
Fulvia SPAGGIARI
Professore Associato Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede exMatematica

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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 3manifolds 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 3sphere, whose (n2) and (n+1)surgeries are toroidal, and (n1) and nsurgeries are Seifert fibered.
2023
 Spines and surgery descriptions of graph manifolds
[Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract
Graph manifolds are compact orientable 3manifolds 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 2disc and $N^2$ denotes the 2disc 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 3dimensional 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 BakerPetersen, Qazaqzeh and MoralesMarcen 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 FC4complexes
[Articolo su rivista]
Cavicchioli, Alberto; Hegenbarth, Friedrich; Spaggiari, Fulvia
abstract
In this paper we continue our investigations of 4dimensional complexes in [A. Cavicchioli,
F.Hegenbarth, F. Spaggiari, Fourdimensional complexes with fundamental class,Mediterr. J. Math. 17 (2020),
175]. We study a class of finite oriented 4complexes which we call FC4complexes, defined as follows. An
FC4complex is a 4dimensional finite oriented CWcomplex X with a single 4cell such that H4(X, ℤ) ≅ ℤ
with a fundamental class [X] ∈ H4(X, ℤ). By wellknown results ofWall, any Poincaré complex is of this type.
We are interested in two questions. First, for which 3complexes 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 Λcoefficients. Here Λ denotes the integral group ring of π1(K). Before,
we give a classification of all FC4complexes based on the finite 3complex K, and make some remarks concerning
ℤ and ΛPoincaré duality.
2021
 New formulas for the character varieties of twobridge links
[Articolo su rivista]
Cavicchioli, A.; Spaggiari, F.
abstract
We consider some families of twobridge 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 2bridge 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 5manifolds
[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 5manifolds up to piecewiselinear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edgecolored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed ndimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 5manifolds of reduced complexity less than or equal to 20. In particular, this gives a combinatorial characterization of S2×S3 among closed connected spin PL 5manifolds.
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 3manifolds. The results have also didactical importance
2020
 FourDimensional Complexes with Fundamental Class
[Articolo su rivista]
Cavicchioli, A.; Hegenbarth, F.; Spaggiari, F.
abstract
This paper continues the study of 4dimensional 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/s0000901811023) 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 4complexes. More precisely, we introduce a new class of oriented fourdimensional 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 4complexes, 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 4complex to be a PD 4complex. Finally, we obtain a partial classification of SFC 4complexes. A future goal will be a classification in terms of algebraic SFC 4complexes similar to the very satisfactory classification result of PD 4complexes obtained by Baues and Bleile (Algebraic Geom. Topol. 8:2355–2389, 2008).
2019
 On graphtheoretical 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 piecewiselinear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edgecolored 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 ndimensional 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 lowdimensional 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 fourdimensional Poincarè duality cobordism groups
[Articolo su rivista]
Cavicchioli, A; Hegenbarth, F; Spaggiari, F
abstract
This paper continues the study of 4dimensional Poincarè duality cobordism theory from our previous work "PD4complexes: constructions, cobordisms and signatures", Homology, Homotopy and Applications 18(2) (2016), 267281. 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 PD4complexes which cannot be homotopy equivalent to manifolds.
2016
 A New Class of Homology and Cohomology 3Manifolds
[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 3manifold with
coefficients in Z_p for p\in P and is not a homology
or cohomology 3manifold with coefficients in Z_q for $q\not\in
P$. In addition, M_p is not a homology or cohomology
3manifold 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 3manifolds, 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 3sphere 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 3manifolds 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 3manifolds. The used techniques are based on various combinatorial representations of 3manifolds, such as polyhedral schemes, Heegaard diagrams, branched coverings, and Dehn surgery.
2016
 PD_4Complexes: 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 nondegenerated 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 wellknown 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 2generator Solmanifolds 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 3sphere branched over specified links. Finally, we give a surgery description of the considered 3manifolds; indeed, they can be obtained by n−2 and 0 Dehn surgeries along the components of the Whitehead link.
2013
 CUSPED HYPERBOLIC 3MANIFOLDS 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
3manifolds obtained from certain triangulated 3cells 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 nfold (nonstrongly) cyclic coverings of the
3sphere 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 3manifolds 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)
2bridge 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 2bridge knots, have Heegaard genus 2 and are 2fold coverings of the 3sphere branched over wellspecified 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 BrowderLivesay
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 BrowderQuinn 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 edgecolored 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 graphtheoretical 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 3manifolds 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 2fold branched coverings of the 3sphere. Finally, we obtain some topological applications on the manifolds given byexceptional surgeries on hyperbolic 2bridge 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 2parametric 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 2fold coverings of the 3sphere 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 edgecolored 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 3manifolds that arise by identifying the faces of a Platonic solid was completely solved in a nice paper of Everitt [B. Everitt, 3manifolds from Platonic solids, Topology Appl. 138 (2004) 253–263]. His work completes the classification begun by Best [L.A. Best, On torsionfree 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 3manifolds from Platonic solids in the spherical and Euclidean cases, and completely classified them, upto homeomorphism. Here we describe many topological properties of closed hyperbolic 3manifolds 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 BrowderLivesay invariants
[Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Muranov, Y. u. V.; Spaggiari, Fulvia
abstract
The BrowderLivesay invariants provide obstructions to the realization of elements of Wall groups by normal maps of closed manifolds. A generalization of the iterated BrowderLivesay 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 BrowderLivesay invariants and the BrowderQuinn 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 HONEYCOMB SPACES
[Articolo su rivista]
E., Barbieri; Cavicchioli, Alberto; Spaggiari, Fulvia
abstract
We study the topology and geometry of some series of closed connected orientable 3manifolds constructed as honeycomb spaces. These manifolds are quotients of certain polyhedral 3cells 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 3sphere 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 BrowderLivesay 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 BrowderQuinn 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 3manifolds that arise by identifying the faces of a Platonic solid was completely solved in a nice paper of Everitt [B. Everitt, 3manifolds from Platonic solids, Topology Appl. 138 (2004) 253–263]. His work completes the classification begun by Best [L.A. Best, On torsionfree 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 3manifolds 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 4dimensional surgery problem for a closed connected orientable topological 4manifold with vanishing second homotopy and whose fundamental group is isomorphic to a free product of an oneended group with a nontrivial free group. Our result is related to a theorem of Krushkal and Lee.
2008
 CLASSIFYING COMBINATORIAL 4MANIFOLDS 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 4manifolds of reduced complexity less than or equal to 14.
2008
 Classifying combinatorial 4manifolds 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 4manifolds 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 edgecolored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed ndimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 4manifolds of reduced complexity less than or equal to 14.
2008
 Graphs encoding 3manifolds 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 3manifolds by a 6parameter 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 3manifold. 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 HellingKimMennicke groups and manifolds
[Articolo su rivista]
E., Barbieri; Cavicchioli, Alberto; Spaggiari, Fulvia
abstract
We construct an infinite family of closed connected orientable 3manifolds by pairwise identifications of faces in the boundary of certain polyhedral 3cells. We determine geometric presentations of the fundamental group, and study the split extension of it. Then we prove that these manifolds are nfold cyclic coverings of the 3sphere 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 nspace whose associated contracted complex is minimal with respect to the number of nsimplexes. 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 3manifolds. These manifolds are constructed from triangulated 3balls by identifying pairs of boundary faces via orientationreversing 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 CassonWalkerLescop 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 3space, 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 3cell consisting of 8n pentagons. The tessellation produces a family of closed connected orientable 3manifolds 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 lowdimensional 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 highdimensional 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 3manifolds 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 polysurface fundamental groups
[Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Spaggiari, Fulvia
abstract
We study closed topological 2ndimensional manifolds M with polysurface fundamental groups. We prove that if M is simple homotopy equivalent to the total space E of a Ybundle over a closed aspherical surface, where Y is a closed aspherical
nmanifold, then M is scobordant to E. This extends a wellknown 4dimensional result of Hillman (1991) to higher dimensions. Our proof is different from that of the quoted paper: we use MayerVietoris techniques and the properties of the Ltheory 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 nmanifold 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 nmanifold 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 3manifolds introduced by Kim and Kostrikin in Sbornik Math. 188 (1997) as quotient spaces of certain polyhedral 3cells by pairwise identifications of their boundary faces. Then we study further classes of closed orientable 3manifolds 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 BrowderQuinn 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 twoparameter family of Heegaard diagrams, constructed by Ochiai, encodes the genuine threesphere. The result is obtained, up to isotopy, by using a sequence of only three moves in this order: a WhiteheadZieschang reduction, a band sum and a cancellation of a handle.
2006
 Topology of fourmanifolds with special homotopy groups
[Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract
We study the homotopy type and the scobordism class of a closed connected topological 4manifold 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 nonrealizable 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 3space, all of whose subfigures are realizable as such projections.
2005
 Connected sums of 4manifolds
[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 Asubmodule. When do there exist closed connected 4manifolds 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 wellknown 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 3manifolds of finite volume. Then we investigate the asphericity for many presentations contained in our family.
2005
 On the PontryaginSteenrodWu theorem
[Articolo su rivista]
D., Repovs; M., Skopenkov; Spaggiari, Fulvia
abstract
We present a short and direct proof (based on the PontryaginThom construction) of the following PontryaginSteenrodWu 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 21 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 3manifolds (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 2fold coverings, and give examples of nonequivalent knots with the same tetrahedron manifold as 2fold 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 3manifolds. 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 3manifolds obtained by Dehn surgery with rational coefficients along the components of certain periodic links. These manifolds were introduced in [Osaka J. Math. 39 (2002), 705721] as natural generalizations of Takahashi manifolds. In this note we reobtain the result of [Osaka J. Math. 39 (2002)] by a different approach based on a grouptheoretic argument from [Tsukuba J. Math. 22 (1998), 723739]. 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 fourmanifolds up to homotopy type
[Articolo su rivista]
Cavicchioli, Alberto; Ruini, Beatrice; Spaggiari, Fulvia
abstract
Let M be a closed connected oriented topological 4manifold. Suppose that there is a degree one map f from M to another closed topological 4manifold 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 simplyconnected closed topological 4manifold 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 4manifolds with special homotopy complete the paper.
2003
 Fourmanifolds with π1free second homotopy
[Articolo su rivista]
Spaggiari, Fulvia
abstract
We study the homotopy type of closed connected orientable topological 4manifolds M with Lambdafree 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 4manifolds with free fundamental group. Other applications on special classes of closed topological manifolds complete the paper.
2003
 On calculation of the Witten invariants of 3manifolds
[Articolo su rivista]
E., Rafikov; D., Repovs; Spaggiari, Fulvia
abstract
In this paper we present a short definition of the Witten invariants of 3manifolds. 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 PrasolovSossinsky 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 3manifolds. 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 fourmanifolds
[Articolo su rivista]
Spaggiari, Fulvia
abstract
We study the stable classification of closed connected oriented spin smooth 4manifolds by using techniques of KervaireMilnor 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_2coefficients.
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 3manifolds obtained by side pairings of boundary faces on certain symmetric polyhedral 3balls. 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 KimKostrikin 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 onesided 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 3dimensional 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 3orbifolds and highdimensional knots, respectively.
2002
 Embedding 4manifolds 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 4manifolds M with trivial H_2(M; Z). Moreover, we investigate the problem of embedding M in 5space or other standard simplyconnected 5manifolds according to Barden's list [Ann. of Math. 82 (1965) 365385]. These results are related with papers [Invent. Math. 77 (1984) 173184; Topology 23 (1984) 257269] 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 Lgroups and natural maps
[Articolo su rivista]
Ruini, Beatrice; Spaggiari, Fulvia
abstract
In this paper we compute surgery (resp. splitting) obstruction groups, here called Lgroups, and natural maps for many diagrams of oriented finite (not necessary abelian) 2groups 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 3manifolds with boundary and their special spines by a class of graphs with extrastructure which are strictly related to ographs. Then we describe a simple algorithm for constructing the boundary of these manifolds by using a list of 6tuples of nonnegative integers. Finally we discuss some combinatorial methods for determining the hyperbolicity equations. Examples of hyperbolic 3manifolds 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 3manifolds, i.e., Heegaard diagrams, branched coverings, and crystallizations (a special class of pseudographs endowed with proper edgecolorings). Exploring the connections between those theories, we prove the validity of a conjecture, stated by Dunwoody in GroupsKorea 1994, Walter de Gruyter, 1995, concerning the class of closed orientable 3manifolds 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 3bridge 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 finiteoriented 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 CWtower of categories due to Baues. This fact allows us to obtain a new formula for the topdimensional obstruction for extending maps to homotopy equivalences. Then we complete the paper with an algebraic characterization of highdimensional handlebodies
1999
 Cyclic branched coverings of 2bridge 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 2bridge 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 4manifolds according to the (regular) genus. In particular, we prove that any closed prime orientable PL 4manifold of genus six is topologically homeomorphic to a lensfiber bundle over the 1sphere. There are good reasons to conjecture that the genus six characterizes the topological product RP^3 X S^1 of the real projective 3space by the 1sphere among closed connected prime orientable 4manifolds.
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 snarksnontrivial cubic graphs of class two, of cyclic edgeconnectivity 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 wellknown 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 3manifolds obtained by Dehn surgeries along certain links, first considered by Takahashi. The interest about such manifolds arises from the fact that they include wellknown families of 3manifolds, 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 2fold coverings of the 3sphere branched along the closures of specified 3string braids. We also describe many of the abovementioned manifolds as nfolds cyclic branched coverings of the 3sphere.
1998
 Topological properties of highdimensional handles
[Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Spaggiari, Fulvia
abstract
We investigate the group of orientationpreserving autohomeomorphisms resp. homotopy selfequivalences of the connected sum X of p copies of S^1 x S^n, for p greater than or equal 1, modulo those pseudoisotopic resp. homotopic to the identity. This result is related to a paper of Hosokawa and Kawauchi on unknotted surfaces in Euclidean 4space, 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 edgecolored 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 3manifolds, dependent on three positive integers and a permutation.
1997
 A splitting theorem for homotopy equivalent 4manifolds
[Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Spaggiari, Fulvia
abstract
We prove a decomposition theorem for closed connected homotopy equivalent smooth fourmanifolds, which partially extends a recent result of Curtis, Freedman, Hsiang and Stong, Invent. Math., 123 (1996), to the nonsimply connected case. Then we study the question of when a homotopy equivalence between closed smooth 4manifolds is homotopic to a topological homeomorphism. In particular, we obtain a new proof of the wellknown uniqueness of closed aspherical smooth 4manifolds with good fundamental groups.
1997
 A splitting theorem for homotopy equivalent smooth
4manifolds
[Articolo su rivista]
Cavicchioli, Alberto; F., Hegenbarth; Spaggiari, Fulvia
abstract
We prove a decomposition theorem for closed connected homotopy equivalent smooth fourmanifolds, which partially extends a recent result of Curtis et al. (1996) to the nonsimply connected case. Then we study the question of when a homotopy equivalence between closed smooth 4manifolds is homotopic to a topological homeomorphism. In particular, we obtain a new proof of the wellknown uniqueness of closed aspherical smooth 4manifolds 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 3manifolds 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 0symmetric 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 RRsystem theory easily implies the theorem of Moser on the classification of 3manifolds 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 5manifolds
[Articolo su rivista]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract
We classify the genus one compact (PL) 5manifolds and prove some results about closed 5manifolds with free fundamental group. In particular, let M be a closed connected orientable smooth 5manifold with free fundamental group. Then we prove that the number of distinct smooth 5manifolds homotopy equivalent to M equals the 2nd Betti number (mod 2) of M.
1992
 The classification of 3manifolds with spines related to Fibonacci groups
[Relazione in Atti di Convegno]
Cavicchioli, Alberto; Spaggiari, Fulvia
abstract
Lecture Notes in Math. The classification of 3manifolds with spines related to Fibonacci groups. We study the topological structure of closed connected orientable 3manifolds which admit spines corresponding to the standard presentation of Fibonacci groups.