, interesting consequences are obtained, which are related with the same problem in the ndimensional setting.
1997
 Classifying PL 5manifolds by regular genus: the boundary case
[Articolo su rivista]
Casali, Maria Rita
abstract
In the present paper we face the problem of classifying classes of orientable PL 5manifolds $M^5$ with $h\ge1$ boundary components, by making use of a combinatorial invariant called “regular genus” $G(M^5)$. In particular, a complete classification up to regular genus five is obtained: if $G(M^5)=\rho \le 5,$ then $M^5= \#_{\rho \rho’} (S^1 x S^4) \# H_{\rho’}^h$where $\rho’=G(\partial M^5}$ denotes the regular genus of the boundary $\partial M^5$ and $H_{\rho’}^h$ denotes the connected sum of h orientable 5dimensional handlebodies $Y_{\alpha_i}$ of genus $\alpha_i \ge 0$ so that $\sum_{i=1,…,h} \alpha_i = \rho’$. Moreover, we give a characterization of orientable PL 5manifolds $M^5$ with boundary satisfying particular conditions related to the “gap” between $G(M^5)$ and either $ G(\partial M^5}$ or the rank of their fundamental group. Further, the paper explains how the above results (together with other known properties of regular genus of PLmanifolds) may lead to a combinatorial approach to 3dimensional Poincarè Conjecture.
1997
 Geometric topology by crystallization theory: results and problems
[Articolo su rivista]
Casali, Maria Rita
abstract
Within geometric (or PL) topology, a representation theory exists, which makes use of a particular class of edgecoloured graphs  called crystallizations  to deal with PLmanifolds of arbitrary dimension, with or without boundary. The present paper is mainly devoted to review some recent developments of crystallization theory, and to show the existingrelationships with other "classical" representation methods for PLmanifolds, such as Heegaard splittings, branched coverings and surgery on framed links.
1997
 Handledecompositions of PL 4manifolds
[Articolo su rivista]
Casali, Maria Rita; L., Malagoli
abstract
The present paper studies the relationship between handledecompositions of PL 4manifolds and the so called "crystallization theory", which represents PL nmanifolds by means of pseudosimplicial triangulations admitting exactly n+1 vertices. Within this theory, the combinatorial invariant "regular genus" plays a central role. In particular, the authors obtain the characterization of PL 4manifolds $M^4$ (with empty or connected boundary $\partial M^4$) satisfying suitable conditions concerning the regular genus $G(M^4)$, the rank of the fundamental group $rk(\pi_1(M^4))$, and the boundary regular genus $G(\partial M^4)$. As a consequence, it is completed the classification of PL 4manifolds with (possibly disconnected) boundary up to regular genus two.
1996
 An infinite class of bounded 4manifolds having regular genus three
[Articolo su rivista]
Casali, Maria Rita
abstract
In the present paper the classification of PL 4manifolds by means of the combinatorial invariant “regular genus” is proved to be not finite to one: indeed, the set of all $D^2$bundles over $S^2$ (i.e. every bundle $\csi_c$ with Euler class $c$ and boundary L(c,1), $c \in Z\{0,1,1}$, together with the trivial bundle $S^2 X D^2$) constitutes an infinite family of PL 4manifolds with the same regular genus (equal to three). Further, general results are obtained, concerning PL 4manifolds with “restricted gap” between their regular genus and the rank of their fundamental group, especially in case of free fundamental group.
1996
 Equivalenze combinatorie di varietà bidimensionali
[Articolo su rivista]
Casali, Maria Rita; R., Panisi
abstract
In this work, a set of combinatorial moves is determined, which realize the homeomorphism of bidimensional manifolds, working on their coloured triangulations, or –equivalently – on the edgecoloured graphs representing them. In particular, the obtained moves result to be equivariant with respect to the manifold boundary.
1996
 Sul genere regolare delle PLvarietà con gruppo fondamentale libero.
[Abstract in Atti di Convegno]
Casali, Maria Rita
abstract
All'interno della teoria di rappresentazione delle nvarietà PL tramite grafi colorati sugli spigoli ([M.Pezzana, Sulla struttura topologica delle varietà compatte, Atti Sem. Mat. Fis. Univ. Modena, 23 (1974), 269277], [M.Ferri  C.Gagliardi  L.Grasselli, A graphtheoretical representation of PLmanifolds. A survey on crystallizations, Aequationes Mat. 31, (1986), 121141], [A.Vince, ngraphs, Discrete Math. 72 (1988), 367380], [A.Costa, Coloured graphs representing manifolds and universal maps, Geom. Dedicata 28 (1988), 349357], [S.Lins, Gems, computers and attractors for 3manifolds, Knots and Everything, World Scientific 5, 1995]…), è nota l'esistenza di un invariante per varietà PL  detto genere regolare  che estende a dimensione arbitraria le classiche nozioni di genere di una superficie e di genere di Heegaard di una 3varietà (si veda [C.Gagliardi, Extending the concept of genus to dimension n, Proc. Amer. Math. Soc. 81 (1981), 473481]). In qualunque dimensione, sia nel caso chiuso che nel caso con bordo, è facile verificare che $G(M^n) \ge rk(M^n)$, ove $G(M^n)$ denota il genere regolare della nvarietà $M^n$ e $rk(M^n)$ denota il rango del suo gruppo fondamentale $\pi_1(M^n)$. Nella presente comuncazione sono esposti alcuni risultati ottenuti in dimensione quattro e cinque sulla classificazione delle PLvarietà con gruppo fondamentale libero, quando sia nota la differenza tra il genere regolare della varietà e il rango del suo gruppo fondamentale. Di seguito è esposto il primo di tali risultati, che riguarda il caso in cui la differenza descritta sia nulla.Sia $M^n$ una nvarietà PL di dimensione n, con n=4 o n=5. Allora: $G(M^n) = m$ e $rk(M^n) se e solo se $M^n = #m(S^1 x S^{n1}$ (se $\partial M^n =\emptyset$) o $M^n = #l(S^1 x S^{n1} # Y^n_{ml}$ (se $\partial M^n \ne \emptyset$), ove $#a(S^1 x S^{n1}$ denota la somma connessa di a copie del fibrato standard su $S^1$ con fibra $S^{n1}$, mentre $Y^n_a$ denota il corpo di manici ndimensionale di genere a. Come conseguenza dei risultati ottenuti, si caratterizzano le varietà $M^n$ con $G(M^n) \le G(\partial M^n) +1$ (per n=4 e per n=5), e si completano le classificazioni delle 4varietà con bordo fino a genere regolare tre, delle 5varietà con bordo fino a genere regolare cinque e delle 5varietà chiuse fino a genere regolare otto.
1995
 A note about bistellar operations on PLmanifolds with boundary
[Articolo su rivista]
Casali, Maria Rita
abstract
In 1990, U. Pachner proved that simplicial triangulations of the same PLmanifold (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, and bistellar 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.
1995
 A note about the closing S3 recognition algorithm
[Articolo su rivista]
Casali, Maria Rita
abstract
In [V2], Vince outlined three potential graph algorithms for $S^3$ recognition, namely shelling, reducing, and closing; on the other hand, he conjectured that the graph $H_0$ of Fig.1  which proves the first two to fail  could be a counterexample for the third one, too. This note shows that the conjecture is false; so, the validity of the closing algorithm is still an open problem.
1995
 Coloured knots and coloured graphs representing 3fold simple coverings of S3
[Articolo su rivista]
Casali, Maria Rita
abstract
It is wellknown that every closed orientable 3manifold $M^3$ is the 3fold simple covering $M^3(K,\omega)$ of $S^3$ branched over a knot K: hence, $M^3$ may be visualized by the associated coloured knot $(K,\omega)$. On the other hand, PLmanifolds of arbitrary dimension may be represented by coloured graphs, via pseudosimplicial triangulations. The present paper produces an algorithm to construct a 4coloured graph representing $M^3(K,\omega)$, directly 'drawn over' the coloured knot $(K,\omega)$.
1994
 Classifying PL 5manifolds up to regular genus seven
[Articolo su rivista]
Casali, Maria Rita; Gagliardi, Carlo
abstract
In the present paper we show that the only closed orientable PL 5manifolds of regular genus less or equal to seven are the 5sphere $S^5$ and the connected sum of m copies of $S^1 X S^4$, with $m \le 7$. As a consequence, the genus of $S^3 X S^2$ is proved to be eight. This suggests a possible approach to the (3dimensional) Poincarè Conjecture, via the wellknown classification of simply connected 5manifolds, obtained by Smale and Barden.
1994
 The average edge order of 3manifold coloured triangulations
[Articolo su rivista]
Casali, Maria Rita
abstract
If K is a triangulation of a closed 3manifold M with $E_0(K)$ edges and $F_0(K)$ triangles, then the average edge order of K is defined to be $\mu_0(K) = 3F_0(K)/E_0(K) In [8], the relations between this quantity and the topology of M are investigated, especially in the case of $\mu_0(K)$ being small (where the study relies on Oda's classification of triangulations of $S^2$ up to eight vertices. In the present paper, the attention is fixed upon the average edge order of coloured triangulations; surprisingly enough, the obtained results are perfectly analogous to LuoStong' ones, and may be proved with little effort by means of edgecoloured graphs representing manifolds.
1993
 A note on the characterization of handlebodies
[Articolo su rivista]
Casali, Maria Rita
abstract
The work is devoted to extend to dimension five the following combinatorial characterization of (orientable and nonorientable) handlebodies, already proved for dimensions three and four by the same author: a compact connected 5manifold $M^5$ is a handlebody (of genus g) iff $ G(M^5)= G(\partial M^5} (=g)$, $G(X) being the regular genus of the manifold X. Moreover, partial results in dimension n induce to conjecture that an analogous characterization also holds for handlebodies of arbitrary dimension.
1993
 A universal branching set for 4dimensional manifolds
[Articolo su rivista]
Casali, Maria Rita
abstract
In this work, a universal branching set K for orientable 4manifolds, such that $\pi_1(S^4  K) = [a, b, c/aca^{1}c^{1} =1]$ is proved to exist. This leads to the possibility of representing every closed connected orientable 4manifold by a suitable transitive set $\{\sigma, \tau, \mu\}$ of permutations, in analogy with known results for dimension three (see [Montesinos] and [CostadelValMelus]).
1992
 A combinatorial characterization of 4dimensional handlebodies
[Articolo su rivista]
Casali, Maria Rita
abstract
In this work, (orientable and nonorientable) 4dimensional handlebodies are proved to be the only 4manifolds whose regular genus equals the one of their boundary. As a consequence, we obtain the classification of all 4manifolds of regular genus g lessthanorequalto 1.
1992
 Twofold branched coverings of S3 have type six
[Articolo su rivista]
Casali, Maria Rita
abstract
In this work we prove that every closed, orientable 3manifold $M^3$ which is a twofold covering of $S^3$ branched over a link, has type six. This implies that $M^3$ is the quotient of the universal pseudocomplex K(4,6) by the action of a finite index subgroup of a fuchsian group with presentation S(4,6)= < a_1, a_2, a_3, a_4 / (a_1)^3 = (a_2)^3 = (a_3)^3 = (a_4)^3 = a_1 a_2 a_3 a_4 =1 >Moreover, the same result is proved to be true in case $M^3$ being an unbranched covering of a twofold branched covering of $S^3$.
1991
 2symmetric crystallizations and 2fold branched coverings of S3
[Articolo su rivista]
Casali, Maria Rita; Grasselli, Luigi
abstract
For each integer g>1, a class $M_g$ of “2symmetric” crystallizations, depending on a 2(g+1)tuple of positive integers satisfying simple conditions is introduced; the “2symmetry” implies that the represented closed, orientable 3manifolds are 2fold covering spaces of $S^3$ branched over a link. Since every closed, orientable 3manifold M of Heegaard genus $g \le 2$ admits a crystallization belonging to $M_g$, we obtain an easy proof og the fact that M is a 2fold covering spaces of $S^3$ branched over a link. Further, the class contains all LinsMandel crystallizations S(b,l,t,c), with l odd, which are thus proved to represent 2fold branched coverings of $S^3$.
1991
 CLASSIFICAZIONE DELLE 5VARIETA' P.L. CON GENERE REGOLARE <= 8
[Abstract in Atti di Convegno]
Casali, Maria Rita
abstract
Il genere regolare di una nvarietà PL $M^n$ è un invariante combinatorio $G(M^n) \ge 0$, definito in C.Gagliardi, Extending the concept of genus to dimension n, Proc. Amer. Math. Soc. 81 (1981), 473481]) all'interno della teoria di rappresentazione delle varietà tramite grafi colorati sugli spigoli, che estende a dimensione arbitraria le classiche nozioni di genere di una superficie e di genere di Heegaard di una 3varietà. E' noto che, in ogni dimensione, $G(M^n)$ assume valore zero se e soltanto se $M^n$ è omeomorfa alla nsfera $S^n.$ Per quanto riguarda la dimensione cinque, si è ottenuta la classificazione completa delle varietà chiuse connesse ed orientabili aventi genere regolare minore o uguale ad otto: Sia $M^5$ una 5varietà PL chiusa connessa orientabile. Allora: (a) $1 \le G(M^n) = m \le 7$ se e solo se $M^5 = #m(S^1 x S^4);$ (b) $G(M^n) = 8$ se e solo se o $M^5 = #8(S^1 x S^4)$ o $M^5 = S^2 x S^3$ (ove $#m(S^1 x S^4)$ denota la somma connessa di m copie di $S^1 x S^4$). Come conseguenza, si è calcolato il genere regolare dello spazio proiettivo reale $RP^5$: $G(RP^5) =9.$
1990
 Una caratterizzazione dei corpi di manici 3dimensionali
[Articolo su rivista]
Casali, Maria Rita
abstract
In this work we produce the following characterization of 3dimensional "handlebodies": a 3manifold is a handlebody if and only if its regular genus equals the genus of its boundary.
1990
 Wave moves on crystallizations
[Articolo su rivista]
Casali, Maria Rita; Grasselli, Luigi
abstract
In this paper, the relations between the notions of “wave move” (by HommaOchiai) and “frame” (by Tsukui) are investigated. A genus three frame of $S^3$ is produced, giving a counterexample to a conjecture of Tsukui; on the contrary, the conjecture is proved to be true in genus two.
1989
 A catalogue of the genus two 3manifolds
[Articolo su rivista]
Casali, Maria Rita
abstract
In this paper, the extension of the notion of “wave move” to crystallizations (by CasaliGrasselli) leads to a “reduced” catalogue of all genus $g\le 2$ 3manifolds, depending on 6tuples of positive integers. The appendix shows a partial output of the computer program which generates the catalogue and gives a presentation of the fundamental group of each element.
1989
 Representing branched coverings by edge coloured graphs
[Articolo su rivista]
Casali, Maria Rita; Grasselli, Luigi
abstract
Given a link L in $S^3$, we describe a standard method for constructing a class $\Gamma_{L,d}$ of 4coloured graphs representing all closed orientable 3manifolds which are dfold coverings of $S^3$ branched over the link L.
1988
 Characterizing crystallizations among LinsMandel 4coloured graphs
[Articolo su rivista]
Casali, Maria Rita; Grasselli, Luigi
abstract
In [Discr. Math. 57 (1985), 261284], Lins and Mandel introduce a class of 3manifolds represented by 4coloured graphs S(b,l,t,c) depending on a 4tuple (b,l,t,c) of positive integers; moreover, they prove that, if the following conditions hold $(b,c)=1$, $(l,t)=1$, $c=(1)^t$ if l odd, then S(b,l,t,c) is a crystallization of an (orientable) 3manifold.In this paper we show that the above conditions are also necessary: hence, they characterize crystallizations among LinsMandel graphs.
1987
 Fundamental groups of branched covering spaces of S^3
[Articolo su rivista]
Casali, Maria Rita
abstract
Given a knot K in $S^3$, it is known a standard method (by Casali and Grasselli) for constructing a 4coloured graph representing the closed orientable 3manifold $M=M(K,d,\omega)$ which is the dfold covering space of $S^3$ branched over K and associated to the transitive drepresentation $\omega$ of the knot group. In this paper we obtain a presentation of the fundamental group of M, directly from the Wirtinger presentation of the knot group and from the transitive drepresentation $\omega$.
1987
 LE 3VARIETA' COME TRIPLI RIVESTIMENTI RAMIFICATI DI S^3
[Abstract in Atti di Convegno]
Casali, Maria Rita
abstract
E' nota la possibilità di rappresentare le nvarietà PL attraverso particolari grafi colorati sugli spigoli mediante n+1 colori, detti cristallizzazioni (si veda [M.Pezzana, Sulla struttura topologica delle varietà compatte, Atti Sem. Mat. Fis. Univ. Modena 23 (1974), 269277] o [M.Ferri  C.Gagliardi  L.Grasselli, A graphtheoretical representation of PLmanifolds. A survey on crystallizations, Aequationes Math. 31 (1986), 121141]). In questo lavoro si presenta un approccio, basato esclusivamente sulla teoria delle cristallizzazioni, alla dimostrazione del seguente teorema (originariamente provato da Hilden e Montesinos): Ogni 3varietà chiusa e orientabile M è triplo rivestimento semplice di $S^3$ ramificato su un nodo K.