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Fulvia SPAGGIARI

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

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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 - A New Class of Homology and Cohomology 3-Manifolds [Articolo su rivista]
Garity, D. J.; Karimov, U. H.; Repovs, D.; Spaggiari, Fulvia
abstract

We show that for any set of primes P there exists a space M_P which is a homology and cohomology 3-manifold with coefficients in Z_p for p\in P and is not a homology or cohomology 3-manifold with coefficients in Z_q for $q\not\in P$. In addition, M_p is not a homology or cohomology 3-manifold with coefficients in Z or Q.

2016 - Isometry Groups of Some Dunwoody Manifolds [Articolo su rivista]
Spaggiari, Fulvia; Telloni, Agnese Ilaria
abstract

Dunwoody manifolds are an interesting class of closed connected orientable 3--manifolds, which are defined by means of Heegaard diagrams having a rotational symmetry. They are proved to be cyclic coverings of lens spaces (possibly S^3) branched over (1,1)--knots. Here we study the Dunwoody manifolds which are cyclic coverings of the 3--sphere branched over two specified families of Montesinos knots. Then we determine the Dunwoody parameters for such knots and the isometry groups for the considered manifolds in the hyperbolic case. A list of volumes for some hyperbolic Dunwoody manifolds completes the paper.

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.

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 - 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 - 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.

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 - 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 - Regular genus and products of spheres [Articolo su rivista]
Spaggiari, Fulvia
abstract

A crystallization of a closed connected PL manifold M is a special edge-colored graph representing M via a contracted triangulation. The regular genus of M is the minimum genus of a closed connected surface into which a crystallization of M regularly embeds. We disprove a conjecture on the regular genus of S^2 X S^n, n >=3, stated in [J. Korean Math. Soc. 41 (2004), n.3, p. 420].

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 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 - 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 geometric study of generalized Neuwirth groups [Articolo su rivista]
Spaggiari, Fulvia
abstract

We define a family of groups with balanced presentations and prove that these groups correspond to spines (or, equivalently, to Heegaard diagrams) of a certain class of Seifert fibered 3-manifolds. These manifolds are constructed from triangulated 3-balls by identifying pairs of boundary faces via orientation-reversing homeomorphisms. Then we describe the manifolds as cyclic branched coverings of certain lens spaces when the groups are cyclically presented. Finally, we give explicit computations of the Casson-Walker-Lescop invariant and the Rohlin invariant for many manifolds in the above class.

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 - Asphericity of symmetric presentations [Articolo su rivista]
Spaggiari, Fulvia
abstract

Using the notion of relative presentation due to Bogley and Pride, we give a new proof of a theorem of Prishchepov on the asphericity of certain symmetric presentations of groups. Then we obtain further results and applications to topology of low-dimensional 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 - An infinite sequence of non-realizable weavings [Articolo su rivista]
D., Repovs; A., Skopenkov; Spaggiari, Fulvia
abstract

A weaving is a number of lines drawn in the plane so that no three lines intersect at a point, and the intersections are drawn so as to show which of the two lines is above the other. For each integer n >= 4 we construct a weaving of n lines, which is not realizable as a projection of a number of lines in 3-space, all of whose subfigures are realizable as such projections.

2005 - Connected sums of 4-manifolds [Articolo su rivista]
F., Hegenbarth; D., Repovs; Spaggiari, Fulvia
abstract

We study the following problem for closed connected oriented manifolds M of dimension 4. Let A be the integral group ring of the fundamental group of M. Suppose that a subset G of H_2(M; A) is a free A-submodule. When do there exist closed connected 4-manifolds P and M´ such that M is homotopy equivalent to the connected sum P # M´, where the fundamental group of P is isomorphic to the fundamental group of M, the fundamental group of M' is isomorphic to 0, and the tensor product of H_2 (M´ ; Z) with A over Z is isomorphic to G. An answer is given in terms of the fundamental group of M and the intersection forms on H_2 (M ; A) and H_2 (M ; Z).

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 - On the Pontryagin-Steenrod-Wu theorem [Articolo su rivista]
D., Repovs; M., Skopenkov; Spaggiari, Fulvia
abstract

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) Let M be a connected orientable closed smooth (n+1)-manifold, n>=3. Define the degree map deg: \pi^n(M) \to H^n(M;Z) by the formula deg f=f*[S^n], where [S^n] \in H^n(M;Z) is the fundamental class. The degree map is bijective if there exists \beta \in H_2(M,Z/2Z) such that \beta \cdot w_2(M)\ne 0. If such \beta does not exist, then deg is a 2-1 map; and (b) Let M be an orientable closed smooth (n+2)-manifold, n>=3. An element \alpha lies in the image of the degree map if and only if \rho_2 \alpha \cdot w_2(M)=0, where \rho_2 :Z \to Z/2Z is reduction modulo 2.

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.

2005 - The combinatorics of some tetrahedron manifolds [Articolo su rivista]
Spaggiari, Fulvia
abstract

We study a family of closed connected orientable 3-manifolds (which are examples of tetrahedron manifolds) obtained by pairwise identifications of the boundary faces of a standard tetrahedron. These manifolds generalize those considered in previous papers due to Grasselli, Piccarreta, Molnar and Sieradski. Then we completely describe our tetrahedron manifolds in terms of Seifert fibered spaces, and determine their Seifert invariants. Moreover, we obtain different representations of our manifolds as 2-fold coverings, and give examples of non-equivalent knots with the same tetrahedron manifold as 2-fold branched covering space.

2004 - On branched coverings of lens spaces [Articolo su rivista]
E., Barbieri; Spaggiari, Fulvia
abstract

We construct some series of polyhedral schemata which represent orientable closed connected 3-manifolds. We show that these manifolds have spines corresponding to certain balanced presentations of their fundamental groups. Then we study some covering properties of such manifolds and prove that many of them are cyclic branched coverings of lens spaces. Our theorems contain a number of published results from various sources as particular cases.

2004 - Periodic links and manifolds [Articolo su rivista]
E., Barbieri; Spaggiari, Fulvia
abstract

We consider orientable closed 3-manifolds obtained by Dehn surgery with rational coefficients along the components of certain periodic links. These manifolds were introduced in [Osaka J. Math. 39 (2002), 705-721] as natural generalizations of Takahashi manifolds. In this note we re-obtain the result of [Osaka J. Math. 39 (2002)] by a different approach based on a group-theoretic argument from [Tsukuba J. Math. 22 (1998), 723-739]. This permits to simplify some proofs of [Osaka J. Math. 39 (2002)] and to obtain some new related results.

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 - Four-manifolds with π1-free second homotopy [Articolo su rivista]
Spaggiari, Fulvia
abstract

We study the homotopy type of closed connected orientable topological 4-manifolds M with Lambda-free second homotopy group, where Lambda is the integral group ring of pi_1(M). This is related with problem N.4.53 of [R. Kirby, Contemporary Math., 35 (1984)], and extends some results proved for the class of closed 4-manifolds with free fundamental group. Other applications on special classes of closed topological manifolds complete the paper.

2003 - On calculation of the Witten invariants of 3-manifolds [Articolo su rivista]
E., Rafikov; D., Repovs; Spaggiari, Fulvia
abstract

In this paper we present a short definition of the Witten invariants of 3-manifolds. We also give simple proofs of invariance of those obtained for r = 3 and r = 4. Our definition is extracted from the 1993 paper of Lickorish and the Prasolov-Sossinsky book, where it is dispersed over 20 pages. We show by several examples that it is indeed convenient for calculations.

2003 - On spines of Seifert fibered manifolds [Articolo su rivista]
Ruini, Beatrice; Spaggiari, Fulvia; A., Vesnin
abstract

We define a family of balanced presentations of groups and prove that they correspond to spines of some Seifert fibered 3-manifolds. These presentations of groups (and manifolds) generalize in a natural way many classes of presentations of groups (and manifolds) previously studied by several authors. Moreover, we construct crystallizations representing the small Seifert manifolds of the considered class.

2003 - On the stable classification of spin four-manifolds [Articolo su rivista]
Spaggiari, Fulvia
abstract

We study the stable classification of closed connected oriented spin smooth 4-manifolds by using techniques of Kervaire-Milnor surgery. Then we reproduce a nice result of Kurazono and Matumoto [I. Kurazono, T. Matumoto, Hiroshima Math. J. 28, 1998] for such manifolds under the assumption that the fundamental group is finitely presentable and has vanishing second and third homology with Z_2-coefficients.

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 - Surgery on triples of manifolds [Articolo su rivista]
Y. V., Muranov; D., Repovs; Spaggiari, Fulvia
abstract

The surgery obstruction groups for a manifold pair were introduced by Wall for the study of the surgery problem on a manifold with a submanifold. These groups are closely related to the problem of splitting a homotopy equivalence along a submanifold and have been used in many geometric and topological applications. In the present paper the concept of surgery on a triple of manifolds is introduced and algebraic and geometric properties of the corresponding obstruction groups are described. It is then shown that these groups are closely related to the normal invariants and the classical splitting and surgery obstruction groups, respectively, of the manifold in question. In the particular case of one-sided submanifolds relations between the newly introduced groups and the surgery spectral sequence constructed by Hambleton and Kharshiladze are obtained.

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 - Esercizi di geometria [Monografia/Trattato scientifico]
Ruini, Beatrice; Spaggiari, Fulvia
abstract

Questo volume e` una raccolta di esercizi risolti che riguardano argomenti di geometria affine, euclidea e proiettiva, di geometria delle coniche e delle quadriche e di geometria delle curve e delle superfici differenziabili. Lo scopo primario del testo consiste nel proporre applicazioni dei risultati teorici relativi agli argomenti che vengono generalmente sviluppati nei corsi di Geometria per gli studenti dei Corsi di Laurea della Facolta`di Ingegneria e dei Corsi di Laurea in Matematica e Fisica della Facolta` di Scienze.

2002 - On the computation of L-groups and natural maps [Articolo su rivista]
Ruini, Beatrice; Spaggiari, Fulvia
abstract

In this paper we compute surgery (resp. splitting) obstruction groups, here called L-groups, and natural maps for many diagrams of oriented finite (not necessary abelian) 2-groups and homomorphisms which preserve orientations.

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 - Manifolds spines and hyperbolicity equations [Articolo su rivista]
Ruini, Beatrice; Spaggiari, Fulvia
abstract

We give a combinatorial representation of compact connected orientable 3-manifolds with boundary and their special spines by a class of graphs with extrastructure which are strictly related to o-graphs. Then we describe a simple algorithm for constructing the boundary of these manifolds by using a list of 6-tuples of non-negative integers. Finally we discuss some combinatorial methods for determining the hyperbolicity equations. Examples of hyperbolic 3-manifolds of low complexity illustrate in particular cases the constructions and algorithms presented in the paper.

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

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 the genus of RP^3 X S^1 [Articolo su rivista]
Spaggiari, Fulvia
abstract

We continue the topological classification of closed connected orientable 4-manifolds according to the (regular) genus. In particular, we prove that any closed prime orientable PL 4-manifold of genus six is topologically homeomorphic to a lens-fiber bundle over the 1-sphere. There are good reasons to conjecture that the genus six characterizes the topological product RP^3 X S^1 of the real projective 3-space by the 1-sphere among closed connected prime orientable 4-manifolds.

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 - On the structure of Takahashi Manifolds [Articolo su rivista]
Ruini, Beatrice; Spaggiari, Fulvia
abstract

We study the topological structure of the closed orientable 3-manifolds obtained by Dehn surgeries along certain links, first considered by Takahashi. The interest about such manifolds arises from the fact that they include well-known families of 3-manifolds, previously studied by several authors, as the Fibonacci manifolds, the Fractional Fibonacci manifolds, and the Sieradski manifolds, respectively. Our main results states that the Takahashi manifolds are 2-fold coverings of the 3-sphere branched along the closures of specified 3-string braids. We also describe many of the above-mentioned manifolds as n-folds cyclic branched coverings of the 3-sphere.

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 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 - On certain classes of finite groups [Articolo su rivista]
Spaggiari, Fulvia
abstract

We prove some algebraic properties of finitely presented groups with two generators and two relators, some of them arising from the study of closed 3-manifolds with finite fundamental group and other introduced by C.M. Campbell, H.S.M. Coxeter and E.F. Robertson in [Proc. Roy. Soc. Lond. A 357, 1977], which are relevant to a search for trivalent and 0-symmetric Cayley diagrams.

1996 - Esercizi di Algebra Lineare [Monografia/Trattato scientifico]
C., Bignardi; Ruini, Beatrice; Spaggiari, Fulvia
abstract

Questo volume raccoglie una serie di esercizi, sia svolti che proposti, che puo`risultare utile non solo agli studenti dei corsi di Laurea e di Diploma in Matematica e Ingegneria, ma anche a tutti coloro che desiderano affrontare argomenti di Algebra Lineare. In Appendice si e`presentata una selezione di temi scritti, proposti nelle sessioni d'esame dei corsi di Geometria presso le Facolta`di Scienze ed Ingegneria dell'Universita`di Modena.

1993 - A note on Generalized Petersen Graphs [Articolo su rivista]
Spaggiari, Fulvia
abstract

We give an alternative constructive proof of a theorem of M. Watkins, F. Castagna and G. Prins, i.e., the generalized Petersen graphs are of class one, except the Petersen graph. As a consequence, we obtain some bounds for the regular genus of these graphs and give a partial solution to a conjecture stated in [F. Castagna, G.Prins, Pacific J. Math. 40, 1972]

1993 - On a theorem of L. Moser [Articolo su rivista]
Spaggiari, Fulvia
abstract

We prove that the RR-system theory easily implies the theorem of Moser on the classification of 3-manifolds obtained by (p,q)-surgery along the torus knot of type (m,n). Then we completely determine the Seifert invariants of the considered manifolds and we obtain simple geometric presentations for their fundamental groups.

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.

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.