Embedded contact homology is an invariant of closed oriented contact 3-manifolds first defined by Hutchings, and is isomorphic to both Heegard Floer homology (by the work of Colin, Ghiggini and Honda) and Seiberg-Witten Floer cohomology (by the work of Taubes). The embedded contact chain complex is defined by counting closed orbits of the Reeb vector field and certain pseudoholomorphic curves in the symplectization of the manifold. As part of their proof that ECH=HF, Colin, Ghiggini and Honda showed that if the contact form is suitably adapted to an open book decomposition of the manifold, then embedded contact homology can be computed by considering only orbits and differentials in the complement of the binding of the open book; this fact was then in turn used to define a knot version of embedded contact homology, denoted ECK, where the (null-homologous) knot in question is given by the binding. In this thesis we start by generalizing these results to the case of rational open book decompositions, allowing us to define ECK for rationally null-homologous knots. In its most general form this is a bi-filtered chain complex whose homology yields ECH of the closed manifold. There is also a hat version of ECK in this situation which is equipped with an Alexander grading equivalent to that in the Heegaard Floer setting, categorifies the Alexander polynomial, and is conjecturally isomorphic to the hat version of knot Floer homology. The main result of this thesis is a large negative $n$-surgery formula for ECK. Namely, we start with an (integral) open book decomposition of a manifold with binding $K$ and compute, for all $n$ greater than or equal to twice the genus of $K$, ECK of the knot $K(-n)$ obtained by performing ($-n$)-surgery on $K$. This formula agrees with Hedden's large $n$-surgery formula for HFK, providing supporting evidence towards the conjectured equivalence between the two theories. Along we the way, we also prove that ECK is, in many cases, independent of the choices made to define it, namely the almost complex structure on the symplectization and the homotopy type of the contact form. We also prove that, in the case of integral open book decompositions, the hat version of ECK is supported in Alexander gradings less than or equal to twice the genus of the knot.

This thesis consists of a summary of two papers dealing with questions related to Legendrian submanifolds of contact manifolds together with exact Lagrangian cobordisms between Legendrian submanifolds. The focus is on studying Legendrian submanifolds from the perspective of their handle decompositions. The techniques used are mainly from Symplectic Field Theory. In Paper I, a series of examples of Legendrian surfaces in standard contact 5-space are studied. For every g > 0, we produce g+1 Legendrian surfaces of genus g, all with g+1 transverse Reeb chords, which lie in distinct Legendrian isotopy classes. For each g, exactly one of the constructed surfaces has a Legendrian contact homology algebra admitting an augmentation. Moreover, it is shown that the same surface is the only one admitting a generating family. Legendrian contact homology with Novikov coefficients is used to classify the different Legendrian surfaces. In particular, we study their augmentation varieties. In Paper II, the effect of a Legendrian ambient surgery on a Legendrian submanifold is studied. Given a Legendrian submanifold together which certain extra data, a Legendrian ambient surgery produces a Legendrian embedding of the manifold obtained by surgery on the original submanifold. The construction also provides an exact Lagrangian handle-attachment cobordism between the two submanifolds. The Legendrian contact homology of the submanifold produced by the Legendrian ambient surgery is then computed in terms of pseudo-holomorphic disks determined by data on the original submanifold. Also, the cobordism map induced by the exact Lagrangian handle attachment is computed. As a consequence, it is shown that a sub-critical standard Lagrangian handle attachment cobordism induces a one-to-one correspondence between the augmentations of the Legendrian contact homology algebras of its two ends.

L'homologie de contact est un invariant homologique pour variétés de contact dont la définition est basée sur l'utilisation de courbes holomorphes. Ce travail de thèse concerne l'étude de cet invariant dans le cas des produits de contact. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

This thesis is devoted to the study of the effect of Legendrian surgery on contact manifolds. In particular, we study the effect of this surgery on the Reeb dynamics of the contact manifold on which we perform such a surgery along Legendrian links. We obtain an exact sequence of cyclic Legendrian homology for the Legendrian links. Then we present the applications in 3-dimension and higher dimensions. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

This thesis consists of three scientific papers dealing with invariants of Legendrian and Lagrangian submanifolds. Besides the scientific papers, the thesis contains an introduction to contact and symplectic geometry, and a brief outline of Symplectic field theory with focus on Legendrian contact homology. In Paper I we give an orientation scheme for moduli spaces of rigid flow trees in Legendrian contact homology. The flow trees can be seen as the adiabatic limit of sequences of punctured pseudo-holomorphic disks with boundary on the Lagrangian projection of the Legendrian. So to equip the trees with orientations corresponds to orienting the determinant line bundle of the dbar-operator over the space of Lagrangian boundary conditions on the punctured disk. We define an  orientation of this line bundle and prove that it is well-defined in the limit. We also prove that the chosen orientation scheme gives rise to a combinatorial algorithm for computing the orientation of the trees, and we give an explicit description of this algorithm. In Paper II we study exact Lagrangian cobordisms with cylindrical Legendrian ends, induced by Legendrian isotopies. We prove that the combinatorially defined DGA-morphisms used to prove invariance of Legendrian contact homology for Legendrian knots over the integers can be derived analytically.  This is proved using the orientation scheme from Paper I together with a count of abstractly perturbed flow trees  of the Lagrangian cobordisms. In Paper III we prove a flexibility result for closed, immersed Lagrangian submanifolds in the standard symplectic plane.

L'objectif de cette thèse est d'investiguer la relation entre l'homologie de contact Legendrienne d'une variété de contact de dimension 3, et l'entropie topologique des flots de Reeb associés à cette variété de contact. Une variété de contact est une variété differentielle M de dimension impaire munie d'un champ d'hyperplan Y maximalement non-intégrable. Les champs de Reeb sont une classe speciale de champs de vecteurs sur M qui sont définis en utilisant la structure de contact; ils préservent la structure de contact et ils préservent aussi une forme de volume sur M.<p><p>L'entropie topologique h est un nombre non-négatif qu'on associe à un système dynamique et qui mesure la complexité de ce système. Si un système dynamique est d'entropie topologique positive, on dit que ce système est chaotique.<p><p>Comme les champs de Reeb sont construits en utilisant la structure de contact Y, il est naturel d'attendre que la topologie de (M,Y) influence la dynamique des champs de Reeb auxquels elle est associée. En particulier, il est naturel de se demander s'il existe des variétés de contact dont tous les champs de Reeb associés ont une entropie topologique positive. Si une varieté de contact a cette propriété, on dira qu'elle est d'entropie positive. <p><p>Macarini et Schlenk ont été les premiers à étudier cette question. Ils ont montré qu'il existe un grand ensemble de variétés différentielles Q, telles que le fibré unitaire T_1 Q muni de sa structure de contact canonique Y_{can} est d'entropie topologique positive. Plus précisement, ils ont utilisé l'homologie de Floer Lagrangienne, qui est un invariant symplectique, pour montrer que si Q est rationnellement hyperbolique alors (T_1 Q,Y_{can}) est d'entropie positive. <p><p>Pour étudier l'entropie topologique dans le cas où M n'est pas un fibré unitaire on substitue à l'homologie de Floer Lagrangienne un invariant plus naturel des variétés de contact: l'homologie de contact Legendrienne à bandes. On demontre dans cette thèse que l'homologie de contact Legendrienne à bandes est bien adaptée pour étudier l'entropie topologique. Plus précisement, on montre que quand l'homologie de contact Legendrienne à bandes est bien définie pour un champ de Reeb associé à (M,Y) et sa croissance est exponentielle, alors (M,Y) est d'entropie positive. <p><p>On utilise ce résultat pour trouver des nouveaux exemples de variétés de contact de dimension 3 qui sont d'entropie positive. On montre même qu'il y a des variétés de dimension 3 qui possèdent une infinité de structures de contact différentes qui sont toutes d'entropie positive. Ces résultats et bien d'autres nous permettent de conjecturer que la ``plupart' des variétés de contact de dimension 3 sont d'entropie positive. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished