Auswahl der wissenschaftlichen Literatur zum Thema „Double Branched Covers“

We prove a folklore theorem of Thurston, which provides necessary and sufficient conditions for primality of a certain class of theta-curves. Namely, a theta-curve in the 3-sphere with an unknotted constituent knot [Formula: see text] is prime, if and only if lifting the third arc of the theta-curve to the double branched cover over [Formula: see text] produces a prime knot. We apply this result to Kinoshita’s theta-curve.

We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and for double covers of projective spaces branched over smooth quadric hypersurfaces is a combination of two operations: one interchanges a quadric hypersurface with its classical projective dual and the other interchanges a quadric hypersurface with the double cover branched along it.

AbstractWe give a topological interpretation of the core group invariant of a surface embedded in S4 [F-R], [Ro]. We show that the group is isomorphic to the free product of the fundamental group of the double branch cover of S4 with the surface as a branched set, and the infinite cyclic group. We present a generalization for unoriented surfaces, for other cyclic branched covers, and other codimension two embeddings of manifolds in spheres.

The double branched cover is a construction which provides a link between problems in knot theory and other questions in low-dimensional topology. Given a knot in a 3-manifold, the double branched cover gives a natural way of associating a 3-manifold to the knot. Similarly, the double branched cover of a properly embedded surface in a 4-manifold is a 4-manifold whose boundary is the double branched cover of the boundary link of the surface. Consequently, whenever a link in S^3 bounds certain types of surfaces, its double branched cover will bound a 4-manifold of an appropriate type. The most familiar situation in which this connection is used is the application to slice knots as the double branched cover of a smoothly slice knot is the boundary of a smooth rational ball. Examples of 3-manifolds which bound rational balls can therefore easily be constructed by taking the double branched covers of slice knots while obstructions to a 3-manifold bounding a rational ball can be interpreted as slicing obstructions. This thesis is primarily concerned with two different extensions of this idea. Given a closed, orientable 3-manifold, it is natural to ask whether it admits a smooth embedding in the four-sphere $S^4$. Examples can be obtained by taking the double branched covers of doubly slice links. These are links which are cross-sections of an unknotted embedding of a two-sphere in S^4. Certain links can be shown to be doubly slice via ribbon diagrams with appropriate properties. Other embeddings can be obtained via Kirby calculus. On the other hand, many obstructions to a 3-manifold bounding a rational ball can be adapted to give stronger obstructions to embedding smoothly in S^4. Using an obstruction based on Donaldson's theorem on the intersection forms of definite 4-manifolds, we determine precisely which connected sums of lens spaces smoothly embed. This method also gives strong constraints on the Seifert invariants of Seifert manifolds which embed when either the base orbifold is non-orientable or the first Betti number is odd. Other applicable methods, also based on obstructions to bounding a rational ball, include the d invariant from Ozsvath and Szabo's Heegaard-Floer homology and the Neumann-Siebenmann mu-bar invariant. These are used, in conjunction with some embedding results derived from doubly slice links, to examine the question of when the double branched cover of a 3 or 4 strand pretzel link embeds. The fact that the double branched cover of a slice knot bounds a rational ball has a second interpretation in terms of knot concordance. In this viewpoint, the double branched cover gives a homomorphism from the concordance group of knots to the rational cobordism group of rational homology 3-spheres. This can be extended to a concordance group of links using a notion of concordance based on Euler characteristic. This yields link concordance groups which contain the knot concordance group as a direct summand with an infinitely generated complement. The double branched cover homomorphism extends to large subgroups containing the knot concordance group.

La première partie du seizième problème de Hilbert traite de la topologie des courbes algébriques réelles régulières dans le plan projectif. Il est bien connu que bon nombre des propriétés topologiques satisfaites par de telles courbes sont également vraies pour la classe plus large des courbes flexibles, introduites par O. Viro en 1984. Le but de cette thèse est d'approfondir les origines topologiques des restrictions sur les courbes réelles, en lien avec le seizième problème de Hilbert. Nous ajoutons une condition naturelle à la définition de courbe flexible, à savoir qu'elles doivent intersecter une conique réelle vide Q comme une courbe algébrique, c'est-à-dire en des points positifs uniquement. Nous voyons CP(2) comme un cylindre sur un espace lenticulaire L(4,1)×R, que l'on compactifie en ajoutant RP(2) et Q aux bords, et nous utilisons la décomposition induite sur S(4)=CP(2)/conj. C'est un fait standard que la surface d'Arnold joue un rôle essentiel dans l'étude des courbes de degré pair. Nous introduisons un analogue de cette surface pour des courbes de degré impair. Nous généralisons également la notion de courbe flexible pour inclure des surfaces non orientables. Nous considérons qu'une courbe flexible est de degré m si son auto-intersection est m² et si elle intersecte la conique Q de manière transverse en exactement 2m points. Notre résultat principal affirme que pour une telle courbe flexible (non nécessairement orientable) de degré impair m=2k+1 ne peut pas posséder plus de -χ(F)/2-k²+k+1, où χ(F) est la caractéristique d'Euler de F. Cette borne supérieure se simplifie en k² dans le cas d'une courbe flexible au sens usuel. Nous généralisons également notre résultat pour des courbes flexibles sur des quadriques, ce qui produit une nouvelle restriction, même pour des courbes algébriques. Dans les chapitres introductifs, un aperçu détaillé de la théorie classique des courbes réelles planes est fait, en s'appuyant aussi bien sur le point de vue réel que complexe. Certains résultats à propos de la théorie des surfaces nouées dans les 4-variétés sont énoncés. Plus précisément, il est question de faits concernant la classe d'Euler du fibré normal d'une surface plongée. Cela nous amène ensuite à considérer la fonction de genre non-orientable d'une 4-variété. Cela constitue un analogue de la conjecture de Thom (résolue par Kronheimer et Mrowka en 1994) pour des surfaces non orientables. Nous calculons presque totalement cette fonction pour CP(2), et nous étudions cette fonction sur d'autres 4-variétés. Enfin, nous digressons autour de la nouvelle notion de courbes flexibles non orientables, où nous dressons une liste de résultats connus qui restent vrai dans ce cadre. Nous nous concentrons aussi sur la classe des courbes algébriques et flexibles qui sont invariantes sous l'action d'une involution holomorphe de CP(2), une notion introduite par T. Fiedler et appelées courbes symétriques. Nous donnons un état de l'art, et nous formulons une succession de petits résultats à propos de la disposition d'une courbe symétrique par rapport aux éléments de symétrie. Nous proposons également une approche pour tenter de généraliser la congruence de Fiedler p-n≡k² [16], valable pour des M-courbes symétriques de degré 2k, à des (M-1)-courbes symétriques de degré 2k
The first part of Hilbert's sixteenth problem deals with the topology of non-singular real plane algebraic curves in the projective plane. As well-known, many topological properties of such curves are shared with the wider class of flexible curves, introduced by O. Viro in 1984. The goal of this thesis is to further investigate the topological origins of the restrictions on real curves in connection with Hilbert's sixteenth problem. We add a natural condition to the definition of flexible curves, namely that they shall intersect an empty real conic Q like algebraic curves do, i.e. all intersections are positive. We see CP(2) as a cylinder over a lens space L(4,1)×R which is compactified by adding RP(2) and Q respectively to the ends, and we use the induced decomposition of S(4)=CP(2)/conj. It is a standard fact that Arnold's surface plays an essential role in the study of curves of even degree. We introduce an analogue of this surface for curves of odd degree. We generalize the notion of flexible curves further to include non-orientable surfaces as well. We say that a flexible curve is of degree m if its self-intersection is m² and it intersects the conic Q transversely in exactly 2m points. Our main result states that for a not necessarily orientable curve of odd degree 2k+1, its number of non-empty ovals is no larger than χ(F)/2-k²+k+1, where χ(F) is the Euler characteristic of F. This upper bound simplifies to k² in the case of a usual flexible curve. We also generalize our result for flexible curves on quadrics, which provides a new restriction, even for algebraic curves. In the introductory chapters, a thorough survey of the classical theory of real plane curves is outlined, both from the real and the complex points of view. Some results regarding the theory of knotted surfaces in 4-manifolds are laid down. More specifically, we review statements involving the Euler class of normal bundles of embedded surfaces. This eventually leads us to consider the non-orientable genus function of a 4-manifold. This forms a non-orientable counterpart of the Thom conjecture, proved by Kronheimer and Mrowka in 1994 in the orientable case. We almost entirely compute this function in the case of CP(2), and we investigate that function on other 4-manifolds. Finally, we digress around the new notion of non-orientable flexible curves, where we survey which known results still hold in that setting. We also focus on algebraic and flexible curves invariant under a holomorphic involution of CP(2), a smaller class of curves introduced by T. Fiedler and called symmetric curves. We give a state of the art, and we formulate a collection of small results results regarding the position of a symmetric plane curve with respect to the elements of symmetry. We also propose a possible approach to generalize Fiedler's congruence p-n≡k² [16], holding for symmetric M-curves of even degree 2k, into one for symmetric (M-1)-curves of even degree

Hurwitz a montre qu’un revêtement ramifié f:M→N de surfaces avec lieu de ramification P⊂N détermine et est déterminé, à un automorphisme intérieur près du groupe symétrique S_m , par un homomorphisme π_1(NP, ∗) → S_m . Ce résultat réduit les questions d’existence et d’unicité à un problème combinatoire. Pour un ensemble de générateurs convenable de π_1(NP, ∗), une représentation π_1(NP, ∗) → S_m détermine et est déterminée par une suite (a_1 , b_1 , . . . , a_g , b_g , z_1, . . . , z_k ) d’éléments de S_m satisfaisant [a_1 , b_1 ] · · · [a_g , b_g ]z_1 · · · z_k = 1. La suite (a_1, b_1 , . . . , a_g , b_g , z_1 , . . . , z_k) de permutations est appelé un système de Hurwitz pour f. Par conséquent, pour comprendre les classes de revêtements ramifiés, on doit étudier les orbites des systèmes de Hurwitz par des actions sur S_m. Une de ces actions est la conjugaison simultanée qui conduit à l’étude de l’ensemble des classes doubles des groupes symétriques. Dans le premier chapitre, nous présentons les travaux récents de Neretin sur la structure multiplicative sur l’ensemble S_∞S^n_∞ /S_∞ . Dans le deuxième chapitre, nous visons étendre les résultats de Neretin au groupe B_∞ des tresses à support fini avec un nombre infini de brins. Nous montrons que B_∞ B^n_∞/B_∞ admet une telle structure multiplicative et expliquons comment cette structure est liée à des constructions similaires dans Aut(F_∞) et GL(∞). Nous définissons également une généralisation à un paramètre de la structure habituelle de monoïde sur l'ensemble des classes doubles de GL(∞) et montrons que la représentation de Burau fournit un foncteur entre les catégories des classes doubles de B_∞ et de GL(∞). Le dernier chapitre est consacré à l'étude des homomorphismes π_1(NP, ∗) → G, où G est un groupe discret. Nous exposons la classification stable de tels homomorphismes selon Samperton et de nouveaux résultats concernant le nombre de stabilisations nécessaires pour les rendre équivalents par rapport aux mouvements de Hurwitz. Nous explorons ensuite une généralisation de la classification des revêtements ramifiés finis en introduisant la monodromie des tresses associée à des surfaces plongées en codimension 2. Suivant des idées de Kamada, nous définissons la monodromie des tresses associée à des surfaces tressées correspondant à G = B_∞ et nous étudions les fonctions sphériques associées aux représentations des groupes des tresses
Hurwitz showed that a branched cover f:M→N of surfaces with branch locus P⊂N determines and is determined, up to inner automorphism of the symmetric group S_m, by a homomorphism π_1(NP, ∗) → S_m . This result reduces the questions of existence and uniqueness of branched covers to combinatorial problems. For a suitable set of generators for π_1(NP, ∗), a representation π_1(NP, ∗) → S_m determines and is determined by a sequence (a_1 , b_1 , . . . , a_g , b_g , z_1, . . . , z_k ) of elements of S_m satisfying [a_1, b_1 ] · · · [a_g , b_g ]z_1 · · · z_k = 1. Thesequence (a_1 , b_1 , . . . , a_g , b_g , z_1 , . . . , z_k ) of permutations is called a Hurwitz system for f .Therefore, to understand the classes of branched covers one need to study the orbits of Hurwitz systems by suitable actions on S^n_m, n = 2g+k. One of such actions is the simultaneous conjugation that leads to the study of the set of double cosets of symmetric groups.In Chapter 1 we bring an exposition of the recent work of Neretin on the multiplicative structure on the set S_∞S^n_∞/S_∞ .In Chapter 2 we aim at extending Neretin’s results to the group B_∞ of finitely supported braids on infinitely many strands. We prove that B_∞B^n_∞/B_∞ admits such a multiplicative structure and explain how this structure is related to similar constructions in Aut(F_∞ ) and GL(∞). We also define a one-parameter generalization of the usual monoid structure on the set of double cosets of GL(∞) and show that the Burau representation provides a functor between the categories of double cosets of B_∞ and GL(∞).The last chapter is dedicated to the study of homomorphisms π_1(NP, ∗) → G, G a discrete group. We give an exposition of the stable classification of such homomorphisms following the work of Samperton and some new results concerning the number of stabilizations necessary to make them equivalent with respect to Hurwitz moves. We also explore a generalization of the classification of finite branched covers by introducing the braid monodromy for surfaces embedded in codimension 2. Following ideas of Kamada we defined a braid monodromy associated to braided surfaces, which correspond to G = B_∞ and study the spherical functions associated to braid group representations

Abstract In Chapter 4 we saw that a dynamical system can have many inequivalent double covers, each with the same symmetry group. This was shown explicitly for double covers of adynamical system exhibiting horseshoe dynamics. The four distinct double covers all possessed R z (n) symmetry but possessed different topological indices (no, n1 ) . The rotation axis of the symmetry group R z (n) linked the image dynamical system in four ways (cf. Fig. 4.2). In creating these four distinct double covers we were careful that the rotation axis did not intersect the strange attractor. This is possible at the level of a branched manifold description of the dynamics and sometimes possible for real strange attractors. But it is not always possible. For example, there is usually no clear gap in a (Rossler-like) strange attractor between orbit segments that project to the branches O and 1 under the Birman-Williams projection.

This chapter looks to the other branch of the legal profession: the attorneys and solicitors. Attorneys and solicitors dealt directly with the client and cover a vast range of legal matters, including wills, property transfers, and other affairs that usually do not need to be tested in court. Attorneys were much more numerous than barristers in Regency England. The social standing of attorneys relative to the gentry, the clergy, and other genteel professions was also open to doubt. They certainly lacked the prestige of barristers. Indeed, attorneys seemed little better than school-teachers or the better sort of shopkeeper: respectable enough in their way, but not a career for the younger sons of the gentry. However, as the chapter shows, attorneys were also considered a rather gentlemanly profession, and many gentlemen indeed prospered by becoming one.

Abstract This chapter examines subject-matter scope, which refers to the scope covered by a measure as regards its subject matter, that is to say the branches and areas of the law to which it applies. In the case of the instruments with which this book is concerned, the relevant provisions are contained in Article 1 of Brussels 2012 and Lugano, and Articles 1 and 2 of Hague. Article 1(1) of all three instruments provides that the instrument in question 'shall apply in civil and commercial matters'. Brussels 2012 and Lugano add 'whatever the nature of the court or tribunal'. This phrase is not found in the Hague Convention, though no doubt it is understood. The importance of the phrase 'whatever the nature of the court or tribunal' is that a civil claim joined to criminal proceedings can nevertheless be within the subject-matter scope of Brussels or Lugano, even though it is ancillary to criminal proceedings. The chapter considers the general concept and then looks at specific problems. The cases explored show that the distinction between a civil matter and a public matter is far from straightforward.