Seifert fibred 3-manifolds were originally defined and classified by Seifert. Scott gives a survey of results connected with these classical Seifert spaces, in particular he shows they correspond to 3-manifolds having one of six of the eight 3-dimensional geometries (in the sense of Thurston). Essentially, a classical Seifert manifold is a S1-bundle over a 2-orbifold. More generally, a Seifert manifold is the total space of a bundle over a 2-orbifold with flat fibres. It is natural to ask if these generalised Seifert manifolds describe geometries of higher dimension. Ue has considered the geometries of orientable Seifert 4-manifolds (which have general fibre a torus). He proves that (with a finite number of exceptions orientable manifolds of eight of the 4-dimensional geometries are Seifert fibred. However, Seifert manifolds with a hyperbolic base are not necessarily geometric. In this paper, we seek to extend Ue's work to the non-orientable case. Firstly, we will show that Seifert spaces over an aspherical base are determined (up to fibre preserving homeomorphism) by their fundamental group sequence. Furthermore when the base is hyperbolic, a Seifert space is determined (up to fibre preserving homeomorphism) by its fundamental group. This generalises the work of Zieschang, who assumed the base has no reflector curves, the fibre was a torus and that a monodromy of a loop surrounding a cone point is trivial. Then we restrict to the 4 dimensional case and find necessary and sufficient conditions for Seifert 4 manifolds over hyperbolic or Euclidean orbifolds to be geometric in the sense of Thurston. Ue proved that orientable Seifert 4-manifolds with hyperbolic base are geometric if and only if the monodromies are periodic, and we will prove that we can drop the orientable condition. Ue also proved that orientable Seifert 4-manifolds with a Euclidean base are always geometric, and we will again show the orientable assumption is unnecessary.

Seifert fibred 3-manifolds were originally defined and classified by Seifert. Scott gives a survey of results connected with these classical Seifert spaces, in particular he shows they correspond to 3-manifolds having one of six of the eight 3-dimensional geometries (in the sense of Thurston). Essentially, a classical Seifert manifold is a S1-bundle over a 2-orbifold. More generally, a Seifert manifold is the total space of a bundle over a 2-orbifold with flat fibres. It is natural to ask if these generalised Seifert manifolds describe geometries of higher dimension. Ue has considered the geometries of orientable Seifert 4-manifolds (which have general fibre a torus). He proves that (with a finite number of exceptions orientable manifolds of eight of the 4-dimensional geometries are Seifert fibred. However, Seifert manifolds with a hyperbolic base are not necessarily geometric. In this paper, we seek to extend Ue's work to the non-orientable case. Firstly, we will show that Seifert spaces over an aspherical base are determined (up to fibre preserving homeomorphism) by their fundamental group sequence. Furthermore when the base is hyperbolic, a Seifert space is determined (up to fibre preserving homeomorphism) by its fundamental group. This generalises the work of Zieschang, who assumed the base has no reflector curves, the fibre was a torus and that a monodromy of a loop surrounding a cone point is trivial. Then we restrict to the 4 dimensional case and find necessary and sufficient conditions for Seifert 4 manifolds over hyperbolic or Euclidean orbifolds to be geometric in the sense of Thurston. Ue proved that orientable Seifert 4-manifolds with hyperbolic base are geometric if and only if the monodromies are periodic, and we will prove that we can drop the orientable condition. Ue also proved that orientable Seifert 4-manifolds with a Euclidean base are always geometric, and we will again show the orientable assumption is unnecessary.

A Seifert surface for a knot in ℝ³ is a compact orientable surface whose boundary is the knot. Seifert surfaces are not unique. In 1934 Herbert Seifert provided a construction of such a surface known as the Seifert Algorithm, using the combinatorics of a projection of the knot onto a plane. This thesis presents another construction of a Seifert surface, using differential geometry and a projection of the knot onto a sphere. Given a knot K : S¹⊂ R³, we construct canonical maps F : ΛdiffS² → ℝ=4πZ and G : ℝ³ - K(S¹) → ΛdiffS² where ΛdiffS² is the space of smooth loops in S². The composite FG : ℝ³ - K(S¹) → ℝ=4πZ is a smooth map defined for each u∈2 ℝ³ - K(S¹) by integration of a 2- form over an extension D² → S² of G(u) : S1 → S². The composite FG is a surjection which is a canonical representative of the generator 1∈H¹(ℝ³- K(S¹)) = Z. FG can be defined geometrically using the solid angle. Given u ∈ ℝ³ - K(S¹), choose a Seifert surface Σu for K with u ∉ Σu. It is shown that FG(u) is equal to the signed area of the shadow of Σu on the unit sphere centred at u. With this, FG(u) can be written as a line integral over the knot. By Sard's Theorem, FG has a regular value t ∈ ℝ=4πZ. The behaviour of FG near the knot is investigated in order to show that FG is a locally trivial fibration near the knot, using detailed differential analysis. Our main result is that (FG)-¹(t)⊂ ℝ³ can be closed to a Seifert surface by adding the knot.

Cette thèse aborde l'étude de quelques propriétés riemanniennes des structures de contact en dimension trois et de leurs relations avec la topologie de ces structures. Dans la première partie on décrit diverses notions de courbure de champs de plans sur des variétés riemanniennes de dimension trois en comparant plusieurs approches préexistantes mais souvent mal connues. Dans la deuxième partie on présente les techniques topologiques d'étude des structures de contact en dimension trois. Enfin la troisième partie, qui rassemble l'essentiel des résultats nouveaux de cette thèse, est une étude des structures de contact géodésibles en dimension trois à l'aide des outils présentés dans la deuxième partie.

Suppose K is a fibered knot with bi-orderable knot group. We perform a topological winding operation to half-twist bands in a free incompressible Seifert surface Σ of K. This results in a Seifert surface Σ' with boundary that is a non-fibered knot K'. We call K a fibered base of K'. A fibered base exists for a large class of non-fibered knots. We prove K' has a bi-orderable knot group if Σ' is obtained from applying the winding operation to only one half-twist band of Σ. Utilizing a Seifert surface gluing technique, we obtain HNN extension group presentations for both knot groups that differ by only one relation. To show the knot group of K' is bi-orderable, we apply the following: Let G be a bi-ordered free group with order preserving automorphism ɑ. It is well known that the semidirect product ℤ ×ɑG is a bi-orderable group. If X is a basis of G, a presentation of ℤ ×ɑG is ⟨ t,X | R ⟩, where the relations are R = {txt-1}ɑ(x)-1 : x ∈ X}. If R' is any subset of R, we prove that the group H =⟨ t,X | R' ⟩ is bi-orderable. H is a special case of an HNN extension of G. Finally, we add new relations to the group presentation of H such that bi-orderability is preserved.

[pt] Iniciamos com o estudo das orbifolds, que são espaços topológicos localmente homeomorfos a quocientes de Rn por grupos finitos. Estudamos em seguida os fibrados de Seifert de dimensão três, que consistem-se de folheações por círculos que podem ser vistas como fibrados sobre orbifolds. Esse material é usado em seguida no estudo das geometrias modelo. Uma geometria modelo (ou geometria de Thurston) é um par (G;X), onde X é uma variedade conexa e simplesmente conexa e G é um grupo de difeomorfismos de X com certas propriedades que nos permite encontrar uma métrica riemanniana em X tal que G é o grupo de todas as isometrias. A classificação das geometrias modelo é muito útil na classificação topológica das variedades que admitem uma métrica localmente homogênea e foi feita por Thurston em Three-Dimensional Geometry and Topology, vol.1, Princeton University Press, 1997. Na seqüência, apresentamos uma breve descrição de cada geometria modelo bem como parte da prova do teorema de classificação das geometrias modelo. / [en] We begin by studying orbifolds, i.e., topological spaces locally homeomorphic to quotients of Rn by finite groups. Then we study Seifert fiber spaces of dimension three which are certain type of foliations by circles that can be seen as fiber bundles over orbifolds. This material is useful in the subsequent study of Thurston model geometries. A Thurston model geometry is a pair (G;X), where X is a connected and simply connected manifold and G is a group of diffeomorfisms of X with certain properties that allow us to find a riemannian metric on X such that G is the group of all isometries. The classification of the model geometries is very useful in the topological classification of manifolds that admit a locally-homogeneous metric and was done by Thurston in Three-Dimensional Geometry and Topology, vol.1, Princeton University Press, 1997. Then we give a brief description of each one of these eight geometries and present part of Thurston s classification theorem.

Berge introduced knots that are primitive/primitive with respect to the standard genus 2 Heegaard surface, F, for the 3-sphere; surgery on such knots at the surface slope yields a lens space. Later Dean described a similar class of knots that are primitive/Seifert with respect to F; surgery on these knots at the surface slope yields a Seifert fibered space. The examples Dean worked with are among the twisted torus knots. In Chapter 3, we show that a given knot can have distinct primitive/Seifert representatives with the same surface slope. In Chapter 4, we show that a knot can also have a primitive/primitive and a primitive/Seifert representative that share the same surface slope. In Section 5.2, we show that these two results are part of the same phenomenon, the proof of which arises from the proof that a specific class of twisted torus knots are fibered, demonstrated in Section 5.1. / text

In this thesis, we study the classification problem of Stein fillable tight contact structures on any Seifert fibered 3&minus / manifold M over S 2 with 4 singular fibers. In the case e0(M) &middot / &minus / 4 we have a complete classification. In the case e0(M) &cedil / 0 we have obtained upper and lower bounds for the number of Stein fillable contact structures on M.

In 2-dimensional topology, quotients of polygons by edge-pairings provide a rich source of examples of closed, connected, orientable surfaces. In fact, they provide all such examples. The 3-dimensional analogue of an edge-pairing of a polygon is a face-pairing of a faceted 3-ball. Unfortunately, quotients of faceted 3-balls by face-pairings rarely provide us with examples of 3-manifolds due to singularities that arise at the vertices. However, any face-pairing of a faceted 3-ball may be slighted modified so that its quotient is a genuine manifold, i.e. free of singularities. The modified face-pairing is called a twisted face-pairing. It is natural to ask which closed, connected, orientable 3-manifolds may be obtained as quotients of twisted face-pairings. In this paper, we focus on a special class of face-pairings called cycle-free twisted face-pairings and give description of their quotient spaces in terms of integer weighted graphs. We use this description to prove that most spherical 3-manifolds can be obtained as quotients of cycle-free twisted face-pairings, but the Poincaré homology 3-sphere cannot. / Master of Science

Dans cette thèse, on étudie les C2-feuilletages de codimension 1, dans les 3-variétés compactes connexes et orientables. Il est bien connu que l’on peut construire explicitement sur de telles variétés un feuilletage qui possède des composantes de Reeb. Vient alors le problème crucial d’existence des feuilletages tendus (toujours ouvert).Rappelons qu’un feuilletage tendu n’admet pas de composante de Reeb, mais que la réciproque est fausse.La première partie de ce travail, consiste à comprendre la différence entre un feuilletage non-tendu sans composante de Reeb et un feuilletage tendu. On verra que l’orientation transverse des feuilles toriques joue un rôle crucial, en donnant une condition nécessaire et suffisante sur cette orientation transverse pour qu’un feuilletage soit tendu. Pour cela on étudiera de près les procédés géométriques de tourbillonement et de spiralement, et on montrera qu’ils apparaissent toujours au voisinage d’une feuille torique.La seconde partie de ce travail se concentre sur le problème d’existence de feuilletages tendu. Rappelons que depuis les travaux de D. Gabai [1983], on sait que si une 3-variété admet une homologie non-triviale, alors elle admet un feuilletage tendu. Mais le problème d’existence est toujours ouvert parmi les sphères d’homologies, et on s’intéresse ici à celles qui sont fibrées de Seifert. On montre que toutes les sphères d’homologie entière fibrées de Seifert sauf S3 et la sphère d’homologie de Poincaré admettent un feuilletage tendu. Par contre, parmi les sphères d’homologie rationnelle non-entière, fibrées de Seifert, il existe une infinité de telles variétés qui admettent un feuilletage tendu, et une infinité qui n’en admettent pas. / In this thesis, we study codimension 1, C2-foliations, in compact, connected and orientable 3-manifolds. It is well known that we can explicitly construct on such manifolds a foliation admitting Reeb components. Then comes the crucial problem of existence of taut foliation (still opened).Recall that a taut foliation does not admit a Reeb component, but the converse is false. The first step of this work focuses on the difference between a non-taut and Reebless foliation, and a taut foliation. We will understand that the transverse orientation of the torus leaves plays a key-role, by giving a necessary and sufficient condition on the transverse orientation, for a foliation to be taut. For this, we will study the geometric processes of turbulization and spiraling with generalizations, and we see that they always appear in a neighborhood of a torus leaf.The second step of this work is concentrated on the problem of existence of taut foliations. Recall that since the work of D. Gabai [1983], we know that if a 3-manifold has non-trivial homology, then it admits a taut foliation. This problem is still opened among homology spheres and we focus here on Seifert fibered ones. We show that all Seifert fibered integral homology spheres (but S3 and Poincar ́e homology sphere) admit a taut foliation. Nevertheless, among Seifert fibered rational (and non-integral) homology spheres, there exist infinitely many which admit a taut foliation and infinitely many which do not admit one.