Alberto CAVICCHIOLI - personale UniMoRe

Nuova ricerca

Alberto CAVICCHIOLI

Professore Ordinario
Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede ex-Matematica

Pubblicazioni

2024 - Exceptional Dehn Surgeries on Some Infinite Series of Hyperbolic Knots and Links [Articolo su rivista]
Cavicchioli, A.; Spaggiari, F.
abstract

We study closed connected orientable 3-manifolds obtained by Dehn surgery along the oriented components of a link, introduced and considered by Motegi and Song (2005) and Ichihara et al. (2008). For such manifolds, we find a finite balanced group presentation of the fundamental group and describe exceptional surgeries. This allows us to construct an infinite family of tunnel number one strongly invertible hyperbolic knots with three parameters, which admit toroidal surgeries and Seifert fibered surgeries. Among the obtained results, we mention that for every integer n>5 there are infinitely many hyperbolic knots in the 3-sphere, whose (n-2) and (n+1)-surgeries are toroidal, and (n-1) and n-surgeries are Seifert fibered.

2023 - Spines and surgery descriptions of graph manifolds [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

Graph manifolds are compact orientable 3--manifolds obtained by gluing several copies of $D^2 \times \mathbb S^1$ and $N^2 \times \mathbb S^1$ together by homeomorphisms of some components of their boundaries ($D^2$ is the 2--disc and $N^2$ denotes the 2--disc with two holes). Here we study spines and surgery representations of orientable graph manifolds, and derive geometric presentations of their fundamental group. Then we determine the homeomorphism type of many Takahashi manifolds and the Teragaito manifolds, showing that they are graph manifolds with specified invariants. Finally, we describe graph manifolds arising from toroidal surgeries on certain classes of hyperbolic knots.

2023 - The character variety of one relator groups [Articolo su rivista]
Cavicchioli, A.; Spaggiari, F.
abstract

We consider some families of one relator groups arising as fundamental groups of 3-dimensional manifolds, and calculate their character varieties in SL(2, C). Then we give simple geometrical descriptions of such varieties, and determine the number of their irreducible components. Our paper relates to the work of Baker-Petersen, Qazaqzeh and Morales-Marcen on the character variety of certain classes of one relator groups, but we use different methods based on the concept of palindrome presentations of given groups.

2022 - All Dehn Fillings of the Whitehead Link Complement are Tetrahedron Manifolds [Articolo su rivista]
Cavicchioli, A.; Spaggiari, F.
abstract

In this paper we show that Dehn surgeries on the oriented components of the Whitehead link yield tetrahedron manifolds of Heegaard genus ≤ 2. As a consequence, the eight homogeneous Thurston 3–geometries are realized by tetrahedron manifolds of Heegaard genus ≤ 2. The proof is based on techniques of Combinatorial Group Theory, and geometric presentations of manifold fundamental groups.

2022 - Fundamental class, Poincaré duality and finite oriented FC4-complexes [Articolo su rivista]
Cavicchioli, Alberto; Hegenbarth, Friedrich; Spaggiari, Fulvia
abstract

In this paper we continue our investigations of 4-dimensional complexes in [A. Cavicchioli, F.Hegenbarth, F. Spaggiari, Four-dimensional complexes with fundamental class,Mediterr. J. Math. 17 (2020), 175]. We study a class of finite oriented 4-complexes which we call FC4-complexes, defined as follows. An FC4-complex is a 4-dimensional finite oriented CW-complex X with a single 4-cell such that H4(X, ℤ) ≅ ℤ with a fundamental class [X] ∈ H4(X, ℤ). By well-known results ofWall, any Poincaré complex is of this type. We are interested in two questions. First, for which 3-complexes K does an element [φ] ∈ π3(K) exist such that K ∪φ D4 is a Poincaré complex? Second, if there exists one, how many others can be constructed from K? The latter question was addressed studied in the above cited previous paper of the authors. In the present paper we deal with the first problem, and give necessary and sufficient conditions on K and [φ] ∈ π3(K) to satisfy Poincaré duality with ℤ- and Λ-coefficients. Here Λ denotes the integral group ring of π1(K). Before, we give a classification of all FC4-complexes based on the finite 3-complex K, and make some remarks concerning ℤ- and Λ-Poincaré duality.

2021 - New formulas for the character varieties of two-bridge links [Articolo su rivista]
Cavicchioli, A.; Spaggiari, F.
abstract

We consider some families of two-bridge links, including the links b(2p; 3), the twisted Whitehead links and the double twist links, and calculate their character varieties. Then we give simple geometrical descriptions of such varieties, and determine the number of their irreducible components. Our paper relates to the nice work of Hilden, Lozano and Montesinos on the character variety of a class of 2-bridge links, but uses very dierent methods based on the concept of palindome presentations of link groups. The obtained formulas for the above character varieties are new, easily programmable, and more simple in many cases than the equations known in the literature.

2021 - On reduced complexity of closed piecewise linear 5-manifolds [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed 5-manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n-dimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 5-manifolds of reduced complexity less than or equal to 20. In particular, this gives a combinatorial characterization of S2×S3 among closed connected spin PL 5-manifolds.

2021 - Recognizing Euclidean Space Forms with Minimal Fundamental Tetrahedra [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We completely recognize the topological structure of the ten compact euclidean space forms with special minimal tetrahedra, constructed by face pairings in nice papers of Molnar. From these polyhedral descriptions we derive special presentations with two generators for the fundamental groups of the considered manifolds. Our proofs also show that such group presentations completely characterize the euclidean space forms among closed connected 3-manifolds. The results have also didactical importance

2020 - Four-Dimensional Complexes with Fundamental Class [Articolo su rivista]
Cavicchioli, A.; Hegenbarth, F.; Spaggiari, F.
abstract

This paper continues the study of 4-dimensional complexes from our previous work Cavicchioli et al. (Homol Homotopy Appl 18(2):267–281, 2016; Mediterr J Math 15(2):61, 2018. https://doi.org/10.1007/s00009-018-1102-3) on the computation of Poincaré duality cobordism groups, and Cavicchioli et al. (Turk J Math 38:535–557, 2014) on the homotopy classification of strongly minimal PD 4-complexes. More precisely, we introduce a new class of oriented four-dimensional complexes which have a “fundamental class”, but do not satisfy Poincaré duality in all dimensions. Such complexes with partial Poincaré duality properties, which we call SFC 4-complexes, are very interesting to study and can be classified, up to homotopy type. For this, we introduce the concept of resolution, which allows us to state a condition for a SFC 4-complex to be a PD 4-complex. Finally, we obtain a partial classification of SFC 4-complexes. A future goal will be a classification in terms of algebraic SFC 4-complexes similar to the very satisfactory classification result of PD 4-complexes obtained by Baues and Bleile (Algebraic Geom. Topol. 8:2355–2389, 2008).

2019 - On graph-theoretical invariants of combinatorial manifolds [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge--colored graphs, called {sl crystallizations}. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n-dimensional PL manifold. Here we consider this invariant, and extend in this context the concept of average order first introduced by Luo and Stong in 1993, and successively investigated by Tamura in 1996 and 1998. Then we obtain some classification results for closed connected smooth low-dimensional manifolds according to reduced complexity and average order. Finally, we answer to a question posed by Trout in 2013.

2019 - The character variety of some classes of rational knots [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We propose a method to determine the character variety of a class J(m,n) of rational knots, which includes the twist knots. The defining polynomials depend only on the variables m and n. This answers for these classes of knots a question posed in a paper of Hilden, Lozano and Montesinos, and allows us to give an easy geometrical description of the considered character variety. Our results are obtained by using special presentations of the knot groups whose relators are palindromes.

2018 - On four-dimensional Poincarè duality cobordism groups [Articolo su rivista]
Cavicchioli, A; Hegenbarth, F; Spaggiari, F
abstract

This paper continues the study of 4-dimensional Poincarè duality cobordism theory from our previous work "PD4-complexes: constructions, cobordisms and signatures", Homology, Homotopy and Applications 18(2) (2016), 267-281. Let P be an oriented finite Poincarè duality complex of dimension 4. Then we calculate the Poincarè duality cobordism group of P ΩPD4(P) . The main result is the existence of the exact sequence of Theorem 1.1. It turns out that ΩPD4(P) depends only on the fundamental group and the assembly map A4. This does not hold in higher dimensions. Then we discuss several examples. The cases in which the canonical map ΩTOP4(P) → ΩPD4(P) is not surjective are of particular interest. Its image coincides with the kernel of the total surgery obstruction map. In these cases there are PD4-complexes which cannot be homotopy equivalent to manifolds.

2016 - HIGHER-DIMENSIONAL GENERALIZED MANIFOLDS: SURGERY AND CONSTRUCTIONS [Monografia/Trattato scientifico]
Cavicchioli, Alberto; Hegenbarth, Friedrich; Repovs, Dusan
abstract

Higher-Dimensional Generalized Manifolds: Surgery and Constructions EMS Series of Lectures in Mathematics Alberto Cavicchioli (Università degli Studi di Modena e Reggio Emilia, Italy) Friedrich Hegenbarth (Università degli Studi di Milano, Italy) Dušan Repovš (University of Ljubljana, Slovenia) Higher-Dimensional Generalized Manifolds: Surgery and Constructions ISBN print 978-3-03719-156-9, ISBN online 978-3-03719-656-4 DOI 10.4171/156 May 2016, 154 pages, softcover, 17 x 24 cm. 32.00 Euro Generalized manifolds are a most fascinating subject to study. They were introduced in the 1930s, when topologists tried to detect topological manifolds among more general spaces (this is nowadays called the manifold recognition problem). As such, generalized manifolds have served to understand the nature of genuine manifolds. However, it soon became more important to study the category of generalized manifolds itself. A breakthrough was made in the 1990s, when several topologists discovered a systematic way of constructing higher-dimensional generalized manifolds, based on advanced surgery techniques. In fact, the development of controlled surgery theory and the study of generalized manifolds developed in parallel. In this process, earlier studies of geometric surgery turned out to be very helpful. Generalized manifolds will continue to be an attractive subject to study, for there remain several unsolved fundamental problems. Moreover, they hold promise for new research, e.g. for finding appropriate structures on these spaces which could bring to light geometric (or even analytic) aspects of higher-dimensional generalized manifolds. This is the first book to systematically collect the most important material on higher-dimensional generalized manifolds and controlled surgery. It is self-contained and its extensive list of references reflects the historic development. The book is based on our graduate courses and seminars, as well as our talks given at various meetings, and is suitable for advanced graduate students and researchers in algebraic and geometric topology. Keywords: Homology manifold, Poincaré duality, degree 1 normal map, boundedly controlled surgey, surgery spectrum, assembly map, Quinn index, Euclidean neighborhood retract, cell-like resolution, disjoint disks property, manifold recognition problem

2016 - On certain classes of closed 3-manifolds with different geometric structures [Capitolo/Saggio]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

In this note, we review some recent results concerning the topology and geometry of closed connected orientable 3-manifolds. The used techniques are based on various combinatorial representations of 3-manifolds, such as polyhedral schemes, Heegaard diagrams, branched coverings, and Dehn surgery.

2016 - PD_4-Complexes: constructions, cobordisms and signatures [Articolo su rivista]
Cavicchioli, Alberto; Hegenbarth, Friedrich; Spaggiari, Fulvia
abstract

The oriented topological cobordism group $\Omega_4 (P)$ of an oriented $\operatorname{PD}_4$--complex $P$ is isomorphic to $\Bbb Z \oplus \Bbb Z$. The invariants of an element $\{ f : X \to P \} \in \Omega_4 (P)$ are the signature of $X$ and the degree of $f$. We prove an analogous result for the Poincar\' e duality cobordism group $\Omega_{4}^{\operatorname{PD}} (P)$: If $\pi_1 (P)$ does not contain nontrivial elements of order $2$, then $\Omega_{4}^{\operatorname{PD}} (P)$ is isomorphic to $L^{0} (\Lambda) \oplus \Bbb Z$, where $L^{0} (\Lambda)$ is the Witt group of non-degenerated hermitian forms on finitely generated stably free $\Lambda$--modules. The component of an element $\{ f : X \to P \} \in \Omega_{4}^{\operatorname{PD}} (P)$ in $L^{0} (\Lambda)$ is related to the symmetric signature of $X$. Then we construct explicitly $\operatorname{PD}_4$--complexes, define the well--known map $L_4 (\pi_1 (P)) \to \Omega_{4}^{\operatorname{PD}} (P)$, and characterize the image of the map $\Omega_{4}^{\operatorname{PD}} (P) \to \Omega_{4}^{N} (P)$. The results are summarized in Theorems 1.1 and 1.2 stated in the introduction.

2016 - SOLIDI PLATONICI e ARCHIMEDEI: classificazione e costruzioni geometriche. Una proposta didattica basata sulla caratteristica di Eulero. [Articolo su rivista]
Cavicchioli, Alberto; Prandini, Monica
abstract

In this paper we present a simple proof of the classification theorem of Archimedean solids (known as Keplero’s theorem), which is based on the concept of Euler characteristic. It extends the classical proof of the classification theorem of Platonic solids. Then we describe simple geometric constructions to obtain all Archimedean solids starting from Platonic ones. A discussion on Johnson solids completes the paper. The proposed techniques and methods are very elementary and can be included in different topics studied by high school students.

2014 - GEOMETRIA delle CURVE [Monografia/Trattato scientifico]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

Questo volume raccoglie i contenuti di varie lezioni che gli autori impartiscono nei corsi di Geometria per i Corsi di Laurea in Matematica, Fisica ed Ingegneria dell' Universita` di Modena e Reggio E. Lo scopo principale del testo e` quello di fornire le conoscenze di base, i metodi e le tecniche operative della Geometria Differenziale e Algebrica delle Curve immerse nel piano e nello spazio euclideo. Inoltre, si sviluppa la teoria classica delle equazioni algebriche e si illustra, in modo approfondito, la Geometria delle Curve Algebriche Piane. In particolare, nell'ultimo capitolo si espone la teoria delle Cubiche Piane.

2014 - ON MINIMAL POINCARE` 4-COMPLEXES [Articolo su rivista]
Cavicchioli, Alberto; Friedrich, Hegenbarth; Dusan, Repovs
abstract

We consider 2 types of minimal Poincare` 4-complexes. One is defined with respect to the degree 1-map order. This idea was already present in our previous papers, and more systematically studied later by Hillman. The second type of minimal Poincare` 4-complexes was introduced by Hambleton, Kreck, and Teichner. It is not based on an order relation. In the present paper we study existence and uniqueness questions.

2014 - Some tetrahedron manifolds with Sol geometry and related groups [Articolo su rivista]
Cavicchioli, Alberto; E., Molnar; Spaggiari, Fulvia; J., Szirmai
abstract

We study a series of 2-generator Sol-manifolds depending on a positive integer n, introduced by Molnar and Szirmai. We construct them as tetrahedron manifolds and show that they are twofold coverings of the 3-sphere branched over specified links. Finally, we give a surgery description of the considered 3-manifolds; indeed, they can be obtained by n−2 and 0 Dehn surgeries along the components of the Whitehead link.

2013 - CUSPED HYPERBOLIC 3-MANIFOLDS FROM SOME REGULAR POLYHEDRA [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia; Telloni, Agnese Ilaria
abstract

We illustrate some topological properties and give surgery descriptions of the cusped hyperbolic orientable 3–manifolds obtained by face pairings of the regular octahedron and the regular cube. Some applications and connections with the work of several authors (quoted in the references) complete the paper.

2013 - CYCLIC BRANCHED COVERINGS OF SOME PRETZEL LINKS [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We construct infinite families of closed connected orientable 3-manifolds obtained from certain triangulated 3-cells by pairwise identifications of their boundary faces. Our combinatorial constructions extend and complete a particular polyhedral scheme which Kim and Kostrikin used in 1995 and 1997 to define a series of spaces denoted M_3(n). Then we determine geometric presentations of the fundamental groups, and prove that many of the constructed manifolds are n-fold (non-strongly) cyclic coverings of the 3-sphere branched over some specified pretzel links.

2013 - Fundamental Group and Covering Properties of Hyperbolic Surgery Manifolds [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia; Telloni, Agnese Ilaria
abstract

We study a family of closed connected orientable 3-manifolds obtained by Dehn surgeries with rational coefficients along the oriented components of certain links. This family contains all the manifolds obtained by surgery along the (hyperbolic) 2-bridge knots. We find geometric presentations for the fundamental group of such manifolds and represent them as branched covering spaces. As a consequence, we prove that the surgery manifolds, arising from the hyperbolic 2-bridge knots, have Heegaard genus 2 and are 2-fold coverings of the 3-sphere branched over well-specified links.

2013 - Knots with the Lens Space Surgery [Articolo su rivista]
Cavicchioli, Alberto; I., Telloni
abstract

In this paper we construct an infinite family of hyperbolic (1, 1)–knots with two parameters, and show that some of them admit exceptional Dehn surgery such as lens space surgery, Seifert surgery, and toroidal surgery. Furthermore, we give simple examples of hyperbolic (1, 1)–knots which admit two toroidal surgeries at distance four such that both toroidal surgeries do not create Klein bottles.

2013 - On the Surgery Theory for Filtered Manifolds [Articolo su rivista]
Cavicchioli, Alberto; Hegenbarth, Friedrich; Yuri, Muranov; Spaggiari, Fulvia
abstract

In this paper we describe some relations between various structure sets which arise naturally for a Browder-Livesay filtration of a closed topological manifold. We use the algebraic surgery theory of Ranicki for realizing the surgery groups and natural maps on the spectrum level. We obtain also new relations between Browder-Quinn surgery obstruction groups and structure sets. Finally we illustrate several examples and applications.

2013 - SOME HYPERBOLIC SPACE FORMS WITH FEW GENERATED FUNDAMENTAL GROUPS [Articolo su rivista]
Cavicchioli, Alberto; Emil, Molnar; Agnese I., Telloni
abstract

We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane reflections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.

2012 - ON SOME CLASSES OF HYPERBOLIC KNOTS WITH THE 3-SPHERE SURGERY [Articolo su rivista]
Cavicchioli, Alberto; T., Saito
abstract

For the distance of (1, 1)-splittings of a knot in a closed orientable 3-manifold, it is an important problem whether a (1, 1)-knot can admit (1, 1)-splittings of different distances.In this paper, we give one-parameter families of hyperbolic (1, 1)-knots such that each (1, 1)-knot admits a Dehn surgery yielding the 3-sphere. It is remarkable that such knots are the first concrete examples each of whose (1, 1)-splittings is of distance three.

2012 - The Combinatorics of Piecewise Linear Manifolds by Colored Graphs [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

Crystallization theory is a combinatorial representation of piecewiselinear (closed connected) manifolds of arbitrary dimension. This theory differs from the most important representation methods for triangulated manifolds as for example Heegaard splittings, standard spines, surgery along framed links and branched coverings, which work well in dimension less than or equal four. Crystallizations form a particular class of edge-colored multigraphs arising from combinatorial triangulations of manifolds which are minimal with respect to the number of vertices. Classical results and techniques on crystallizations are reviewed from a graph-theoretical point of view, especially to pay attention to certain new combinatorial invariants as regular genus, complexity and average order. These invariants are shown to be related with the topology of manifolds. Several open problems and conjectures concerning them complete the survey paper.Mathematics Subject Classification: 57M15, 57Q15, 05C10

2011 - DEHN SURGERIES ON SOME CLASSICAL LINKS [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia; Telloni, Agnese Ilaria
abstract

We consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. Wefind geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given byexceptional surgeries on hyperbolic 2-bridge knots.

2011 - EXCEPTIONAL SURGERIES ON CERTAIN (1,1)-KNOTS [Articolo su rivista]
Cavicchioli, Alberto; Telloni, Agnese Ilaria
abstract

We determine certain exceptional surgeries on a 3--parametric family of hyperbolic 1--bridge genus one knots ((1,1)-knots, in short). In particular, we show that such knots admit two infinite series of lens space surgeries. Our work is related to a nice paper of Teragaito [16], since we represent his toroidal manifolds as 2--fold coverings of the 3-sphere branched over well--specified links.

2011 - TETRAHEDRON MANIFOLD SERIES OF HEEGAARD GENUS TWO WITH KNOT PRESENTATION AND DEHN SURGERY [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We construct a 2-parametric family of tetrahedron manifolds, mainly with hyperbolic structure. We give geometric presentations of their fundamental groups with two generators. Furthermore, we show that the constructed manifolds are 2-fold coverings of the 3-sphere branched over specified knots, and give a surgery description of them.

2010 - TOPOLOGY OF COMPACT SPACE FORMS FROM PLATONIC SOLIDS. II [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia; Telloni, Agnese Ilaria
abstract

The problem of classifying, up to isometry, the orientable 3-manifolds that arise by identifying the faces of a Platonic solid was completely solved in a nice paper of Everitt [B. Everitt, 3-manifolds from Platonic solids, Topology Appl. 138 (2004) 253–263]. His work completes the classification begun by Best [L.A. Best, On torsion-free discrete subgroups of PSL2(C) with compact orbit space, Canad. J. Math. 23 (1971) 451–460], Lorimer [P.J. Lorimer, Four dodecahedral spaces, Pacific J. Math. 156 (2) (1992) 329–335], Prok [I. Prok, Classification of dodecahedral space forms, Beiträge Algebra Geom. 39 (2) (1998) 497–515; I. Prok, Fundamental tilings with marked cubes in spaces of constantcurvature, Acta Math. Hungar. 71 (1–2) (1996) 1–14], and Richardson and Rubinstein [J. Richardson, J.H. Rubinstein, Hyperbolic manifolds from a regular polyhedron, preprint].In a previous paper we investigated the topology of closed orientable 3-manifolds from Platonic solids in the spherical and Euclidean cases, and completely classified them, upto homeomorphism. Here we describe many topological properties of closed hyperbolic 3-manifolds arising from Platonic solids. As a consequence of our geometric and topologicalmethods, we improve the distinction between the hyperbolic “Platonic” manifolds with the same homology, which up to this point was only known by computational means.

2009 - ASSEMBLY MAPS AND REALIZATION OF SPLITTING OBSTRUCTIONS [Articolo su rivista]
Cavicchioli, Alberto; Y. V., Muranov; Spaggiari, Fulvia
abstract

In 1987 Kharshiladze introduced the concept of type for an element in a Wall group, and proved that the elements of first and second type cannot be realized by normal maps of closed manifolds. This approach is sufficiently easy for computingthe assembly maps and sometimes very effective. Here we give a geometrical interpretation of this approach by using the Browder–Quinn surgery obstruction groups for filtered manifolds. To understand the obtained relations we give algebraic definitionsof the element types which are based on the algebraic surgery theory of Ranicki. Our approach describes a level of indeterminate for the algebraic passing to surgery ona codimension k submanifold of a given Browder–Livesay filtration. Then we study the realization of splitting obstructions by simple homotopy equivalences of closed manifolds, and compute the assembly maps for some classes of groups.

2009 - ON FOOTBALL MANIFOLDS OF E. MOLNAR [Articolo su rivista]
Cavicchioli, Alberto; Telloni, Agnese Ilaria
abstract

A closed 3-manifold M is said to be hyperelliptic if it has an involution \tau such that the quotient space of M by the action of \tau is homeomorphic to the standard 3-sphere. We show that the hyperbolic football manifolds of Emil Molnár (1988) are hyperelliptic. Then we determine the isometry groups of suchmanifolds. Another consequence is that the unique hyperbolic dodecahedral and icosahedral 3-space forms with first homology group Z_35 (constructed by I. Prok in 1998, on the basis of a principal algorithm due to Emil Molnár, and byRichardson and Rubinstein in 1982) are also hyperelliptic.

2009 - On iterated Browder-Livesay invariants [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Muranov, Y. u. V.; Spaggiari, Fulvia
abstract

The Browder-Livesay invariants provide obstructions to the realization of elements of Wall groups by normal maps of closed manifolds. A generalization of the iterated Browder-Livesay invariants is proposed and properties of the invariants obtained are described. The generalized definition makes it possible to investigate the relationship between a normal map and its restriction to a submanifold and clarifies the relationship between the Browder-Livesay invariants and the Browder-Quinn groups of obstructions to surgery on filtered manifolds. Several theorems describing a relationship between a normal map and its restriction to a submanifold are proved.

2009 - PALINDROME PRESENTATIONS OF RATIONAL KNOTS [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia; D., Repovs
abstract

We give explicit palindrome presentations of the groups of rational knots, that is, presentations with relators which read the same forwards and backwards. This answers a question posed by Hilden, Tejada and Toro in 2002. Using such presentations we obtain simple alternative proofs of some classical results concerning the Alexander polynomial of all rational knots and the character variety of certain rational knots. Finally, we derive a new recursive description of the SL(2,C) character variety of twist knots.

2009 - SOME SERIES OF HONEY-COMB SPACES [Articolo su rivista]
E., Barbieri; Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We study the topology and geometry of some series of closed connected orientable 3-manifolds constructed as honey-comb spaces. These manifolds are quotients of certain polyhedral 3-cells by pairwise identification of their boundary faces. We determine geometric presentations of the fundamental group and study the split extension of it. Then we describe geometric structures, homeomorphism type and covering properties of our manifolds which are shown to be cyclic coverings of the 3-sphere branched over known links with two components. Finally, we answer open questions on certain manifolds, defined by Kim and Kostrikin, and give a complete classification of them.

2009 - SURGERY ON PAIRS OF CLOSED MANIFOLDS [Articolo su rivista]
Cavicchioli, Alberto; Y. V., Muranov; Spaggiari, Fulvia
abstract

To apply surgery theory to the problem of classifying pairs of closed manifolds, it is necessary to know the subgroup of the group LP_∗ generated by those elements which are realized by normal maps to a pair of closed manifolds. This closely relates to the surgery problem for a closed manifold and to the computation of the assembly map. In this paper we completely determine such subgroups for many cases of Browder-Livesay pairs of closed manifolds. Moreover, very explicit results are obtained in the case of an elementary fundamental group. Then we generalize them, and obtain several further results about the realization of elements in the Browder-Quinn surgery obstruction groups by means of normal maps to a closed manifold filtered by closed submanifolds.

2009 - Topology of compact space forms from Platonic solids. I. [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia; Telloni, Agnese Ilaria
abstract

The problem of classifying, up to isometry, the orientable 3-manifolds that arise by identifying the faces of a Platonic solid was completely solved in a nice paper of Everitt [B. Everitt, 3-manifolds from Platonic solids, Topology Appl. 138 (2004) 253–263]. His work completes the classification begun by Best [L.A. Best, On torsion-free discrete subgroups of PSL_2(C) with compact orbit space, Canad. J. Math. 23 (1971) 451–460],Lorimer [P.J. Lorimer, Four dodecahedral spaces, Pacific J. Math. 156 (2) (1992) 329–335], Prok [I. Prok, Classification of dodecahedral space forms, Beiträge Algebra Geom. 39 (2)(1998) 497–515], and Richardson and Rubinstein [J. Richardson, J.H. Rubinstein, Hyperbolic manifolds from a regular polyhedron, Preprint]. In this paper we investigate the topology of closed orientable 3-manifolds from Platonic solids. Here we completely recognize those manifolds in the spherical and Euclidean cases, and state topological properties for many of them in the hyperbolic case. The proofs of the latter will appear in a forthcoming paper.

2008 - A result in surgery theory [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We study the topological 4-dimensional surgery problem for a closed connected orientable topological 4-manifold with vanishing second homotopy and whose fundamental group is isomorphic to a free product of an one-ended group with a non-trivial free group. Our result is related to a theorem of Krushkal and Lee.

2008 - CLASSIFYING COMBINATORIAL 4-MANIFOLDS UP TO COMPLEXITY [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

The goal of this paper is to give some theorems which relate tothe problem of classifying smooth 4{manifolds up to piecewise -linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge{colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reducedcomplexity, for any closed n{dimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 4-manifolds of reduced complexity less than or equal to 14.

2008 - Classifying combinatorial 4-manifolds up complexity [Articolo su rivista]
Cavicchioli, A.; Spaggiari, F.
abstract

The goal of this paper is to give some theorems which relate to the problem of classifying smooth 4-manifolds up to piecewise -linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed n-dimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 4-manifolds of reduced complexity less than or equal to 14.

2008 - Graphs encoding 3-manifolds of genus 2 [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We give a simple alternative proof of the representation theorem of all genus two 3-manifolds by a 6-parameter family of integers, due to casali and Grasselli (Discrete Math. 87 (1991)). Our approach is different from that of the quoted paper, and it is based on the concept of extended Heegaard diagram. This permits to obtain new topological meanings of the arithmetic conditions which the parameters must satisfy for representing closed manifolds. Applications on homology spheres of genus 2 complete the paper.

2008 - ON SOME QUESTIONS ABOUT A FAMILY OF CYCLICALLY PRESENTED GROUPS [Articolo su rivista]
Cavicchioli, Alberto; E. A., O'Brien; Spaggiari, Fulvia
abstract

We study various questions about the generalised Fibonacci groups, a family of cyclically presented groups, which includes as special cases the Fibonacci, Sieradski, and Gilbert and Howie groups.

2008 - TRIGONOMETRIA EUCLIDEA, SFERICA e IPERBOLICA [Articolo su rivista]
Cavicchioli, Alberto
abstract

Questa nota di tipo didattico e divulgativo presenta in modo semplice ed unitario le formule classiche di trigonometria (euclidea, sferica ed iperbolica) ottenute, come caso particolare, dallo studio della trigonometria su una superficie differenziabile (di classe C^{\infty}, per comodita` di trattazione) con curvatura gaussiana costante.

2008 - Universal presentations for manifold groups [Articolo su rivista]
E., Barbieri; Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We construct a universal presentation for the fundamental group of a closed connected orientable 3-manifold. This result may be of considerable interest not only for topologists but also for experts in combinatorial group theory. Some applications on periodic manifolds and geometric presentations complete the paper.

2007 - A generalization of Helling-Kim-Mennicke groups and manifolds [Articolo su rivista]
E., Barbieri; Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We construct an infinite family of closed connected orientable 3-manifolds by pairwise identifications of faces in the boundary of certain polyhedral 3-cells. We determine geometric presentations of the fundamental group, and study the split extension of it. Then we prove that these manifolds are n-fold cyclic coverings of the 3-sphere branched over some pretzel links.

2007 - On the genus of real projective spaces [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We construct a crystallization of the real projective n-space whose associated contracted complex is minimal with respect to the number of n-simplexes. Then we compute the regular genus of such a space, which is the minimum genus of a closed connected surface into which a crystallization of it regularly embeds.

2006 - A note on irreducible Heegaard diagrams [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

AbstractWe construct a Heegaard diagram of genus three for the real projective 3-space, which has no waves and pairs of complementary handles. The first example was given by Im and Kim but our diagram has smaller complexity. Furthermore the proof presented here is quite different to that of the quoted authors, and permits also to obtain a simple alternative proof of their result. Examples of irreducible Heegaard diagrams of certain connected sums complete the paper.

2006 - A topological study of some groups arising from cellular quotients [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia; Mo, Wang
abstract

Kim and Kostrikin constructed in Sbornik Math. 188 (1997) a tessellation on the boundary of a polyhedral 3-cell consisting of 8n pentagons. The tessellation produces a family of closed connected orientable 3-manifolds with spines corresponding to certain presentations of their fundamental groups. We investigate the topological and algebraic properties of such groups together with their derived quotients and split extensions, and completely classify the considered manifolds.

2006 - Certain cyclically presented groups with the same polynomial [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We consider three infinite families of cyclic presentations of groups, depending on a finite set of integers and having the same polynomial. Then we prove that the corresponding groups with the same parameters are isomorphic, and that the groups are almost all infinite. Finally, we completely compute the maximal Abelian quotients of such groups, and show that their HNN extensions are high-dimensional knot groups. Our results contain as particular cases the main theorems obtained in two nice articles: Johnson et al. (1999) and Havas et al. (2001).

2006 - Dehn surgeries on periodic links [Articolo su rivista]
E., Barbieri; Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We consider orientable closed connected 3-manifolds obtained by Dehn surgeries with rational coefficients along the components of certain periodic links. These manifolds extend many classes of (hyperbolic) manifolds considered by several authors (see the references). We find geometric presentations of the fundamental group of such manifolds, and study some covering properties of them. Then we obtain results on their geometric structures in many cases.

2006 - Manifolds with poly-surface fundamental groups [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Spaggiari, Fulvia
abstract

We study closed topological 2n-dimensional manifolds M with poly-surface fundamental groups. We prove that if M is simple homotopy equivalent to the total space E of a Y-bundle over a closed aspherical surface, where Y is a closed aspherical n-manifold, then M is s-cobordant to E. This extends a well-known 4-dimensional result of Hillman (1991) to higher dimensions. Our proof is different from that of the quoted paper: we use Mayer-Vietoris techniques and the properties of the L-theory assembly maps for such bundles.

2006 - Mixed structures on a manifold with boundary [Articolo su rivista]
Cavicchioli, Alberto; Y. V., Muranov; Spaggiari, Fulvia
abstract

For a closed topological n-manifold X, the surgery exact sequence contains the set of manifold structures and the set of tangential structures of X. In the case of a compact topological n-manifold with boundary (X, \partial X), the classical surgery theory usually considers two different types of structures. The first one concerns structures whose restrictions are fixed on the boundary. The second one uses two similar structures on the manifold pair. In this paper we introduce new mixed structures on a topological manifold with boundary, and describe their properties. Then we obtain connections between these structures and the classical ones, and prove that they fit in some surgery exact sequences. Finally, we discuss several geometric examples.

2006 - On the classification of Kim and Kostrikin manifolds [Articolo su rivista]
Cavicchioli, Alberto; L., Paoluzzi; Spaggiari, Fulvia
abstract

We completely classify the topological and geometric structures of some series of closed connected orientable 3-manifolds introduced by Kim and Kostrikin in Sbornik Math. 188 (1997) as quotient spaces of certain polyhedral 3-cells by pairwise identifications of their boundary faces. Then we study further classes of closed orientable 3-manifolds arising from similar polyhedral schemata, and describe their topological properties.

2006 - On the elements of the second type in surgery groups [Monografia/Trattato scientifico]
Cavicchioli, Alberto; Y. V., Muranov; Spaggiari, Fulvia
abstract

In 1987 Kharshiladze introduced the concept of type for an element in a Wall group, and proved that the elements of the first and second type cannot be realized by normal maps of closed manifolds. In the present paper we give a geometrical interpretation of this approach by using the Browder-Quinn surgery obstruction groups for filtered manifolds. Then we study some algebraic and geometrical properties of the elements of the second type, and apply the obtained results for computing the assembly map for some classes of groups. Further applications about the realization problem of the surgery and splitting obstructions complete the paper.

2006 - Remarks on a paper of M. Ochiai [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

This note is related to a nice short paper of M. Ochiai. We prove in a very fast way that the two-parameter family of Heegaard diagrams, constructed by Ochiai, encodes the genuine three-sphere. The result is obtained, up to isotopy, by using a sequence of only three moves in this order: a Whitehead-Zieschang reduction, a band sum and a cancellation of a handle.

2006 - Topology of four-manifolds with special homotopy groups [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We study the homotopy type and the s-cobordism class of a closed connected topological 4-manifold with vanishing second homotopy group. Our results are related to problem 4.53 of Kirby in Geometric Topology, Studies in Advanced Math. 2 (1997), and give a partial answer to a question stated by Hillman in Bull. London Math. Soc. 27 (1995)

2005 - Families of group presentations related to topology [Articolo su rivista]
Cavicchioli, Alberto; D., Repovs; Spaggiari, Fulvia
abstract

We study some algebraic properties of a class of group presentations depending on a finite number of integer parameters. This class contains many well-known groups which are interesting from a topological point of view. We find arithmetic conditions on the parameters under which the considered groups cannot be fundamental groups of hyperbolic 3-manifolds of finite volume. Then we investigate the asphericity for many presentations contained in our family.

2005 - Geometric topology of generalized 3-manifolds [Articolo su rivista]
Cavicchioli, Alberto; Repovs, D; Thickstun, T. L.
abstract

English translation in Journal of Mathematical Sciences 144 (5) (2007), 4413-4422We describe the history and the present status of one of the main classical problems in low-dimensional geometric topology: the recognition of topological 3-manifolds in the class of all generalized 3-manifolds (that is, ANR homology 3-manifolds). This problem naturally splits into the cell-like resolution problem for 3-manifolds by means of homology 3-manifolds and the general-position problem for topological 3-manifolds. we have also included some open problems.

2005 - Relative groups in surgery theory [Articolo su rivista]
Cavicchioli, Alberto; Yv, Muranov; Spaggiari, Fulvia
abstract

In this paper we consider various types of relative groups which naturally arise in surgery theory, and describe algebraic properties of them. Then we apply the obtained results to investigate the splitting obstruction groups LS* and the surgery obstruction groups LP* for a manifold pair. Finally, we introduce the lower LS*- and LP*-groups, and describe connections between them and the corresponding lower L-*-groups and surgery exact sequence.

2004 - Secondo modulo di geometria [Monografia/Trattato scientifico]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

Questo volume raccoglie le lezioni che gli autori impartiscono nel secondo modulo di Geometria per i Corsi di Laurea (di primo livello) in Matematica, Fisica ed Ingegneria dell'Universita` di Modena e Reggio Emilia. Lo scopo principale del testo e` quello di fornire le basi tecniche e gli strumenti algebrici necessari per comprendere e affrontare problemi tipici di Algebra Lineare (con particolare riguardo alla teoria degli autovalori e delle forme quadratiche), di Geometria Proiettiva e di Geometria delle Coniche e delle Quadriche (e piu` in generale delle Iperquadriche).

2004 - Varieties of Fibonacci type [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We study algebraic systems, or briefly algebras, which are groups with an additional unary operation satisfying certain laws. These laws are directly suggested by the relators of a new family of cyclically presented groups depending on four positive integers, which is defined and studied in the paper.

2003 - Decomposing four-manifolds up to homotopy type [Articolo su rivista]
Cavicchioli, Alberto; Ruini, Beatrice; Spaggiari, Fulvia
abstract

Let M be a closed connected oriented topological 4-manifold. Suppose that there is a degree one map f from M to another closed topological 4-manifold P, which induces an isomorphism between the fundamental groups. We give a homological condition on the integer intersection form of M and on the intersection form of M over the integral group ring of its fundamental group under which M is homotopy equivalent to a connected sum of P with a simply-connected closed topological 4-manifold M'. This gives a partial solution to a conjecture of Hillman stated in Bull. London Math. Soc. 27 (1995). Then some splitting results for closed 4-manifolds with special homotopy complete the paper.

2003 - On a certain surgery spectral sequence [Articolo su rivista]
Cavicchioli, Alberto; Y. V., Muranov; D., Repovs
abstract

Given a morphism between twisted quadratic extensions of antistructures, we construct a spectral sequence which generalizes, in a natural way, the surgery spectral sequence introduced by Hambleton and Kharshiladze in Mat. sbornik 183 (1992). Our spectral sequence allows us to obtain some additional information about the Browder-Livesay invariants. Then we study the relations between the above-mentioned spectral sequences, and discuss some examples and applications.

2003 - On the construction of 4k-dimensional generalized manifolds [Relazione in Atti di Convegno]
Cavicchioli, Alberto; F., Hegenbarth; D., Repovs
abstract

We construct 4k-dimensional generalized manifolds, for k greater than 1, which have no resolutions. The construction proceeds as in a paper of Bryant, Ferry, Mio and Weinberger, Ann. of Math. 143 (1996), but does not use their controlled (\epsilon, \delta)-surgery sequence.

2003 - Seifert hyperelliptic manifolds [Articolo su rivista]
E., Barbieri; Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We study some series of finite presentations of groups and fibered closed 3-manifolds obtained by side pairings of boundary faces on certain symmetric polyhedral 3-balls. These spaces cyclically extend some classical examples, and contain a family of manifolds constructed by Kim and Kostrikin in Sbornik Math. 188 (1997). Then we describe some covering properties of our manifolds, and show that many of them are hyperelliptic. Finally, we completely classify Kim-Kostrikin manifolds, and determine their Seifert invariants.

2003 - Special classes of snarks [Articolo su rivista]
Cavicchioli, Alberto; Te, Murgolo; Ruini, Beatrice; Spaggiari, Fulvia
abstract

We report the most relevant results on the classification, up to isomorphism, of nontrivial simple uncolorable (i.e., the chromatic index equals 4) cubic graphs, called snarks in the literature. Then we study many classes of snarks satisfying certain additional conditions, and investigate the relationships among them. Finally, we discuss connections between the snark family and some significant conjectures of graph theory, and list some problems and open questions which arise naturally in this research.

2003 - Topological properties of cyclically presented groups [Articolo su rivista]
Cavicchioli, Alberto; D., Repov; Spaggiari, Fulvia
abstract

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3-dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3-orbifolds and high-dimensional knots, respectively.

2002 - Embedding 4-manifolds with vanishing second homology [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Spaggiari, Fulvia
abstract

In this paper we study algebraic and geometric properties of closed oriented smooth 4-manifolds M with trivial H_2(M; Z). Moreover, we investigate the problem of embedding M in 5-space or other standard simply-connected 5-manifolds according to Barden's list [Ann. of Math. 82 (1965) 365-385]. These results are related with papers [Invent. Math. 77 (1984) 173-184; Topology 23 (1984) 257-269] of Cochran.

2002 - Primo modulo di geometria [Monografia/Trattato scientifico]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

Questo volume raccoglie le lezioni che gli autori impartiscono nel primo modulo di Geometria per i Corsi di Laurea (di primo livello) in Matematica, Fisica ed Ingegneria dell'Universita` di Modena e Reggio Emilia. Lo scopo principale del testo e` quello di fornire le basi tecniche e gli strumenti algebrici necessari per comprendere e risolvere i piu` svariati problemi di Algebra Lineare e di Geometria Euclidea.

2001 - Algebraic properties of decorated splitting obstruction groups [Articolo su rivista]
Cavicchioli, Alberto; Muranov, Yv; Repovs, D.
abstract

We review the basic concepts and properties of decorated obstruction groups of types LS and LP. Then we establish new relations among these groups and describe the properties of natural maps between different decorated groups. We construct several spectral sequences which contain these decorated groups, and study their connections with the spectral sequences in K-theory for certain quadratic extensions of antistructures. Finally, we introduce the concept of geometric diagram of groups and calculate explicitly the obstruction groups for a diagram formed by finite 2-groups.

2001 - Algebraic properties of surgery obstruction groups [Articolo su rivista]
Cavicchioli, Alberto; Muranov, Y. V.; Repovs, D.
abstract

In questo articolo si riassumono le definizioni e le principali proprieta`dei gruppi di ostruzione con decorazione di tipo LS e LP. Si stabiliscono nuove relazioni fra questi gruppi e si descrivono le proprieta`delle mappe naturali fra differenti gruppi con decorazione. Si costruiscono varie successioni spettrali, contenenti questi gruppi con decorazione, e si studiano la loro connessione con le successioni spettrali in K-teoria per certe estensioni quadratiche di antistrutture. Infine, si introduce il concetto di diagramma geometrico di gruppi e si calcolano esplicitamente i gruppi di ostruzione per un diagramma formato da 2-gruppi finiti.

2001 - Families of structures on spherical fibrations [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth
abstract

Let SF(n) be the usual monoid of orientation- and base point-preserving self-equivalences of the n-sphere S-n. If Y is a (right) SF(n)-space, one can construct a classifying space B(Y, SF(n), *)=B_n for S^n-fibrations with Y-structure, by making use of the two-sided bar construction. Let k: B_n--> BSF(n) be the forgetful map. A Y-structure on a spherical fibration corresponds to a lifting of the classifying map into B_n. Let K_i=K(Z_2, i) be the Eilenberg-Mac Lane space of type (Z_2, i). In this paper we study families of structures on a given spherical fibration. In particular, we construct a universal family of Y-structures, where Y=W_n is a space homotopy equivalent to \Pi _(i greater than or equal to1) K_i. Applying results due to Booth, Heath, Morgan and Piccinini, we prove that the universal family is a spherical fibration over the space map{B_n, B_n} x B_n. Furthermore, we point out the significance of this space for secondary characteristic classes. Finally, we calculate the cohomology of B_n.

2001 - On a conjecture of M. J. Dunwoody [Articolo su rivista]
Cavicchioli, Alberto; Ruini, Beatrice; Spaggiari, Fulvia
abstract

We deal with three combinatorial representations of closed orientable 3-manifolds, i.e., Heegaard diagrams, branched coverings, and crystallizations (a special class of pseudo-graphs endowed with proper edge-colorings). Exploring the connections between those theories, we prove the validity of a conjecture, stated by Dunwoody in Groups-Korea 1994, Walter de Gruyter, 1995, concerning the class of closed orientable 3-manifolds represented by symmetric Heegaard diagrams. As a consequence, we classify the topological and geometric structures of many interesting classes of cyclic branched coverings of (hyperbolic) knots encoded by cyclic presentations of groups. In all cases, we show that the polynomial associated with the cyclic presentation coincides (up to a multiplicative unit) with the Alexander polynomial of the considered knot. Finally, we include a partial output of a computer program which generates symmetric Heegaard diagrams of cyclic branched coverings of 3-bridge knots up to nine crossings.

2001 - On the homotopy type of Poincare` spaces [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We study the homotopy type of finite-oriented Poincare' spaces (and, in particular, of closed topological manifolds) in even dimension. Our results relate polarized homotopy types over a stage of the Postnikov tower with the concept of CW-tower of categories due to Baues. This fact allows us to obtain a new formula for the top-dimensional obstruction for extending maps to homotopy equivalences. Then we complete the paper with an algebraic characterization of high-dimensional handlebodies

2001 - Recent results on the topology of three-manifolds [Articolo su rivista]
Cavicchioli, Alberto; D., Repovs; A., Vesnin
abstract

We present several results on the classification of the topological and geometrical structures of closed 3-manifolds. In particular, we describe many important families of hyperbolic 3-manifolds and obtain combinatorial representations of them as branched coverings of the 3-sphere and by means of Dehn surgeries along knots and links. Finally, we determine the isometry groups, the volumes and the main algebraic invariants of the considered manifolds.

2001 - Special classes of closed four-manifolds [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; D., Repovs
abstract

We present several results and state some open problems on the classification of topological and geometric structures of closed connected oriented (smooth) four-manifolds. In particular, we discuss many interesting classes of closed four-manifolds satisfying additional properties, that is, spin manifolds, manifolds with special homology (resp. homotopy), exact manifolds, geometric manifolds, and smooth manifolds. The results are obtained by using methods and techniques from algebraic and differential topology, and homological algebra.

2000 - An extension of the Bolsinov-Fomenko theorem on orbital classification of integrable Hamiltonian systems [Articolo su rivista]
Cavicchioli, Alberto; D., Repovs; Ab, Skopenkov
abstract

The main result of the paper is an extension of the Bolsinov-Fomenko theorem on topological orbital classification of nondegenerate integrable Hamiltonian systems with two degrees of freedom on three-dimensional constant energy manifolds (1994). Namely, it is shown that their restriction that the integral has no critical circles with nonorientable separatrix diagrams can be omitted. Our proof is based on an analogue of obstruction theory for certain types of Seifert fibrations.

2000 - Omotopia, Cobordismo e Decomposizioni di 4-Varieta` [Capitolo/Saggio]
Cavicchioli, Alberto
abstract

Si espone una panoramica sui risultati piu` importanti relativi alla classificazione del tipo di omotopia e del tipo topologico e sulla determinazione della classe di s-cobordismo di molte classi interessanti di varieta` 4-dimensionali, chiuse e connesse TOP (oppure PL = DIFF). Si usano metodi e tecniche di Topologia Algebrica e Differenziale come, ad esempio, la teoria dell'ostruzione, omologia a coefficienti locali, forme di intersezione, k-invarianti, sistemi di Postnikov, successione di chirurgia, L-gruppi, spazi di moduli e polinomi di Donaldson.

2000 - On certain classes of hyperbolic 3-manifolds [Articolo su rivista]
Cavicchioli, Alberto; L., Paoluzzi
abstract

We study the topological structure and the homeomorphism problem for closed 3-manifolds M(n, k) obtained by pairwise identifications of faces in the boundary of certain polyhedral 3-balls. We prove that they are (n/d)-fold cyclic coverings of the 3-sphere branched over certain hyperbolic links of d + 1 components, where d = (n, k). Then we study the closed 3-manifolds obtained by Dehn surgeries on the components of these links.

2000 - On realization of splitting obstructions in Browder-Livesay groups for closed manifold pairs [Articolo su rivista]
Pm, Akhmetiev; Cavicchioli, Alberto; D., Repovs
abstract

We consider some new types of realization problem for obstructions in the Browder-Livesay groups by homotopy equivalences of closed manifold pairs. We give several examples of calculations. We also consider relations with classical surgery problems.

2000 - On the homotopy classification of 4-manifolds having the fundamental group of an aspherical 4-manifold [Articolo su rivista]
Cavicchioli, Alberto; Hegenbarth, F.
abstract

We study the homotopy type of closed connected oriented topological 4-manifolds whose fundamental groups are isomorphic to that of an aspherical 4-manifold. We give algebraic conditions for which such manifolds are homotopy equivalent to non-trivial connected sums.

1999 - Cyclic branched coverings of 2-bridge knots [Articolo su rivista]
Cavicchioli, Alberto; Ruini, Beatrice; Spaggiari, Fulvia
abstract

In this paper we study the connections between cyclic presentations of groups and the fundamental group of cyclic branched coverings of 2-bridge knots. Then we show that the topology of these manifolds (and knots) arises, in a natural way, from the algebraic properties of such presentations.

1999 - On cyclic branched coverings of torus knots [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; A. C., Kim
abstract

We prove that the n-fold cyclic coverings of the 3-sphere branched over the torus knots K(p,q), p > q > or = 2, admit spines corresponding to cyclic presentations of groups if p is congruent with 1 mod q.

1998 - A geometric study of Sieradski groups [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Ac, Kim
abstract

This paper is devoted to the study of orientable closed 3-manifolds uniformized by cyclically presented groups, introduced by Sieradski in Invent. Math. 84 (1986) and by certain generalizations of them. It is shown that these manifolds are cyclic coverings of the 3-sphere branched over torus knots. This answers in the affirmative an open question suggested by the referee of the quoted paper and recovers the main theorem of a paper of R. Thomas published in Bull. Korean Math. Soc. 28 (1991).

1998 - A survey on snarks and new results: Products, reducibility and a computer search [Articolo su rivista]
Cavicchioli, Alberto; Meschiari, Mauro; Ruini, Beatrice; Spaggiari, Fulvia
abstract

In this paper we survey recent results and problems of both theoretical and algorithmic character on the construction of snarks-non-trivial cubic graphs of class two, of cyclic edge-connectivity at least 4 and with girth greater than or equal to 5. We next study the process, also considered by Cameron, Chetwynd, Watkins, Isaacs, Nedela, and Skoviera, of splitting a snark into smaller snarks which compose it. This motivates an attempt to classify snarks by recognizing irreducible and prime snarks and proving that all snarks can be constructed from them. As a consequence of these splitting operations, it follows that any snark (other than the Petersen graph) of order less than or equal to 26 can be built as either a dot product or a square product of two smaller snarks. Using a new computer algorithm we have confirmed the computations of Brinkmann and Steffen on the classification of all snarks of order less than 30. Our results recover the well-known classification of snarks of order not exceeding 22. Finally, we prove that any snark G of order less than or equal to 26 is almost Hamiltonian, in the sense that G has at least one vertex v for which G\v is Hamiltonian.

1998 - Manifold crystallization [Capitolo/Saggio]
Cavicchioli, Alberto
abstract

We present the main results and basic definitions on the combinatorial representation of closed connected triangulated n-manifolds by means of special classes of edge-colored graphs, called crystallizations.

1998 - On 4-manifolds fibering over surfaces [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; D., Repovs
abstract

We study closed connected topological or smooth 4-manifolds fibering over a surface in terms of classifying spaces, characteristic classes, and intersection forms.

1998 - On manifold spines and cyclic presentations of groups [Capitolo/Saggio]
Cavicchioli, Alberto; F., Hegenbarth; D., Repovs
abstract

This is a survey of results and open problems on compact connected 3-manifolds which admit spines corresponding to cyclic presentations of groups. We also discuss questions concerning spines of knot manifolds and regular neighborhoods of homotopically PL embedded compacta in 3-manifolds.

1998 - On pseudo-isotopy classes of homeomorphisms of #_p (S^1 x S^n) [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth
abstract

We study self-homotopy equivalences and diffeomorphisms of the (n+1)-dimensional manifold X given by the connected sum of p copies of S^1 x S^n for any n greater or equal 3. Then we completely determine the group of pseudo-isotopy classes of homeomorphisms of X and extend to dimension n well-known theorems due to F. Laudenbach and V. Poenaru (see Bull. Soc. Math. france 100 (1972)) and J.M. Montesinos (Geometric Topology, Proceed. 1977 Georgia Topology Conf., 1979, Academic Press, 219-238).

1998 - On the effective Yang-Mills Lagrangian and its equation of motion [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth
abstract

The path integral quantization of gauge theories leads to an effective Lagrangian denoted by L_eff. Its associated equation of motion is of the form d_A^* F_A + (1/2) J_A = 0, where J_A is the vacuum polarization current arising from propagation of quantum fields around the background field A. In this paper we will give a more detailed description of the current J_A.

1998 - Open problems on graphs arising from geometric topology [Articolo su rivista]
Cavicchioli, Alberto; D., Repovs; Ab, Skopenkov
abstract

We have collected several open problems on graphs which arise in geometric topology, in particular in the following areas: (1) basic embeddability of compacta into the plane R^2; (2) approximability of maps by embeddings; (3) uncountable collections of continua in R^2 and their span; and (4) representations of closed PL manifolds by colored graphs. These problems should be of interest to both topologists and combinatorists.

1998 - Spectral sequences in K-theory for a twisted quadratic extension [Articolo su rivista]
Cavicchioli, Alberto; Muranov, Y. V.; Repovs, D.
abstract

We describe spectral sequences of K-groups which can be constructed for a twisted quadratic extension of antistructures. These spectral sequences are based on Tate cohomology groups of the groups K_i(R) for i = 0, 1. The existence of such spectral sequences follows from an earlier work by Muranov if one uses the methods of construction of spectral sequences originally due to Hambleton and Kharshiladze. We describe the first differentials in these spectral sequences and then give some examples related to surgery.

1998 - Topological properties of high-dimensional handles [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Spaggiari, Fulvia
abstract

We investigate the group of orientation-preserving auto-homeomorphisms resp. homotopy self-equivalences of the connected sum X of p copies of S^1 x S^n, for p greater than or equal 1, modulo those pseudo-isotopic resp. homotopic to the identity. This result is related to a paper of Hosokawa and Kawauchi on unknotted surfaces in Euclidean 4-space, published in Osaka J. Math. 16 (1979), extending it (in greater generality) for embeddings of X into Euclidean (n+3)-space. Finally, we classify the homotopy type of the complement of an embedded copy of X into the Euclidean (n+3)-space, giving examples of manifolds homotopy equivalent to a bouquet of spheres which cannot be fibered over a circle.

1997 - A graph theoretical algorithm for computing the (co)homology of polyhedra [Articolo su rivista]
Cavicchioli, Alberto; Meschiari, Mauro; Spaggiari, Fulvia
abstract

We give a simple algorithm for the computational determination of the (co)homology groups of a compact polyhedron (and, in particular, of a triangulated manifold) starting from a combinatorial representation of it by edge-colored graphs. Several examples illustrate the validity of our algorithm in particular cases. Finally, we obtain a partial catalogue of the (co)homological characters for a family of closed connected orientable 3-manifolds, dependent on three positive integers and a permutation.

1997 - A note on four-manifolds with free fundamental groups [Articolo su rivista]
Cavicchioli, Alberto; Hegenbarth, F.
abstract

We study the homotopy decomposition problem for closed connected 4-manifolds with free fundamental groups. For this we apply obstruction theory and give a detailed description of Whitehead's exact sequence for the named class of manifolds.

1997 - A splitting theorem for homotopy equivalent 4-manifolds [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Spaggiari, Fulvia
abstract

We prove a decomposition theorem for closed connected homotopy equivalent smooth four-manifolds, which partially extends a recent result of Curtis, Freedman, Hsiang and Stong, Invent. Math., 123 (1996), to the non-simply connected case. Then we study the question of when a homotopy equivalence between closed smooth 4-manifolds is homotopic to a topological homeomorphism. In particular, we obtain a new proof of the well-known uniqueness of closed aspherical smooth 4-manifolds with good fundamental groups.

1997 - A splitting theorem for homotopy equivalent smooth 4-manifolds [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Spaggiari, Fulvia
abstract

We prove a decomposition theorem for closed connected homotopy equivalent smooth four-manifolds, which partially extends a recent result of Curtis et al. (1996) to the non-simply connected case. Then we study the question of when a homotopy equivalence between closed smooth 4-manifolds is homotopic to a topological homeomorphism. In particular, we obtain a new proof of the well-known uniqueness of closed aspherical smooth 4-manifolds with good fundamental groups.

1997 - Four-manifolds with surface fundamental groups [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; D., Repovs
abstract

We study the homotopy type of closed connected topological 4-manifolds whose fundamental group is that of an aspherical surface F. Then we use surgery theory to show that these manifolds are s-cobordant to connected sums of simply-connected manifolds with an S^2-bundle over F.

1997 - Lezioni di Topologia Algebrica e Differenziale [Monografia/Trattato scientifico]
Cavicchioli, Alberto; F., Hegenbarth
abstract

Questo volume raccoglie vari argomenti che gli autori sviluppano nelle loro lezioni di Topologia Algebrica e Differenziale per il secondo biennio del Corso di Laurea in Matematica nelle Universita` di Modena e Reggio Emilia e di Milano. Lo scopo principale del testo, destinato in modo naturale agli studenti, e` quello di fornire le basi tecniche e gli strumenti algebrico-differenziali necessari per comprendere ed affrontare i piu` svariati problemi riguardanti la classificazione e lo studio delle proprieta` geometriche delle varieta` topologiche, lineari a tratti o differenziabili.

1997 - On the equivalent spines problem [Articolo su rivista]
Cavicchioli, Alberto; Lickorish, Wbr; Repovs, D.
abstract

We construct examples of compact connected orientable 3-manifolds (regularly embedded into the 3-sphere) with connected boundary of genus greater or equal 2 which are not homeomorphic but they admit the same spine.

1997 - Peripheral acyclicity and homology manifolds [Articolo su rivista]
Cavicchioli, Alberto; Repovs, D.
abstract

We study various concepts of peripheral acyclicity and local homotopical properties, generalizing and extending earlier work due to R.C. Lacher, D.R. McMillan jr. and the authors. Then we obtain some applications for cell-like decompositions of manifolds and construct exotic factors (i.e., homology manifolds) of certain Euclidean spaces.

1996 - Lezioni di Geometria: seconda parte [Monografia/Trattato scientifico]
Cavicchioli, Alberto; Meschiari, Mauro
abstract

Questo volume raccoglie diversi argomenti che gli autori sviluppano nelle loro lezioni di Geometria e di Istituzioni di Geometria Superiore per i Corsi di Laurea in Matematica, Fisica ed Ingegneria dell'Universita` di Modena e Reggio Emilia. Si trattano in modo unitario la geometria differenziale delle curve piane e sghembe, delle superfici immerse nello spazio euclideo tridimensionale e delle varieta` differenziabili e riemanniane. Si sviluppano anche diversi argomenti di topologia generale, di algebra tensoriale e di calcolo differenziale, utili per le applicazioni di tipo geometrico.

1996 - Manifolds with highly connected universal covers [Relazione in Atti di Convegno]
Cavicchioli, Alberto; Hegenbarth, F.
abstract

We apply the homology theory with local coefficients to study closed even-dimensional manifolds with highly connected universal covering spaces. Then we obtain simple algebraic characterizations of aspherical manifolds and discuss some properties regarding the minimality of their Euler characteristics.

1996 - Surgery on 3-manifolds with S-1-actions [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth
abstract

We study the topological structure of all 3-manifolds obtained by surgery along principal fibers of a closed orientable connectedS^1-manifold. As a consequence, we give alternative proofs of some classical results due to W. Hell and L. Moser. Moreover, we completely specify the Seifert invariants for the considered manifolds. Finally we classify the manifolds obtained by surgery along certain Seifert links and determine geometric presentations of their fundamental groups.

1996 - Topologia delle varieta`: problemi e risultati [Articolo su rivista]
Cavicchioli, Alberto
abstract

In questo articolo, di carattere generale, si espone una panoramica sui principali risultati classici e moderni, ottenuti da diversi autori, sul problema del riconoscimento delle varieta` topologiche e sul problema della classificazione omotopica e topologica delle varieta` TOP (PL, DIFF). Una serie di problemi aperti completano il testo dell'articolo.

1995 - A homology theory for colored graphs [Articolo su rivista]
Cavicchioli, Alberto; Meschiari, Mauro
abstract

We introduce a homology theory for colored graphs (G, C_G) which is motivated by topological reasons. Indeed, it translates the usual one of the polyhedron |G| canonically associated to (G, C_G). Then we obtain combinatorial analogs to exact homology sequences, cohomology groups, products, duality, etc. and prove some classic results of algebraic topology by graph-theoretic tools. Finally we study some combinatorial invariants of (G, C_G) which are useful to describe the topological structure of |G|.

1995 - On the stable classification of certain 4-manifolds [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; D., Repovs
abstract

We study the s-cobordism type of closed orientable (smooth or PL) 4-manifolds with free or surface fundamental groups. We prove stable classification theorems for these classes of manifolds by using surgery theory.

1995 - Some open problems in geometric topology of low dimensions [Articolo su rivista]
Cavicchioli, Alberto; D., Repovs
abstract

We study some open problems in geometric topolofy of low dimensions: the 2-sphere approximation conjecture, controlled isotopy, equivalent spines, cell-like resolutions, geometric presentations of manifold fundamental groups, classification of triangulated manifolds according to genus.

1995 - Sulla classificazione topologica delle varieta` [Articolo su rivista]
Cavicchioli, Alberto
abstract

In this survey I describe the history and the most important results about the classification problem of topological manifolds.

1994 - Knot manifolds with isomorphic spines [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth
abstract

We study the relation between the concept of spine and the representation of orientable bordered 5-manifolds by Heegaard diagrams. As a consequence, we show that composing invertible non-amphicheiral knots yields examples of topologically different knot manifolds with isomorphic spines. These results are related to some questions listed in Kirby, Proc. Sympos. Pure Math. 32, Amer. Math. Soc., 1978, and Repovs, Suppl. Rend. Circ. Mat. Palermo 18 (1988), and recover the main theorem of Mitchell, Przytycki and Repovs, Bull. Polish Acad. Sci. 37 (1989) as a corollary. Finally, an application concerning knot manifolds of composite knots with h prime factors completes the paper.

1994 - Lezioni di Geometria: prima parte [Monografia/Trattato scientifico]
Cavicchioli, Alberto; Meschiari, Mauro
abstract

Questo volume raccoglie alcuni argomenti che gli autori sviluppano nelle loro lezioni di Geometria per i Corsi di Laurea in Matematica, Fisica ed Ingegneria dell'Universita` di Modena e Reggio Emilia. Lo scopo primario del testo, destinato in modo naturale agli studenti, e` quello di fornire gli strumenti necessari per risolvere i piu` svariati problemi di geometria affine, euclidea e proiettiva. Si illustra anche la teoria delle iperquadriche e delle ipersuperfici e si sviluppano alcuni argomenti sulle varieta` algebriche di uno spazio proiettivo.

1994 - On 4-manifolds with free fundamental group [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth
abstract

We study the homotopy type of closed 4-manifolds (or oriented Poincare spaces) with free fundamental group. This gives a partial solution to problem N. 4.53 of Kirby, Contemporary Math. 35 (1984). Then we extend the Whitehead-Novikov-Wall theorem for this class of manifolds by using surgery techniques.

1994 - Special representations of n-bridge links [Articolo su rivista]
Cavicchioli, Alberto; Ruini, Beatrice
abstract

We describe a simple algorithm to obtain a catalogue of 3-bridge links by computer, depending upon 6-tuples of positive integers. This permits us to represent the genus two 3-manifolds by standardly constructed graphs with colored edges. Finally, we prove some results about the topological structure of these manifolds and extend the combinatorial representation to n-bridge links.

1993 - Lezioni di Algebra Lineare [Monografia/Trattato scientifico]
Cavicchioli, Alberto; Meschiari, Mauro
abstract

Questo volume raccoglie le lezioni di Algebra Lineare che gli autori impartiscono nei corsi di Geometria per il primo anno dei Corsi di Laurea in Matematica, Fisica ed Ingegneria dell'Universita` di Modena e Reggio Emilia. La scelta degli argomenti, trattati spesso in modo nuovo, e` stata guidata soprattutto dall'esigenza di fornire una base operativa per poter sviluppare temi piu` squisitamente geometrici che saranno oggetto di ulteriori volumi. Lo scopo primario del testo, destinato in modo naturale agli studenti, e` quello di fornire i fondamenti algebrici necessari per affrontare i piu` svariati problemi di Algebra Lineare e di Geometria.

1993 - On classification of 4-manifolds according to genus [Articolo su rivista]
Cavicchioli, Alberto; M., Meschiari
abstract

We study the topological structure of closed connected 4-manifolds according to regular genus. In particular, we prove that the topological product of two copies of the 2-sphere (respectively, the real projective 4-space) is the unique prime closed orientable (respectively, non-orientable) 4-manifold of regular genus 4 (respectively, 6).

1993 - On the determination of PL-manifolds by handles of lower dimension [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth
abstract

We extend the Montesinos theorem about handle presentations of PL 4-manifolds to dimension n. For this we consider closed PL (n + 1)-manifolds of the same homotopy type as X = #p (S^1 x S^n). Then we extend PL homeomorphisms of X over the solid handlebody.

1993 - On the topological structure of compact 5-manifolds [Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

We classify the genus one compact (PL) 5-manifolds and prove some results about closed 5-manifolds with free fundamental group. In particular, let M be a closed connected orientable smooth 5-manifold with free fundamental group. Then we prove that the number of distinct smooth 5-manifolds homotopy equivalent to M equals the 2-nd Betti number (mod 2) of M.

1993 - Splittings of manifolds with boundary and related invariants [Articolo su rivista]
Cavicchioli, Alberto; Ruini, Beatrice
abstract

We construct special handle decompositions for a compact connected PL manifold with non empty boundary and study the associated topological invariants. As a consequence, we characterize the unknot in the (n+2)-sphere (n less or equal 2) as the unique n-knot whose complement has genus one. Then we obtain a simple geometric proof of the non cancellation theorem for tame n-knots in the (n+2)-sphere, for n less or equal 2.

1992 - Idempotent comultiplications on graded algebras [Articolo su rivista]
Cavicchioli, Alberto; Meschiari, Mauro
abstract

We classify all idempotent comultiplications on a graded anticommutative algebra up to degree 3, provided its components are torsion free, and topologically realize all algebraic possibilities. Then we extend some results to dimension n and obtain topological consequences about closed n-manifolds with cohomology of special type.

1992 - Imbeddings of polyhedra in 3-manifolds [Articolo su rivista]
Cavicchioli, Alberto
abstract

Si studiano le immersioni di un poliedro compatto in una 3-varieta`e si estende la rappresentazione delle 3-varieta` chiuse, data da Neuwirth nel 1968, al caso con bordo. Infine, si provano due congetture formulate da Repovs nel 1988 e si ottengono alcune applicazioni sui complementari di nodi.

1992 - Neuwirth manifolds and colourings of graphs [Articolo su rivista]
Cavicchioli, Alberto
abstract

We use the representation of PL manifolds via graphs with colored edges. Then we describe a very fast construction to obtain a 4-colored graph representing a closed orientable 3-manifold from its Heegaard diagram. This construction allows us to completely classify a countable class of closed orientable 3-manifolds introduced by L. Neuwirth in Proc. Camb. Phil. Soc. 64 (1968).

1992 - On the genus of smooth 4-manifolds [Articolo su rivista]
Cavicchioli, Alberto
abstract

The projective complex plane and the twisted S^3 bundle over S^1 are proved to be the unique closed prime connected (smooth or PL) 4-manifolds of genus two. Then the classification of the nonorientable 4-manifolds of genus 4 is given. Finally, the genus of a manifold M is shown to be related with the 2nd Betti number of M and some applications are proved in the general (resp. simply-connected) case.

1992 - On the intersection forms of closed 4-manifolds [Articolo su rivista]
Cavicchioli, Alberto; Hegenbarth, F.
abstract

Given a closed 4-manifold M, let M^* be the simply-connected 4-manifol obtained from M by killing the fundamental group. We study the relation between the intersection forms \lambda_M and \lambda_(M^*). Finally, some topological consequences and examples are described.

1992 - The classification of 3-manifolds with spines related to Fibonacci groups [Relazione in Atti di Convegno]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract

Lecture Notes in Math. The classification of 3-manifolds with spines related to Fibonacci groups. We study the topological structure of closed connected orientable 3-manifolds which admit spines corresponding to the standard presentation of Fibonacci groups.

1991 - Mobius transformations and surface groups [Articolo su rivista]
Cavicchioli, Alberto
abstract

We describe a simple geometric construction of the fundamental group of the closed connected orientable surface of genus g (g greater or equal 2) presented in terms of 2g complex Mobius transformations as generators and of one relation among them. Our construction allows us to write the equation of any transformation of the above group. Finally, we give a recurrence formula to obtain all the vertices of the tessellation induced by the above group in the open unit disc endowed by the hyperbolic metric.

1990 - Covering numbers of manifolds and critical points of a Morse function [Articolo su rivista]
Cavicchioli, Alberto
abstract

For a closed connected triangulated n-manifold M, we study some numerical invariants (named category and covering numbers) of M which are strictly related to the topological structure of M. We complete the classical results of 3-manifold topology and then we prove some characterization theorems in higher dimensions. Finally, some applications are given about the minimal number of critical points (resp. values) of Morse functions defined on a closed connected smooth n-manifold.

1990 - Finding nice crystallizations for handle free n-manifolds [Articolo su rivista]
Cavicchioli, Alberto
abstract

The paper deals with combinatorial structures (pseudo-complexes, crystallizations) giving a direct link between the topology of triangulated manifolds and the theory of edge-colored multigraphs. We define the concept of regular crystallization of a manifold and prove that every non-trivial handle free closed n-manifold has a regular crystallization. Then we study some applications of regular crystallizations and give a counter-example to a conjecture of Y. Tsukui about strong frames of the 3-sphere.

1990 - Manifolds of type C(p, q) [Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth
abstract

We study the topology of smooth closed 4-manifolds with free fundamental group. An important subclass consists of the manifolds of type C(p,q) introduced by H. Ikeda and M. Yamashita in Kobe Math. J. 4 (1976). We prove some topological properties and homeomorphism results for such a class of closed 4-manifolds.

1990 - On the automorphism groups of coloured graphs arising from manifolds [Articolo su rivista]
Cavicchioli, Alberto
abstract

We study the isomorphism problem for colored graphs which represent closed connected (pseudo) manifolds. Then we determine the automorphism groups of these graphs and prove some topological properties of the represented spaces.

1989 - A combinatorial characterization of S^3 x S^1 among closed 4-manifolds [Articolo su rivista]
Cavicchioli, Alberto
abstract

The topological product S^3 x S^1 is proved to be the unique closed connected 4-manifold of regular genus one. As a consequence, the complex projective plane has regular genus two.

1988 - Heegaard diagrams, graphs and 3-manifold spines [Articolo su rivista]
Bandieri, Paola; Cavicchioli, Alberto; Grasselli, Luigi
abstract

We study the relations between some well-known graph-theoretical representations of closed 3-manifolds as P-graphs, RR-systems, Heegaard diagrams and crystallizations. As a consequence of the interplay between the theories, we prove some theorems which improve the geometric knowledge of the named representations. Finally, we show the strict connection of the results with the theory of 3-manifold spines.

1988 - Lins-Mandel 3-manifolds and their groups: a simple proof of the homology sphere conjecture [Articolo su rivista]
Cavicchioli, Alberto
abstract

The Lins-Mandel manifolds are combinatorially represented by a 4-parametric family of colored graphs S(n,\ell,t,c) defined in Discrete Math. 57 (1985). In the present paper we obtain a finite presentation of the fundamental group of such manifolds for t = \ell -1 and c=1. As a consequence, the homology sphere conjecture 2, stated in the quoted paper, can be easily proved by a direct algebraic calculus on the homology groups.

1988 - Minimal dissections of bordered manifolds [Articolo su rivista]
Cavicchioli, Alberto; Grasselli, Luigi
abstract

We construct minimal (with respect to vertices) pseudo-dissections for any connected piecewise-linear n-manifold with boundary. As a consequence, we obtain ball coverings for bordered manifolds with nice intersection properties. Then we give a graph-theoretical description of these results.

1988 - On some properties of the groups G(n, l) [Articolo su rivista]
Cavicchioli, Alberto
abstract

In S. Lins-A. Mandel, Discrete Math. 57 (1985) it was constructed a family of closed connected 3-manifolds by means of edge-colored 4-graphs. In the present paper, we describe several algebraic properties satisfied by the fundamental groups of such manifolds.

1987 - Generalized handles in graphs and connected sums of manifolds [Articolo su rivista]
Cavicchioli, Alberto
abstract

We introduce the concept of generalized handles in (n+1)-colored graphs, and determine many combinatorial and topological properties of them. Then the operations of cancelling and cutting a generalized handle are studied. As a consequence, some decomposition theorems about manifolds are proved. In particular, a graph-theoretical condition is obtained to recognize S^(n-1)-bundles over S^1 as factors of a decomposition of a manifold in connected sum.

1986 - A countable class of non-homeomorphic homology spheres with Heegaard genus two [Articolo su rivista]
Cavicchioli, Alberto
abstract

We construct a countable set of prime non-homeomorphic homology 3-spheres with Heegaard genus 2.

1986 - Sfere omologiche fibrate [Relazione in Atti di Convegno]
Cavicchioli, Alberto
abstract

Si studiano le proprieta` algebriche e topologiche di certe classi numerabili di sfere omologiche fibrate con genere di Heegaard 2.

1985 - A genus for n-dimensional knots and links [Articolo su rivista]
Cavicchioli, Alberto
abstract

We introduce the concept of regular genus for n-dimensional links and knots, and study the topological properties of it. Some characterization theorems of the trivial knot are given. In particular, the only genus zero n-dimensional knot is proved to be homeomorphic with the trivial knot. Then the regular genus of a knot is proved to be related to the one-dimensional homology of the universal abelian covering of its complement. Some applications to low-dimensional links and a final section about connected sums of links complete the paper.

1985 - Lins-Mandel crystallizations [Articolo su rivista]
Cavicchioli, Alberto
abstract

We prove that each closed connected 3-manifold admits particular crystallizations with nice properties. As a consequence, all closed 3-manifolds can be represented by a positive integer b and a fixed-point-free involutory permutation on Z_b x Z_6. Other 3-manifold representations are also obtained. Finally, an interesting family of closed connected 3-manifolds is completely studied.

1985 - Minimal atlases of manifolds [Articolo su rivista]
Cavicchioli, Alberto; Grasselli, Luigi
abstract

Given a connected compact n-manifold M, a natural invariant of M is the minimal number of balls which are needed to cover M. If the intersection of any number of balls has again balls as connected components, we get the notion of ball-intersection atlas of M. We prove that each minimal ball-intersection atlas of a connected piecewise-linear n-manifold M has exactly n balls if the boundary of M is non-void.

1984 - Contracted triangulations as branched coverings [Articolo su rivista]
Cavicchioli, Alberto; Grasselli, Luigi
abstract

We prove that every closed connected orientable n-dimensional pseudomanifold, which is n-colored on its vertex set, is a covering of the n-sphere branched over the (n-2)-skeleton of an n-simplex. This extends a theorem of Alexander (Bull. Amer. Math. Soc. 26 (1920)) and a theorem of Ramirez (An. Inst. Mat. Univ. Nac. Autonoma Mexico 15 (1975)). Using the concept of contracted triangulation, every closed connected orientable 3-manifold M is represented as a covering of the 3-sphere branched over an universal graph G, so that the cardinality of the fiber of each point of G depends only on the number of the 3-simplexes of a contracted triangulation of M.

1984 - Gruppi di omografie propriamente discontinui e trivarieta` [Articolo su rivista]
Cavicchioli, Alberto; Grasselli, Luigi
abstract

Si ottiene una rappresentazione di tutte le trivarieta` chiuse, connesse ed orientabili a partire da gruppi propriamente discontinui di trasformazioni reali proiettive.

1983 - Generating all closed 3-manifolds from handlebodies and poligonal schemes [Relazione in Atti di Convegno]
Bandieri, P.; Cavicchioli, Alberto; Grasselli, Luigi
abstract

We obtain a new combinatorial description of all closed connected 3-manifolds from handlebodies and polygonal schemes. Then we illustrate several geometrical and topological consequences of our representation results.

1983 - Generating all closed 3-manifolds from handlebodies and polygonal schemes [Relazione in Atti di Convegno]
Bandieri, Paola; Cavicchioli, Alberto; Grasselli, Luigi
abstract

In this paper we prove that every closed, connected and orientable 3-manifold of genus g can be obtained by identifying the boundaries of two closed domains, whose union is the boundary of a 3-dimensional handlebody of genus (g-1) and whose interiors are disjoint open discs .

1983 - Manici e fattorizzazioni delle trivarieta` [Articolo su rivista]
Cavicchioli, Alberto
abstract

In questa nota si presenta una breve dimostrazione (di carattere topologico) del teorema di riconoscimento dei manici combinatori nell'ambito dei grafi 4-colorati sugli spigoli.

1982 - A three-manifold invariant independent of the homotopy type [Articolo su rivista]
Cavicchioli, Alberto
abstract

We introduce a three-manifold invariant, named ultracontracted genus, which is proved to be independent of the homotopy type. Then we completely classify all closed connected 3-manifolds of ultracontracted genus one.

1982 - Cohomological products and transversality [Articolo su rivista]
Cavicchioli, Alberto; Grasselli, Luigi
abstract

By making use of geometric cocycles we give a definition of transversality for maps between polyhedra, which extends the mock transversality given by Buoncristiano, Rourke and Sanderson in London Math. Soc. Lect. Note ser. 18 (1975).This definition allows us to introduce a product between PL maps that, in the particular case of the geometric cycles and cocycles, gives a description of the cup and cap products in terms of transversality.

1981 - A new handlebody decomposition of 3-manifolds with connected boundary and their fundamental group [Articolo su rivista]
Cavicchioli, Alberto
abstract

We give a new theorem of handlebody decomposition for compact 3-manifolds with connected boundary. The decomposition type, given here, allows us to calculate with great facility a presentation of the fundamental group of a 3-manifold with connected boundary.

1981 - Remarks on Heegaard splitting theory [Articolo su rivista]
Cavicchioli, Alberto
abstract

We introduce a new topological invariant of closed connected 3-manifolds. Then we study a new polyhedral construction, called double-cone, and show that every closed connected 3-manifold can be obtained from a 2-dimensional cell complex by double-cone.

1980 - Crystallizations of PL-manifolds with connected boundary [Articolo su rivista]
Cavicchioli, Alberto; Gagliardi, Carlo
abstract

We extend the contracted triangulation theorem, established for closed PL manifolds by Pezzana in Atti Sem. Mat. Fis. Univ. Modena 23 (1975), to PL manifolds with connected boundary. The result is derived from a general triangulation theorem for quasi-manifolds. Then we found, as a corollary of our main theorem, a graph-theoretical representation for every PL manifold with connected boundary by means of suitable edge-colored graphs, which we still call a "crystallization".

1980 - Su una decomposizione normale per le n-varietà chiuse [Articolo su rivista]
Cavicchioli, Alberto; Grasselli, Luigi; M., Pezzana
abstract

Si prova l'esistenza di triangolazioni ultracontratte per ogni n-varietà chiusa: il risultato è un raffinamento del tentativo di estendere le forme normali delle superficie alle varietà di dimensione superiore. Ciò consente di rappresentare le varietà mediante opportuni grafi colorati sugli spigoli: si prova infine che le varietà risultano orientabili se e solo se i grafi che le rappresentano sono bipartiti.

1980 - Una rappresentazione delle trivarieta` orientabili mediante atlanti minimali di superficie [Articolo su rivista]
Cavicchioli, Alberto
abstract

A representation theorem for orientable closed 3-manifolds is proved. Some applications about lens spaces, 3-manifolds of genus two, and connected sums are studied.

1978 - Pseudo-dissezioni e triangolazioni contratte di spazi con singolarita` isolate [Articolo su rivista]
Cavicchioli, Alberto
abstract

I prove a theorem of contracted triangulation for spaces with "few" points of singularity. These spaces with PL maps form a Category that contains the Category of PL manifolds. Then I prove some theorems of characterization and comparison among the new introduced spaces and the classically well known spaces.

Università degli studi di Modena e Reggio Emilia

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