Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.32 (2001) s1

An (n + 1)-coloured graph $\left(\Gamma,\gamma\right)$ is said to be $m-bipartite$ if m is the maximum integer so that every m-residue of $\left(\Gamma,\gamma\right)$ (i.e. every connected subgraph whose edges are coloured by only m colours) is bipartite; obviously, every (n + 1)-coloured graph, with n $\geq$ 2, results to be m-bipartite for some m, with 2 $\leq$ m $\leq$ n + 1. In this paper, a numerical $code$ of length (2n \textminus{} m + 1) $\times$ q is assigned to each m-bipartite (n + 1)-coloured graph of order 2q. Then, it is proved that$any\; two\; such\; graphs\; have\; the\; same\; code\; if\; and\; only\; if\; they\; are\; colour-isomorphic$, i.e. if a graph isomorphism exists, which transforms the graphs one into the other, up to permutation of the edge-colouring. More precisely, if H is a given group of permutations on the colour set, we face the problem of algorithmically recognizing H-isomorphic coloured graphs by means of a suitable defi{}nition of H-code.

Satellite measurements of the cosmic microwave back-ground radiation will soon provide an opportunity to test whether the universe is multiply connected. This paper presents a new algorithm for deducing the topology of the universe from the microwave background data. Unlike an older algorithm, the new algorithm gives the curvature of space and the radius of the last scattering surface as outputs, rather than requiring them as inputs. The new algorithm is also more tolerant of errors in the observational data.

We consider homotopy classes of non-singular vector fields on three-manifolds with boundary and we define for these classes torsion invariants of Reidemeister type. We show that torsion is well-defined and equivariant under the action of the appropriate homology group using an elementary and self-contained technique. Namely, we use the theory of branched standard spines to express the difference between two homotopy classes as a combination of well-understood elementary catastrophes. As a special case we are able to reproduce Turaev’s theory of Reidemeister torsion for Euler structures on closed manifolds of dimension three.

The space of marked n distinct points on the complex projective line up to projective transformations will be called a configuration space. There are two families of complex hyperbolic structures on the configuration space constructed by Deligne- Mostow and by Thurston. We review that these families are the same, and then exhibit the families for n = 4, 5 in constrast with the deformation theory of real hyperbolic cone 3-manifolds.

We calculate all degrees of selfmaps f : M $\rightarrow$ M of a Seifert manifold M with fundamental group of fi{}nite order N, provided that f induces an automorphism of $\pi_{1}$(M). The answer is given in terms of N, namely, the possible degrees $degf$ form the set $\left\{ k^{2}+\ell N\mid gcd\left(k,N\right)=1,k,\ell,\epsilon\mathbb{Z}\right\} $

There is an extensive literature on the fi{}nite index subgroups and the fi{}nite quotient groups of the Picard group $PSL\left(2,\mathbb{Z}\mid i\mid\right)$. The main result of the present paper is the classifi{}cation of all linear fractional groups $PSL\left(2,p^{m}\right)$ which occur as fi{}nite quotients of the Picard group. We classify also the fi{}nite quotients of linear fractional type of various related hyperbolic tetrahedral groups which uniformize the cusped orientable hyperbolic 3-orbifolds of minimal volumes. Also these cusped tetrahedral groups are of Bianchi type, that is of the form $PSL\left(2,\mathbb{Z}\mid\omega\mid\right)$ or $PGL\left(2,\mathbb{Z}\mid\omega\mid\right)$, for suitable $\omega\epsilon\mathbb{C}.$ It turns out that all fi{}nite quotients of linear fractional type of these tetrahedral groups are obtained by reduction of matrix coeffi{}cients mod p whereas for the Picard group most quotients do not arise in this way (as in the case of the classical modular group $PSL\left(2,\mathbb{Z}\right)$. From a geometric point of view, we are looking for hyperbolic 3-manifolds which are regular coverings, with covering groups isomorphic to $PSL\left(2,q\right)$ or $PGL\left(2,q\right)$ and acting by isometries, of the cusped hyperbolic 3-orbifolds of minimal volumes. So these are the cusped hyperbolic 3-manifolds of minimal volumes admitting actions of linear fractional groups. We also give some application to the construction of closed hyperbolic 3-manifolds with large group actions. We are concentrating in this work on quotients of linear fractional type because all fi{}nite quotients of relatively small order of the above groups are of this or closely related types (similar to the case of Hurwitz actions on Riemann surfaces), so the linear fractional groups are the fi{}rst and most important class of fi{}nite simple groups to take into consideration.

Any fi{}nite group admits actions on closed 3-manifolds, and in particular free actions. For actions with fi{}xed points, assumptions on the type of the fi{}xed point sets of elements drastically reduce the types of the possible groups. Concentrating on the basic case of fi{}nite simple groups we show in the present paper that, if some involution of a fi{}nite simple group G acting orientation-preservingly on a closed orientable 3-manifold has nonempty connected fi{}xed point set, then G is isomorphic to a projective linear group PSL(2, q), and thus of a very restricted type. The question was motivated by our work on the possible types of isometry groups of hyperbolic 3-manifolds occuring as cyclic branched coverings of knots in the 3-sphere. We characterize also fi{}nite groups which admit actions on $\mathbb{Z}_{2}$-homology spheres, generalizing corresponding results for integer homology spheres.

A genus reducing knot is a knot that has infinitely many surgeries after which the Heegaard genus of the manifold reduces. We study certain aspects of this question, in particular solving it for totally orientable Seifert Fibered Spaces, where we find examples of manifolds of arbitrarily high genus containing no such knot.

There is a natural way of indexing the difference between two collections of reducing spheres for the standard genus g Heegard splitting of S3. Any two collections are related by a sequence of collections, any two of which differ by low index, hence in a simple way.

We consider hyperbolic 3-manifolds $M_{n}\left(k\right)$, which are n-fold cyclic branched coverings of 2-bridge knots K. We show that for n $\geq$ 5 the orientation-preserving isometry group of $M_{n}\left(k\right)$ either is a lift of a symmetry group of K or has a very special structure.

In the Kirby list is presented the following problem: describe the equivalence classes in the set of knots under the relation K$_{1}$ is equivalent to K$_{2}$ if their 2-fold cyclic branched coverings are homeomorphic 3-manifolds. In this paper we consider the basic case of hyperbolic manifold. In the fi{}rst part of this paper we want to present briefl{}y the results, yet available in some previous works, which solve this problem. In the second part we present examples of knots with the same 2-fold branched covering which show that the theorem, which describes the possible relations between two knots in the same equivalence class, is the best possible.

In this paper we study the homeomorphisms of B$^{2}$ that are liftable with respect to a simple branched covering. Since any such homeomorphism maps the branch set of the covering onto itself and liftability is invariant up to isotopy fi{}xing the branch set, we are dealing in fact with liftable braids. We prove that the group of liftable braids is fi{}nitely generated by liftable powers of half-twists around arcs joining branch points. A set of such generators is explicitly determined for the special case of branched coverings B$^{2}$ $\rightarrow$ B$^{2}$ . As a preliminary result we also obtain the classifi{}cation of all the simple branched coverings of B$^{2}$.

Evaluating the intersection of the trajectory of the exploring finger, with the virtual surface representing the scene, is a key problem in the VIDET project of an aid for the visually impaired. A substitute for Delaunay triangulation, which permits of local computation for that goal, is proposed.

We give a combinatorial representation of compact connected orientable 3-dimensional manifolds with boundary and their special spines by a class of graphs with extrastructure which are strictly related to o-graphs defined and studied in [3] and [4]. 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.

We give a bound on the number of hyperbolic knots which are double covered by a fixed (non hyperbolic) manifold in terms of the number of tori and of the invariants of the Seifert fibred pieces of its Jaco-Shalen-Johannson decomposition. We also investigate the problem of finding the non hyperbolic knots with the same double cover of a hyperbolic one and give several examples to illustrate the results.

We investigate the topology of the space of characters of a free group into $SL_{2}\mathbb{R}$, $SL_{2}\mathbb{C}$, $SO_{2}$, and $SU_{2}$.

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

Accenniamo al comportamento per $t \rightarrow \infty$ di soluzioni della NLS vicino a groud states e alla dimostrazione di stabilità asintotica dei ground states nel caso di minore “complessità interna”. I risultati sono simili a quelli per sistemi integrabili.

We show that every orientation-preserving periodic diffeomorphism f on a closed orientable 3-manifold M has a \textquotedblleft{}surgery description\textquotedblright{}, that is, there is a framed link $\mathcal{L}\;\textrm{in}\; S^{3}$ which is invariant by a standard rotation $\varphi$ around a trivial knot, such that M is obtained by surgery on $\varphi$ and that f is conjugate to the periodic diff{}eomorphism induced by $\varphi$. We will illustrate this result, by visualizing isometries of the complements of 2-component hyperbolic links with $\leq$ 9 crossings which do not extend to periodic maps of $S^{3}$.

It has been proved by Accola that any 3-fold unbranched covering of a Riemann surface of genus two is hyperelliptic (a 2-fold branched covering of the 2-sphere) if the covering is non- regular, and 1-hyperelliptic (a 2-fold branched covering of a torus) if it is regular. In the present paper, we show that the corresponding result holds for closed 3-manifolds when replacing the genus by the Heegaard genus.