First and foremost, I would like to heartily thank my advisor Patrick Massot. He has been extremely supportive and available since the start of my Master thesis, and throughout all my PhD thesis. There's no doubt that I would have never made it through this long (and sometimes arduous) journey without both his patient academical guidance, animated by very enthusiastic and enlightening discussions, and his constant encouragement, which motivated me even during the tougher periods. I am also thankful for the excellent example he has provided me as a successful researcher and professor; it has been a honor for me to be his first PhD student.

I thank Vincent Colin and Hansjörg Geiges, for accepting to be referees of this thesis, for the time that they dedicated to read this manuscript and for the comments and suggestions that improved it, as well as Frédéric Bourgeois, Paul Gauduchon, Klaus Niederkrüger, Claude Viterbo and Thomas Vogel, for accepting to participate to the jury in the day of the defense.

I would also like to show my gratitude to Sylvain Courte and Emmy Murphy, for useful discussions and remarks which contributed to the improvement of this work, as well as to an anonymous referee, for suggesting many improvements on (a submitted version of) arXiv:1704.07278v2, which are now part of Chapter 4.

As far as the mathematical acknowledgements are concerned, I also owe a big thank to Agustin Moreno, Klaus Niederkrüger and Thomas Vogel, for the many interesting mathematical discussions and the time spent sharing ideas and points of view, not only on the content of this thesis, but also on new research directions. I also wish to thank all the mathematicians working in symplectic and contact geometry in the Parisian area, for being extremely active and organizing regularly very interesting seminars and workshops.

As far as the everyday laboratory life is concerned, I am very grateful to all the members of the Centre de mathématiques Laurent Schwartz for the very welcoming and supportive ambiance. Despite my late lunch hours, I had the pleasure to enjoy many coffee talks, mathematical and not, at the everyday post-lunch or afternoon-break coffee talks in the reunion room; this surely made each working day much more pleasurable and enjoyable. I also owe a big thank to all my PhD colleagues (the current and old ones), for many lunches and coffee breaks together and for giving me such a good example of how passion, determination and hard work can literally do miracles. From a more practical point of view, I also wish to thank the secretary and the informatics team of the CMLS, for taking care of every bureaucratic or technical problem and ensuring an efficient working environment.

I would also like to thank all the friends that I met during my Master and PhD, who shared with me the joys and preoccupations of a PhD student's life, as well as all my old friends in Italy, who were always there to cheer me up when needed.

A special final thank of course goes to my family and my girlfriend: they supported me enormously in these last years and they always managed to keep my enthusiasm and hopes for the future very high, despite all the difficulties (work-related and not). i Chapter 1

Introduction générale

Cette thèse porte sur quelques aspects topologiques de la géométrie de contact. Cette dernière est une branche de la géométrie qui étudie les propriétés des structures de contact, i.e. des distributions d'hyperplans ξ sur une variété différentielle de dimension impaire V 2n+1 qui sont définies localement par une 1-forme α telle que α ∧ dα n est une forme volume (sur son domaine). Plus précisément, on se focalise ici sur les structures de contact coorientées, i.e. sur celles qui sont globalement définies par une 1-forme.

Quelques exemples de variétés de contact fermées viennent naturellement du monde symplectique. Par exemple, si W 2n est une variété compacte à bord V 2n-1 , on peut considérer les formes symplectiques ω sur W qui sont "compatibles" avec V dans le sens suivant : il y a, dans un voisinage de V , un champ de Liouville Z (i.e. un champ de vecteurs Z) tel que L Z ω = ω) qui est positivement transverse à V . Dans cette situation, ξ := ker(ι Z ω| V ) est une structure de contact sur V , et (W, ω) est appelée remplissage symplectique (fort) de (V, ξ).

Ceci dit, pas toutes les variétés de contact sont le bord convexe d'une variété symplectique. Des exemples remarquables sont les structures de contact vrillées, définies dans [START_REF] Eliashberg | Classification of overtwisted contact structures on 3manifolds[END_REF] pour le cas 3-dimensionnel et généralisées à toutes dimensions impaires dans [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF] : leur non-remplissabilité suit des résultats dans [START_REF] Gromov | Pseudo holomorphic curves in symplectic manifolds[END_REF][START_REF] Eliashberg | Filling by holomorphic discs and its applications[END_REF] (voir aussi [START_REF] Zehmisch | The Gromov-Eliashberg tightness-theorem[END_REF]) pour le cas de dimension 3 et de [START_REF] Niederkrüger | The plastikstufe-a generalization of the overtwisted disk to higher dimensions[END_REF][START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF] dans le cas général. Les structures de contact remplissables sont alors une sous-classe des structures non vrillées, qui sont aussi appelées tendues.

Les structures de contact vrillées sont une manifestation importante de la nature topologique de la géométrie de contact. En fait, [START_REF] Eliashberg | Classification of overtwisted contact structures on 3manifolds[END_REF][START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF] montre que le problème géométrique de la construction des structures de contact peut être réduit au problème formel de la construction de leur équivalent homotopique : toute structure presque de contact peut être déformée en structure de contact vrillée. De plus, deux structures de contact vrillées sont isotopes si et seulement si elle sont homotopes parmi les structures presque de contact.

Les structures de contact vrillées manifestent donc de la flexibilité concernant le problème de construction géométrique. Au contraire, les structures tendues peuvent présenter des comportements rigides dans ce contexte : par exemple, deux structures tendues qui sont homotopes parmi les structures presque contact ne sont pas nécessairement isotopes. Ceci dit, des résultats récents suggèrent que, selon le problème considéré, les comportements flexible et rigide peuvent interagir de façon inattendue. Par exemple, [Laz] montre que la cohomologie d'un domaine de Weinstein flexible est "codée" dans la variété de contact qu'il remplit, donc que le bord d'un variété de Weinstein a une certaine rigidité. 1 1.1. LA TOPOLOGIE DU GROUPE DES CONTACTOMORPHISMES À cause de ces interactions non triviales, il n'y a pas de séparation nette entre les comportements flexible et rigide flexible en géométrie de contact. Donc, le problème suivant est très intéressant dans ce contexte : Problème A. Étudier les interactions entre flexibilité et rigidité sous points de vue différents.

La réduction du problème géométrique de la construction des structures de contact à son équivalent formel dans [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF] n'est pas seulement importante en vue de l'étude des interactions flexibilité-rigidité, mais aussi puisqu'il répond à la question fondamentale de quelles variétés de grandes dimensions admettent des structures de contact (ou, plus précisément, il montre que la topologie algébrique peut donner une réponse complète). Comme la littérature montre (voir par exemple [Lut79, Gei91, Gei97a, GT98, GT01, Bou02, CPP15, Etn12]), ceci a été pour longtemps un problème fondamental en géométrie de contact, qui a motivé en particulier la recherche de constructions explicites de variétés de contact en grandes dimensions. Bien sûr, depuis le travail [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF], la question de l'existence des structures de contact n'est plus une motivation pour la recherche de constructions explicites. Ceci dit, la question suivante a sans doute encore un certain intérêt : Problème B. Trouver (et étudier) des constructions de variétés de contact avec des propriétés "remarquables", e.g. qui donnent des structures vrillées, tendues, remplissables, etc.

La Partie I de cette thèse porte principalement sur le Problème A. Plus précisément, il contient un étude de la topologie de l'espace des contactomorphismes, avec une analyse de quelques comportements rigides dans le cas des structures tendues et vrillées en grandes dimensions; ceci sera présenté plus en détail dans la Section 1.1. La Partie II porte sur le Problème B et contient, plus précisément, une réinterprétation de quelques constructions explicites déjà connues en grandes dimensions et l'étude des propriétés des variétés de contact qui en résultent. Ceci est présenté en détail dans la Section 1.2.

Avant de présenter les résultats contenus dans Parties I, II de cette thèse, le lecteur qui n'a pas beaucoup de familiarité avec la géométrie de contact pourrait lire Section 2, où on rappelle les objets principaux qui apparaîtront dans les énoncés présentés dans Sections 1.1, 1.2.

La topologie du groupe des contactomorphismes

Dans Partie I de cette thèse, on s'interesse à la topologie de l'espace des contactomorphismes D (V, ξ) d'une variété de contact (V, ξ), en rapport à celle de l'espace des difféomorphismes D (V ) de la variété différentielle V sous-jacente. Plus précisément, on va étudier les propriétés de l'application j * : π k (D (V, ξ)) → π k (D (V )) induite par l'inclusion naturelle j : D (V, ξ) → D (V ).

Soit Ξ (V ) l'espace de toutes les structures de contact sur V . Dans le cas des variétés de contact fermées, l'application naturelle D (V ) → Ξ (V ), donnée par φ → φ * ξ, aide à comprendre les propriétés de j * , et montre que Ξ (V ) joue le rôle d'intermédiaire pour la relation entre la topologie de D (V, ξ) et celle de D (V ). En fait, la preuve du théorème de Gray implique, modulo un critère général de fibration, que cette application est une fibration localement triviale avec fibre D (V, ξ); voir [START_REF] Giroux | On the contact mapping class group of Legendrian circle bundles[END_REF] pour une explication de ce résultat ou [START_REF] Massot | Natural fibrations in contact topology[END_REF] pour une preuve détaillée. (On remarque que, dans [START_REF] Geiges | On the topology of the space of contact structures on torus bundles[END_REF], il est CHAPTER 1. INTRODUCTION GÉNÉRALE demontré que cette application est une fibration de Serre, ce qui est suffisant pour les considérations suivantes.) Donc, la suite exacte longue en homotopie . . . → π k+1 (Ξ (V )) → π k (D (V, ξ)) j * -→ π k (D (V )) → π k (Ξ (V )) → . . . associé à la fibration donne une relation entre les topologies des trois espaces D (V ), D (V, ξ) et Ξ (V ).

Concernant le cas de dimension 3, la disponibilité de résultats de classifications pour les classes d'isotopie des structures de contact tendues sur des 3-variétés spécifiques V donne quelques résultats explicites sur les groupes d'homotopie en dégrée 0 et 1 sur quelques variétés spécifique. Le lecteur intéressé peut consulter [Gom98, GGP04, Bou06, DG10, GK14, GM17] pour des résultats sur π 1 (Ξ (V ) , ξ) et [START_REF] Giroux | Sur les transformations de contact au-dessus des surfaces[END_REF][START_REF] Giroux | On the contact mapping class group of Legendrian circle bundles[END_REF] pour des résultats sur π 0 (D (V, ξ)); bien sûr, les résultats sur ces deux groupes peuvent être reliés entre eux via la suite exacte longue ci-dessus. Plus précisément, dans [START_REF] Geiges | On the topology of the space of contact structures on torus bundles[END_REF], les auteurs démontrent l'existence d'un élément d'ordre infini dans le groupe fondamental de l'espace des structures de contact sur les fibrés en tores sur S 1 ; une preuve indépendante, qui se généralise au cas du tore de dimension 5, est donnée dans [START_REF] Bourgeois | Contact homology and homotopy groups of the space of contact structures[END_REF], en utilisant l'homologie de contact. [START_REF] Geiges | The fundamental group of the space of contact structures on the 3-torus[END_REF] démontre que le groupe fondamental est en effet isomorphe à Z dans le cas de la variété T 3 et point base la structure de contact standard. Ceci a été ensuite généralisé dans [START_REF] Giroux | On the contact mapping class group of Legendrian circle bundles[END_REF] (voir aussi [START_REF] Giroux | Sur les transformations de contact au-dessus des surfaces[END_REF]), où il est démontré que, pour tout fibré V en S 1 sur une surface fermée Σ et toute structure de contact ξ pour laquelle les fibres sont legendriennes, π 1 (Ξ (V ) , ξ) est cyclique infini et ker(j * | π0 ) est cyclique fini. Dans [START_REF] Gompf | Handlebody construction of Stein surfaces[END_REF], l'auteur observe que S 2 × S 1 , avec son unique (à isotopie près) structure de contact tendue ξ std , admet une mapping class de contact d'ordre infini; ceci réapparaît dans [START_REF] Ding | The diffeotopy group of S 1 × S 2 via contact topology[END_REF], où c'est aussi utilisé pour démontrer que ker(j * | π0 ) et π 1 (Ξ S 2 × S 1 , ξ std ) sont isomorphes à Z.

La situation en grandes dimensions est moins comprise, à cause du manque de résultats de classification. Les seuls résultats connus à ce moment sont dans [START_REF] Bourgeois | Contact homology and homotopy groups of the space of contact structures[END_REF][START_REF] Massot | Examples of non-trivial contact mapping classes in all dimensions[END_REF][START_REF] Lanzat | On the contact mapping class group of the contactization of the A m -Milnor fiber[END_REF]. Dans le premier papier, Bourgeois donne des résultats sur quelques groupes d'homotopie π k (Ξ (V ) , ξ), pour des variétés de contact (V, ξ) spécifiques, en utilisant des outils d'homologie de contact. Dans [START_REF] Massot | Examples of non-trivial contact mapping classes in all dimensions[END_REF], Massot et Niederkrüger donnent des exemples de variétés de contact (V, ξ) pour lesquelles ker (π 0 (D (V, ξ)) → π 0 (D (V ))) n'est pas trivial; ces exemples utilisent des constructions dans [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF], qu'on utilisera aussi dans la suite. Le papier [START_REF] Lanzat | On the contact mapping class group of the contactization of the A m -Milnor fiber[END_REF], qui porte sur le cas non compact, contient des exemples de plongements de groupes de tresses dans le groupe des contactomorphismes des contactisations de quelques variétés symplectiques non compactes.

Tous les exemples rappelés jusqu'ici sont des variétés de contact tendues. On remarque aussi que quelques uns de ces résultats montrent que la rigidité, qui caractérise la classe des structures tendues pour le problème des déformations des structures de contact, entraîne, dans certains cas, de la rigidité aussi pour le problème des déformations des contactomorphismes.

Pour ce qui concerne la classe des structures vrillées, le seul résultat connu à ce moment en dimension 3 est la classification des composantes connexes de l'espace des contactmorphismes pour toutes les structures vrillées sur la 3-sphère. Ce résultat, sans preuve publiée jusqu'à récemment, est attribué à Chekanov dans [START_REF] Eliashberg | Topologically trivial Legendrian knots[END_REF]Remark 4.16]. Vogel donne une preuve complète de cette classification dans [START_REF] Vogel | Non-loose unknots, overtwisted discs, and the contact mapping class group of S 3[END_REF], où il prouve aussi, en utilisant des techniques de topologie de contact en dimension 3, que l'espace des plongements des disques vrillées est non connexe par arcs pour une des structures vrillées sur S 3 . Ceci donne en particulier le premier exemple connu de contactomorphisme d'une 3-variété de contact vrillée qui est isotope à l'identité lissement mais pas parmi 1.1. LA TOPOLOGIE DU GROUPE DES CONTACTOMORPHISMES les contactomorphismes; on rappelle que, d'après [Cer68], chaque difféomorphisme de la 3-sphère qui respecte l'orientation est lissement isotope à l'identité.

Dans Chapitre 3, on s'intéresse au problème de l'existence de sous-groupes cycliques infinis dans ker(j * | π0 ), dans le cas tendu. Comme remarqué ci dessus, le seul exemple connu de ce phénomène est donné dans [START_REF] Gompf | Handlebody construction of Stein surfaces[END_REF][START_REF] Ding | The diffeotopy group of S 1 × S 2 via contact topology[END_REF] pour le cas de l'unique (à isotopie près) structure de contact tendue sur S 1 × S 2 . On généralise ici leur résultat au cas de la structure tendue standard sur S 1 × S 2n et, plus généralement, au cas des structures tendues standard sur le produit DW × S 1 , où DW est le double W ∪ ∂W W d'un domaine de Liouville (W 2n , λ). Plus précisément, on travaille dans le cadre suivant.

Soit (F 2n-2 , ω F , Z F , ψ F ) un domaine de Weinstein et on considère sa stabilisation

(F × C, ω F ⊕ ω 0 , Z F + Z 0 , ψ F + |.| 2 C
), où ω 0 = rdr ∧ dϕ et Z 0 = 1 2 r∂ r , avec coordonnées z = re iϕ ∈ C. On suppose que c > min ψ F est une valeur régulière de ψ := ψ F + |.| 2 C , et soit W le domain compact ψ -1 ((-∞, c]). On suppose aussi qu'il y a une structure presque complexe J F sur F entamée par ω F et telle que (T F, J F ) est trivial en tant que fibré complexe sur F . On considère alors le domaine de Weinstein (F × C × R × S 1 , ω , Z , ψ ), où, en utilisant des coordonnées (s, θ) ∈ R × S 1 , ω = ω F + ω 0 + 2ds ∧ dθ, Z = Z + s∂ s et ψ (p, z, s, θ) = ψ(p, z) + s 2 . La préimage (ψ ) -1 (c), qui est difféomorphe au produit du double DW := W ∪ ∂W W de W et de S 1 , est naturellement munie de la structure de contact ξ = ker α, où α = (ι Z ω )| DW ×S 1 . De plus, le difféomorphisme de F × C × R × S 1 donné par (q, z, s, θ) → (q, e iθ z, s, θ) se restreint à un difféomorphisme bien défini Ψ de DW × S 1 . Dans ce contexte, on prouve le résultat suivant : Théorème I.A. Le difféomorphisme Ψ de DW × S 1 est lissement isotope à un contactomorphisme Ψ c de (DW × S 1 , ξ) tel que, pour tout entier k = 0, sa k-ème itérée n'est pas contacto-isotope à l'identité.

Une application directe du Théorème I.A avec F = C n-1 , ω F = 2 n-1 i=1 r i dr i ∧ dϕ i , Z F = 1 2 n-1 i=1 r i ∂ ri , ψ F (z 1 , . . . , z n-1 ) = r 2 1 + . . . r 2 n-1 et c = 1, où on utilise des coordonnées polaires (z 1 = r 1 e iϕ1 , . . . , z n-1 = r n-1 e iϕn-1 ) sur F = C n-1 , donne la généralisation suivante de [START_REF] Gompf | Handlebody construction of Stein surfaces[END_REF][START_REF] Ding | The diffeotopy group of S 1 × S 2 via contact topology[END_REF] à toutes dimensions :

Corollaire I.B. Soient (x 1 , y 1 , . . . , x n , y n , z, θ) des coordonnées sur la variété R 2n+1 ×S 1 et ξ la structure de contact tendue sur V := S 2n × S 1 définie par la restriction de λ = n i=1 (x i dy i -y i dx i ) + 2zdθ sur R 2n+1 × S 1 à S 2n × S 1 = {z 2 + n i=1 x 2 i + y 2 i = 1}. On considère le difféomorphisme Ψ de S 2n × S 1 donné par la restriction de

R 2n+1 × S 1 → R 2n+1 × S 1 (x i , y i , z, θ) → (ϕ θ (x i , y i , z), θ)
où, pour tout θ ∈ S 1 , ϕ θ : R 2n+1 → R 2n+1 est l'application linéaire qui se restreint à la rotation d'angle θ sur le sous-espace R 2 engendré par (x n , y n ) et fixe les coordonnées (x 1 , y 1 , . . . , x n-1 , y n-1 , z). Alors, Ψ est lissement isotope à un contactomorphisme Ψ c de (V, ξ) tel que [Ψ 2 c ] engendre un sous-groupe cyclique infini de ker (π 0 D (V, ξ) → π 0 D (V )).

On remarque que tout itérée d'ordre pair de Ψ c dans Corollaire I.B est effectivement lissement isotope à l'identité : puisque le groupe fondamentale de SO(m) est isomorphe à Z 2 pour tout entier m ≥ 3, il y a, pour tout k ∈ N, une isotopie lisse de S 2n × S 1 , préservant (globalement) chaque sous-variété S 2n × {pt}, qui relie Ψ 2k à l'identité; en particulier, Ψ 2k c est aussi lissement isotope à l'identité.

CHAPTER 1. INTRODUCTION GÉNÉRALE

De façon analogue au Corollaire I.B, une application du Théorème I.A dans le cas

F = T * T n , ω F = n i=1 dp i ∧ dq i , Z F = 1 2 n i=1 p i ∂ pi et ψ F (q i , p i ) = n i=1 p 2
i donne, pour tout entier n ≥ 1, un autre exemple explicite de (V 2n+1 , ξ) tendue telle que ker (π 0 D (V, ξ) → π 0 D (V )) admet un sous-groupe cyclique d'ordre infini. En fait, dans ce cas aussi, chaque itérée d'ordre pair de Ψ est lissement isotope à l'identité. Ceci suit des faits que T * T n T n × R n , que DW × S 1 T n × S n+2 × S 1 et que, pour tout θ ∈ S 1 , Ψ : T n × S n+2 × S 1 → T n × S n+2 × S 1 agit trivialement sur le premier et troisième facteurs et comme une rotation d'angle θ autour d'un axe donné sur chaque {pt} × S n+2 × {θ}; puisque π 1 (SO(m)) Z 2 pour tout m ≥ 3, on peut conclure, comme déjà fait dans le cas du Corollaire I.B, que Ψ 2k c est lissement isotope à l'identité, pour tout k = 1. On remarque aussi que le fait que ψ F sur T * T n ci-dessus n'est pas Morse n'est pas important : en fait, la condition de Morse pour ψ F ne joue aucun rôle dans la preuve du Théorème I.A.

Dans le Chapitre 4, on étudie la présence d'éléments non-triviaux dans le noyau de l'application π 0 (D (V, ξ)) → π 0 (D (V )) pour quelques variétés de contact vrillées explicites en grandes dimensions, en généralisant le résultat dans [START_REF] Massot | Examples of non-trivial contact mapping classes in all dimensions[END_REF] au cas vrillé. Plus précisément, on commence par démontrer le résultat suivant: Théorème I.C. On considère une variété lisse W de dimension 2n ≥ 2 et une structure de contact ξ sur la variété V := S 1 × W . On suppose que la première classe de Chern c 1 (ξ) ∈ H 2 (V ; Z) est toroidale et que, pour tout entier k ≥ 2, la tirée en arrière π * k ξ de ξ via le revêtement à k-feuilles π k : S 1 × W → S 1 × W donné par π k (s, p) = (ks, p) satisfait c 1 (π * k ξ) = k • c 1 (ξ) modulo le sous-module H 2 ator (V ; Z) des classes atoroidales. Alors, la transformation de contact f : (S 1 ×W, π * k ξ) → (S 1 ×W, π * k ξ) définie par f (s, p) = (s + 2π k , p) est isotope à l'identité lissement mais pas parmi les contactomorphismes. On rappelle qu'une classe c ∈ H 2 (V ; Z) est dite toroidale s'il existe f : T 2 → V telle que f * c = 0 ∈ H 2 (T 2 ; Z), et atoroidale sinon.

Remarque. Le Théorème I.C est aussi vrai (avec preuve similaire) si on remplace ( * ) c 1 (ξ) est toroidale et, pour tout k ≥ 2, c 1 (π * k ξ) = k • c 1 (ξ) mod H 2 ator (V ; Z), par la condition ( * ) c 1 (ξ) n'est pas de torsion et, pour tout k ≥ 2, c 1 (π * k ξ) = k • c 1 (ξ).

On remarque que a ∈ H 2 (V ; Z) est toroidal si et seulement si [a] ∈ H 2 (V ; Z) H 2 ator (V ; Z) n'est pas torsion, puisque H 2 (T 2 ; Z) Z. En particulier, ( * ) est equivalent à "c 1 (ξ) n'est pas torsion modulo H 2 ator (V ; Z) et c 1 (π * k ξ) = k • c 1 (ξ) mod H 2 ator (V ; Z)", donc elle est seulement une variation modulo H 2 ator (V ; Z) de ( * ) (et elle n'est pas plus forte ni plus faible que la ( * )). On remarque aussi que les structures de contact données par les Propositions I.D et I.F (ainsi que la Proposition I.E.i.) ci-dessous satisfont les Conditions ( * ) et ( * ) en même temps; d'autre part, travailler modulo H 2 ator (V ; Z), i.e. avec ( * ), est nécessaire pour la Proposition I.E.ii. On a donc décidé de tout formuler en termes de ( * ), même si ( * ) donnerait (partout sauf dans Proposition I.E.ii.) des preuves un peu plus directes.

On donne ensuite, pour tout n ≥ 1, un nombre infini de variétés de contact (S 1 × W 2n , ξ) explicites qui satisfont les hypothèses du Théorème I.C : Proposition I.D. Soit (M 2n-1 , α + , α -) une des paires de Liouville construites (en nombre infini) dans [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF]. On considère la structure de contact

η = ker 1 + cos (s) 2 α + + 1 -cos (s) 2 α -+ sin (s) dt sur V := T 2 (s,t) × M (ici, la notation T 2 (s,t)
dénote le choix de coordonnées (s, t) sur T 2 ) et soit ξ la structure de contact vrillée obtenue par η via un demi twist de Lutz-Mori le long de {(0, 0)} × M , comme défini dans [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF].

Alors, c 1 (ξ) ∈ H 2 (V ; Z) est toroidale et, pour tout entier k ≥ 2, on a c 1 (π * k ξ) = k • c 1 (ξ) mod H 2 ator (V ; Z), où π k : T 2 (s,t) × M → T 2 (s,t) × M est donné par π k (s, t, q) = (ks, t, q).
Exemple. Si n = 3, (M, α ± ) = (S 1 , ±dθ). De plus, si k = 2, la structure de contact π * 2 ξ sur V := T 2 × M est l'unique structure de contact qui est invariante par l'action à gauche par multiplication de M = S 1 sur V , invariant par f (s, t, θ) = (s + π, t, θ), et telle que chaque tore T 2 (s,t) × {θ 0 } est convexe avec découpage comme dans la Figure 1.1. La Proposition I.D et le Théorème I.C entraînent alors que f n'est pas contacto-isotope à l'identité. On remarque que, même dans ce cas très simple, ce résultat n'était pas présent dans la littérature.

Si le seul but est celui de donner des exemples, en chaque dimension impaire, d'éléments non triviaux dans le noyau de l'application π 0 (D (V, ξ)) → π 0 (D (V )), sans vouloir forcement que la variété de contact vrillée sous-jacente (V, ξ) soit très explicite, comme dans le cas de la Proposition I.D, le résultat suivant peut aussi être démontré en utilisant l'existence, démontrée dans [START_REF] Giroux | Sur les transformations de contact au-dessus des surfaces[END_REF], des décompositions en livres ouverts qui supportent les structures de contact :

Proposition I.E. Soit (V := S 1 × W 2n , η) une variété de contact vrillée telle que c 1 (η) est toroidale et telle que, pour tout k ≥ 2, la tirée en arrière ξ via le revêtement à k- feuilles π k : S 1 s × W → S 1 s × W , donné par π k (s, p) = (ks, p), satisfait c 1 (π * k η) = k • c 1 (η) mod H 2
ator (V ; Z). Alors, on a que : i. La première classe de Chern de chaque structure de contact ξ sur V × T 2 obtenue via la construction de Bourgeois [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] sur (V, η) satisfait aussi les conditions ci-dessus, par rapport au revêtement

µ k := (π k , Id) : V × T 2 → V × T 2 .
ii. Soit ν : V × Σ g → V × T 2 induite par un recouvrement Σ g → T 2 qui est ramifié au dessus de deux points (ici, Σ g est une surface fermée de genre g ≥ 2). Alors, la

CHAPTER 1. INTRODUCTION GÉNÉRALE

première classe de Chern de chaque revêtement ramifié ξ g de ξ sur V × Σ g satisfait aussi les conditions ci-dessus, par rapport au revêtement µ g k := (π k , Id) : V × Σ g → V × Σ g . De plus, si η est vrillée et g est assez grand, ξ g est aussi vrillé.

Un recouvrement ramifié Σ g → T 2 comme dans le point ii. ci dessus peut être obtenu, par exemple, en regardant la surface Σ g comme la 2-sphère avec g anses attachées de façon symétrique par rapport à une rotation de 2π/g autour de l'axe qui passe par les pôles nord et sud. Par induction sur la dimension, la Proposition I.E donne, pour tout entier n ≥ 2, des exemples de (S 1 ×W 2n , ξ) ayant première classe de Chern qui vérifie les propriétés voulue. Concernant le point ii., le lecteur peut consulter [START_REF] Geiges | Constructions of contact manifolds[END_REF] pour une construction des revêtements ramifiés de contact, et Section 5.2 pour une définition. On remarque aussi que l'entier g optimal pour garantir le fait que la structure η g est vrillée est 2, selon une observation due à Niederkrüger (voir l'Observation 9.10).

En utilisant le h-principe de [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF], une classe d'exemples (non explicites) encore plus grande peut être obtenue ainsi : Proposition I.F. On considère une variété lisse W de dimension 2n qui est presque complexe, spin et satisfait H 1 (W ; Z) = {0}. Alors, il y a une structure de contact vrillée

ξ sur V := S 1 × W telle que c 1 (ξ) ∈ H 2 (V ; Z) est toroidale et c 1 (π * k ξ) = k • c 1 (ξ) mod H 2 ator (V ; Z), où π k : S 1 s × W → S 1 s × W est le revêtement à k-feuilles π k (s, p) = (ks, p).

Organisation de la Partie I

Le Chapitre 3 est organisé comme suit. Dans la Section 3.1, on décrit une variété de Liouville explicite qui admet DW ×S 1 comme bord convexe, où DW est le double W ∪ ∂W W d'un domaine de Liouville W (ici, W dénote W avec orientation opposée). On décrit aussi une façon naturelle de construire des contactomorphismes stricts du bord convexe.

La Section 3.2 introduit la notion de famille de bases Lagrangiennes et décrit comment elles donnent un invariant calculable de (classes d'isotopie de contact de) contactomorphismes dans le cas des structures de contact qui sont stablement triviales.

Enfin, la Section 3.3 contient la preuve du Théorème I.A.

Le Chapitre 4 est organisé ainsi. La Section 4.1 contient une preuve par l'absurde du Théorème I.C. En supposant par l'absurde que f est contacto-isotope à l'identité, on construit un contactomorphisme entre deux structures de contact ξ 1 et ξ 2 ; d'autre côté, l'hypothèse sur la première classe de Chern de ξ entraîne que ξ 1 et ξ 2 ne sont même pas isomorphes en tant que structures presque de contact.

La Section 4.2 montre comment obtenir des exemples de variétés de contact (S 1 × W 2n , ξ) qui satisfont les hypothèses de Théorème I.C en utilisant les constructions dans [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF]. Plus précisément, on rappelle la définition de demi twist de Lutz-Mori et les constructions explicites de paires de Liouville dans [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF]. Ensuite, en utilisant une interprétation géométrique des classes de Chern, qu'on rappelle dans l'Appendice A, on décrit les effets d'un demi twist de Lutz-Mori sur les classes de Chern dans ce contexte, et on démontre la Proposition I.D.

Dans la Section 4.3 on décrit comment obtenir des exemples de contactomorphismes qui sont isotopes à l'identité lissement mais pas parmi les contactomorphismes en utilisant l'existence des décompositions adaptés en livres ouverts due à [START_REF] Giroux | Géométrie de contact: de la dimension trois vers les dimensions supérieures[END_REF] et l'h-principe dans [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF]; on prouve en particulier les Propositions I.E et I.F.

QUELQUES CONSTRUCTIONS DE VARIÉTÉS DE CONTACT REVISITÉES

Quelques constructions de variétés de contact revisitées

Les articles [START_REF] Geiges | Constructions of contact manifolds[END_REF] par Geiges et [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] par Bourgeois donnent quelques constructions explicites de structures de contact sur des variétés de grandes dimensions. Plus précisément, dans le premier papier, en développant des idées dans [START_REF] Gromov | Partial differential relations[END_REF], l'auteur transpose quelques constructions du monde symplectique au cadre de contact, en construisant en particulier les revêtements ramifiés de contact et les sommes fibrées de contact; les réductions de contact sont aussi traitées, mais on ne s'intéressera pas à cette construction dans la suite. (Le lecteur intéressé peut aussi consulter [START_REF] Gonzalo | Branched covers and contact structures[END_REF] pour le cas des revêtements ramifiés de contact dans le cas de la dimension 3.) Dans le deuxième papier, en prenant inspiration de [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF], l'auteur demontre que, étant donné une variété de contact (M 2n-1 , ξ) et un livre ouvert (B, ϕ) de M qui porte ξ, il y a une structure de contact η sur M × T 2 qui est invariant sous l'action naturelle de T 2 , qui se restreint à ξ sur chaque sous-variété M × {pt} et qui se déforme naturellement sur la distribution d'hyperplans ξ ⊕ T T 2 sur M × T 2 . On remarque que le livre ouvert (B, ϕ) existe toujours d'après [START_REF] Giroux | Géométrie de contact: de la dimension trois vers les dimensions supérieures[END_REF].

La motivation principale derrière [START_REF] Geiges | Constructions of contact manifolds[END_REF][START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] était le problème de l'existence des structures de contact, i.e. la question de quelles variétés en grandes dimensions admettent des structures de contact. Ce (grand) problème en géométrie de contact a été résolu par [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF], et, comme déjà remarqué dans l'introduction générale ci dessus, on s'intéresse maintenant au problème de trouver des exemples "interessants" de structures de contact.

Les papiers [START_REF] Geiges | Constructions of contact manifolds[END_REF][START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] s'inscrivent bien dans cette perspective, parce qu'ils donnent des variétés de contact plutôt explicites, qu'on peut étudier en détail et qui, sous certains hypothèses, manifestent des propriétés intéressantes, i.e. sont tendues, remplissables, vrillées, etc. Par exemple, ces deux papiers donnent les premiers procèdures explicites pour construire des variétés de contact en grands dimensions qui sont PS-vrillées (donc vrillées, d'après le travaux postérieurs [START_REF] Casals | Geometric criteria for overtwistedness[END_REF][START_REF] Huang | On plastikstufe, bordered Legendrian open book and overtwisted contact structures[END_REF]). Le lecteur peut consulter [START_REF] Presas | A class of non-fillable contact structures[END_REF] pour le cas de la construction dans [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF], [NP10, page 724] pour le cas des revetements ramifiés de contact et [Nie13, Theorem I.5.1], attribué à Presas, pour le cas des sommes fibrées de contact (comparer aussi avec l'Observation 9.10 dans la Section 9.1.2).

Le but de la Partie II est alors de construire des variétés de contact avec des propriétés spécifiques en utilisant [START_REF] Geiges | Constructions of contact manifolds[END_REF][START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF]. Pour cela, on passe des constructions donnés par Geiges et Bourgeois à des définitions; ensuite, on peut étudier les propriétés de ces structures de contact, indépendamment des choix auxiliaires faites dans leur constructions explicites dans [START_REF] Geiges | Constructions of contact manifolds[END_REF][START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF].

Concernant les revêtements ramifiés et les sommes fibrées de contact, on remarque que le problème de leur unicité, i.e. la question si les structures de contact obtenues sont indépendantes des choix auxiliaires faites dans leur construction, n'est pas explicitement traité dans [START_REF] Geiges | Constructions of contact manifolds[END_REF]. On propose donc dans cette thèse une définition de revêtements ramifiés et sommes fibrées de contact, qui permet d'obtenir naturellement un énoncé d'unicité (à isotopie près).

On remarque que dans la littérature il y a déjà une définition de revêtement ramifié de contact qui va dans cette direction. Plus précisément, dans [START_REF] Niederkrüger | Brieskorn manifolds as contact branched covers of spheres[END_REF] les auteurs définissent cette notion en termes de déformations en structures de contact qui satisfont une condition additionnelle le long du lieu de ramification. On retire ici cette hypothèse additionnelle et on prouve :

CHAPTER 1. INTRODUCTION GÉNÉRALE

Proposition II.A. Soit (V 2n-1 , η) une variété de contact et π : V → V un revêtement ramifié avec lieu de ramification (en bas) M . On suppose que η ∩ T M est une structure de contact sur M . Alors :

1. il y a une famille, indexée par [0, 1], de distributions d'hyperplans η t sur V telle que η 0 = π * η et η t est une structure de contact pour tout t ∈ (0, 1];

2. si η t et η t sont comme dans le point 1, alors η r est isotope à η s pour tout r, s ∈ (0, 1].

De plus, dans le point 1, η t peut être choisie invariant par les automorphismes (locales) du revêtement π, pour tout t ∈ (0, 1]; de façon analogue, l'isotopie du point 2 peut être choisie parmi les structures de contact invariant par les automorphismes (locales) de revêtement, si η t et η t le sont aussi.

On appelle alors revêtement ramifié de contact une structure de contact η sur V qui est le point d'arrivée d'un chemin η t comme ci dessus; la Proposition II.A dit exactement que cet objet existe et est bien défini à isotopie près.

À ce point, on peut donner des énoncés précis concernant les propriétés des revêtements ramifiés de contact. Par exemple, on prouve le résultat suivant :

Théorème II.B. On considère un revêtement ramifié π : V → V et une structure de contact ξ sur V , et soit η un revêtement ramifié de contact de η. On suppose que (V, η) est faiblement rempli par (W, Ω), de sorte que le lieu de ramification en bas M de π est rempli par une sous-variété symplectique X de (W, Ω). On suppose aussi que π s'étend à un revêtement ramifié π : W → W , avec lieu de ramification en bas X. Alors, il y a une structure symplectique Ω sur W qui remplit faiblement η sur V = ∂ W .

On analyse ensuite la construction de Bourgeois sous différents points de vue. Premierment, on étudie en détail les liens entre [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] et le papier [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF] par Lutz, duquel le premier prends inspiration. En particulier, on rappelle une partie du travail de Lutz dans [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF], où il définit une application

   structures de contact T 2 -invariants sur le fibré M × T 2 → M    →    (ξ, B, ϕ) ξ champ singulier d'hyperplans sur M et (B, ϕ) décomposition en livre ouvert sur M   
Ensuite, on remarque que, en utilisant la notion de décomposition en livres ouverts porteurs due à Giroux, les calculs dans [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF] montrent (modulo un lemme sur les livres ouverts porteurs par Giroux) que l'image (ξ η , B η , ϕ η ) de η via l'application ci dessus est telle que (B η , ϕ η ) porte ξ η , si cette dernière est une structure de contact sur M . En utilisant ce point de vue des structures de contact T 2 -invariant sur le fibré en tores M × T 2 → M , on montrera ensuite que, si on part d'une variété de contact (M, ξ) munie d'une décomposition en livre ouvert porteur (B, ϕ), la composition de la construction de Lutz après celle de Bourgeois redonne la donnée initiale de (M, ξ) et (B, ϕ).

On analyse aussi la concaténation opposée des deux constructions, et on prouve que, sous certains hypothèses additionnelles, cela donne aussi les données initiales. Dans une deuxième partie on adopte une perspective qui est, en un certain sens, "orthogonale" à celle ci dessus, en regardant la projection M × T 2 → T 2 au lieu de M × T 2 → M . Grâce à la notion de fibré de contact introduite par Lerman dans [START_REF] Lerman | Contact fiber bundles[END_REF], on peut en fait voir les structures de contact construites par Bourgeois comme des structures de fibrés de contact sur les fibré M × T 2 → T 2 , qui admet une connexion plate naturelle. On remarque que cette structure de fibré de contact des exemples dans [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] a déjà été utilisée avec succès dans [Pre07, KN07, NP10] pour obtenir des variétés de contact avec des propriétés remarquables en grandes dimensions.

QUELQUES CONSTRUCTIONS DE VARIÉTÉS DE CONTACT REVISITÉES

Dans cette thèse on utilise donc la théorie des fibrés de contact dans [START_REF] Lerman | Contact fiber bundles[END_REF] pour généraliser la construction de Bourgeois et définir la notion de structure de contact de Bourgeois. Plus précisément, sur un fibré π : V 2n+1 → Σ 2 muni d'une structure de fibré de contact η 0 de référence, chaque fibré de contact η admet un potentiel A par rapport à η 0 , avec une forme de courbature R A ; dans le cas où le fibré de contact de référence η 0 est plat, on appelle structure de contact de Bourgeois chaque structure de fibré de contact sur π : V → Σ qui est aussi une structure de contact sur V et satisfait 1 R A → 0 pour → 0.

En plus de la nécessité de passer de la procédure de construction dans [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] à une définition, une autre motivation pour l'introduction de cette notion est la suivante : la condition sur la courbure est, d'un côté, suffisamment faible pour être satisfaite par une classe de structures de contact qui contient strictement les résultats de la construction dans [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] et, de l'autre côté, suffisamment forte pour assurer, par exemple, des bonnes propriétés du point de vue des remplissages faibles et des décompositions en livre ouvert porteur. Concernant la remplissabilité faible, on prouve le résultat suivant :

Proposition II.C. Soit (M 2n-1 , ξ) une variété de contact et η une structure de Bour- geois sur le fibré trivial M × T 2 → T 2 , qui se restreint à ξ sur M × {pt} = M . Si (M, ξ) est faiblement rempli par (X 2n , ω), alors (M × T 2 , η) est faiblement rempli par (X × T 2 , ω + ω T 2 ), où ω T 2 est une forme volume sur T 2 .
On remarque que le résultat est déjà connu dans le cas de la construction de Bourgeois [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] : l'énoncé et l'idée principale de la preuve sont présentés dans [MNW13, Example 1.1]; voir aussi [LMN18, Theorem A.a] pour une preuve explicite.

Du point de vue des livres ouverts porteurs, les structures de Bourgeois ont implicitement de l'information sur des décompositions en livres ouverts qui portent les structures de contact sur chaque fibre :

Proposition II.D. Soit η une structure de contact de Bourgeois sur π : V → Σ. Il existe une application ψ η qui associe à chaque point b ∈ Σ une classe d'isotopie de livres ouverts porteurs sur (M b , ξ b ) := π -1 (b) , η ∩ T π -1 (b) . De plus, si γ(t), avec t ∈ (-, ), est un chemin dans un ensemble ouvert U de Σ sur lequel π est trivialisée , i.e. sur lequel π devient la projection sur le premier facteur pr U : U × M → U , alors le chemin de classes d'isotopies ψ η • γ(t) vient d'un chemin de livres ouverts (B t , ϕ t ) de {γ(t)} × M tel que son image via pr M : U × M → M est une isotopie de livres ouverts sur M . Dans le cas des exemples dans [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF], via la projection globale pr M : M × T 2 → M , l'application ψ η donne la classe d'isotopie du livre ouvert (B, ϕ) utilisé dans la construction.

Autrement dit, si d'un côté la construction de Lutz donne une inverse à celle de Bourgeois du point de vue des structures de contact T 2 -invariantes sur le fibré principal M ×T 2 → M , de l'autre côté ce résultat donne une inverse à la construction de Bourgeois du point de vue des fibrés de contact sur M × T 2 → T 2 .

Pour démontrer la Proposition II.D, on donne une reinterprétation des livres ouverts porteurs en termes de paires de champs de vecteurs de contact :

Théorème II.E. Sur une variété de contact (M 2n-1 , ξ), chaque paire de champs de vecteurs de contact X, Y , telle que [X, Y ] est partout transverse à ξ, donne un livre ouvert explicite de M qui porte ξ. Vice versa, un livre ouvert qui porte ξ donne une paire X, Y comme ci dessus.

La deuxième partie de ce résultat a été énoncé par Giroux dans des exposés pour le Yashafest de Juin 2007 et pour le AIM workshop de Mai 2012 (voir [Gir12, Claim on page 19]). Un énoncé plus détaillée (ainsi qu'une preuve) du Théorème II.E sera donné dans la Chapter 6. On remarque que ce résultat n'est pas seulement utile à la preuve de la Proposition II.D mais donne aussi un autre point de vue sur les livres ouverts porteurs, ce qui a un intérêt en soi.

Ces réinterprétations et généralisations de [START_REF] Geiges | Constructions of contact manifolds[END_REF][START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] amènent à quelques exemples de variétés de contact en grandes dimensions avec des propriétés intéressantes, i.e. qui sont tendues, remplissables ou vrillées. On obtient ainsi deux nouveaux résultats, l'un concernant les variétés tendues virtuellement vrillées et l'autre concernant les plongements des 3-variétés de contact en codimension 2 avec fibré normal trivial.

Concernant le premier résultat, on rappelle que une structure de contact tendue ξ sur M est dite virtuellement vrillée si sa tirée en arrière ξ sur un revêtement fini M de M est vrillée. Dans cette thèse, on demontre le résultat suivant :

Théorème II.F. Les structures virtuellement vrillée existent en toutes dimensions ≥ 3.

La preuve de ce résultat est par induction sur la dimension. Pour ce qui concerne l'initialisation, l'existence des structures de contact tendues virtuellement vrillées est connue en dimension 3 depuis [START_REF] Gompf | Handlebody construction of Stein surfaces[END_REF]; le lecteur peut aussi consulter [START_REF] Giroux | Structures de contact en dimension trois et bifurcations des feuilletages de surfaces[END_REF][START_REF] Honda | On the classification of tight contact structures[END_REF], qui presentent une classification de ce type de structures de contact sur des 3-variétés particulières. L'étape inductive utilise les Propositions II.C et II.A ci-dessus, i.e. le fait que la construction dans [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] et les revêtements ramifiés de contact préservent la remplissabilité faible, et s'appuie sur l'existence des livres ouverts porteurs démontrée par Giroux [START_REF] Giroux | Géométrie de contact: de la dimension trois vers les dimensions supérieures[END_REF], sur la construction e Bourgeois [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] et sur le critère des "grandes" voisinages [CMP15, Theorem 3.1].

L'autre application concerne la question suivante : étant donné une variété de contact (M, ξ = ker α), existe-t-il un > 0 tel que M × D 2 , ker α + r 2 dϕ est tendue? Ici, D 2 est le disque de rayon et centré à l'origine de R 2 , et (r, ϕ) sont ses coordonnées polaires.

Ceci est lié au problème de trouver des plongements de contact en codimension 2 avec fibré normal trivial dans des variétés ambiantes tendues. En fait, en codimension 2, le fibré normal est trivial si et seulement si le fibré conforme symplectique normal l'est. Donc, d'après le théorème des voisinages de contact [Gei08, Theorem 2.5.15], si (M 2n-1 , ξ = ker α) se plonge dans (V 2n+1 , η) avec fibré normal trivial, alors il admet un voisinage M × D 2 r0 , ker α + r 2 dϕ , pour un certain r 0 > 0. En particulier, si (V, η) est tendue, ce voisinage l'est aussi.

Historiquement, la première motivation pour s'intéresser à cette question sur la "taille" d'un voisinage d'une sous-variété de contact en codimension 2 est donnée par [START_REF] Niederkrüger | Some remarks on the size of tubular neighborhoods in contact topology and fillability[END_REF], où il est démontré que des voisinages "larges" de sous-variétés de contact vrillée sont une obstruction pour la remplissabilité de la variété de contact ambiante. Comme signalé dans [START_REF] Niederkrüger | On fillability of contact manifolds[END_REF], ceci a amené Niederkrüger et Presas à conjecturer que la présence d'un domaine contactomorphe au produit d'un R 3 vrillé et d'un "grand" voisinage dans R 2n avec la structure de Liouville standard aurait pu être la bonne généralisation de la notion de vrillé aux dimensions impaires plus grandes que 3. Après l'introduction dans [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF] d'une définition de structure de contact vrillée en toutes dimensions impaires, [START_REF] Casals | Geometric criteria for overtwistedness[END_REF] à confirmé cette conjecture, en démontrant que la présence d'un tel domaine est en fait équivalent à être vrillé. Plus précisément, ceci suit de [CMP15, Theorem 3.1], qui affirme que, si (M, ξ = ker α) est vrillé, alors M × D 2 R , ker α + r 2 dϕ l'est aussi, si R > 0 est suffisamment grand. En particulier, ceci motive la question ci dessus sur l'existence, pour une variété de contact (M, ξ = ker α) donnée, d'un > 0 tel que M × D 2 , ker α + r 2 dϕ est tendu.

Le problème de trouver des plongements en codimension 2 dans des variétés tendues a Théorème II.G. Chaque 3-variété de contact fermée (M, ξ) avec π 1 (M ) = {1} peut être plongée avec fibré normal trivial dans une 5-variété de contact (V 5 , η) fermée (hyper)tendue.

Corollaire II.H. Pour tout (M 3 , ξ = ker α) avec π 1 (M ) = {1}, il existe > 0 tel que M × D 2 , ker α + r 2 dϕ est tendu.

On rappelle qu'une structure de contact est dite hypertendue si elle admet une forme de contact sans orbites de Reeb fermées contractiles. Toute variété de contact hypertendue est en particulier tendue, d'après [START_REF] Hofer | Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three[END_REF][START_REF] Albers | On the Weinstein conjecture in higher dimensions[END_REF][START_REF] Casals | Geometric criteria for overtwistedness[END_REF].

On observe aussi que, d'après la preuve de Perelman de la conjecture de Poincaré, la condition π 1 (M ) = {1} est équivalente au fait que M n'est pas difféomorphe à S 3 . Or, le cas M = S 3 est déjà connu. En fait, le cas des 3-sphères vrillées est traité dans [CPS16, Proposition 11], déjà cité ci dessus, et la 3-sphère tendue standard (qui est l'unique structure de contact tendue sur S 3 à isotopie près d'après [START_REF] Eliashberg | Contact 3-manifolds twenty years since J. Martinet's work[END_REF]), se plonge de façon naturelle dans la 5-sphère standard avec fibré normal trivial.

Les ingrédients principaux qu'on utilise dans la preuve du Théorème II.G sont l'existence des livres ouverts porteurs pour les 3-variétés de contact démontrée par Giroux, et une preuve détaillée de la dynamique du flot de Reeb des formes de contact construites dans [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF]. Plus précisément on montre que, si π 1 (M ) = {1}, chaque livre ouvert (B, ϕ) de M peut être modifié, via des stabilisations positives, de sorte que chaque composante connexe de B soit homotopiquement non triviale (dans M ). On démontre ensuite que ceci permet d'obtenir, en utilisant [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF], des formes de contact hypertendues sur M × T 2 . Enfin, (M, ξ) se plonge naturellement dans la variété de contact construite par Bourgeois comme une fibre de la fibration M × T 2 → T 2 donnée par la projection sur le deuxième facteur.

Concernant le Corollaire II.H, on remarque qu'il a été récemment généralisé à toutes dimensions, avec des techniques complètement différentes, dans [START_REF] Hernández-Corbato | Tight neighborhoods of contact submanifolds[END_REF].

Organisation de la Partie II

Dans le Chapitre 5, on donne les nouveaux approches annoncés aux revêtements ramifiés de contact et aux sommes fibrées de contact introduites dans [START_REF] Geiges | Constructions of contact manifolds[END_REF], et on démontre en particulier la Proposition II.A et le Théorème II.B.

Le Chapitre 6 décrit la formulation équivalente, basée sur une idée par Giroux [START_REF] Giroux | The existence problem, contact geometry in high dimensions[END_REF], de décompositions en livres ouverts porteurs en termes de paires de champs de vecteurs de contact, et contient donc la preuve du Théorème II.E.

Le Chapitre 7 rappelle la construction de Bourgeois dans [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] et décrit l'étude des structures de contact invariantes sur les fibrés principaux en T 2 donné dans [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF].

Ensuite, on reformule et généralise dans le Chapitre 8 la construction de Bourgeois en utilisant la notion de fibré de contact introduite dans [START_REF] Lerman | Contact fiber bundles[END_REF]. En particulier, on donne la définition de structure de contact de

Background

Basic definitions

In all this section, V denotes a (2n + 1)-dimensional oriented smooth manifold.

Definition 2.1. A hyperplane field ξ on V is called a contact structure if, for each p ∈ V , there is an open neighborhood U of p and α ∈ Ω 1 (U ) such that ξ| U = ker α and dα| ξ| U is a symplectic structure on ξ| U .

One can easily see that this is equivalent to the definition given at the beginning of Chapter 1 in terms of the the volume form α ∧ dα n ; both points of view will be adopted in this thesis. Moreover, we will actually focus only on cooriented contact structures in the following, i.e. those for which there is a global defining form α, also called contact form for the contact structure. Definition 2.2. Let (V, ξ) and (V , ξ ) be two (2n + 1)-dimensional contact manifold. A diffeomorphism ϕ : V → V is called contactomorphism, and denoted ϕ : (V, ξ) → (V , ξ ), if ϕ * ξ = ξ (i.e. if ker(ϕ * α ) = ξ for each contact form α for ξ ). Moreover, a smooth isotopy (ϕ t ) t∈[0,1] from V to V is called contact isotopy if ϕ t is a contactomorphism for all t ∈ [0, 1]. Similarly, in the case dim(V ) < dim(V ), an embedding ϕ : V → V such that ϕ * ξ = ξ is called contact embedding.

Here's a fundamental criterion to find contactomorphisms between contact structures, which will be used many times in this thesis: Theorem 2.3 (Gray). Let V be a closed manifold and (ξ t ) t∈[0,1] a smooth family of contact structures on V , i.e. a family of contact structures defined by a smooth family (α t ) t∈[0,1] of contact forms on V . Then, there is an isotopy

ϕ t : V → V such that ϕ * t ξ t = ξ 0 .
For the proof of this result we invite the reader to consult [Gei08, Section 2.2]. As we will use it in the following, we just point out that the isotopy ϕ t is generated by the time-dependent vector field X t such that X t ∈ ξ t and ι Xt dα t | ker αt = -( d dt α t )| ker αt ; notice that this X t is unique by the contact condition. We also point out that the hypothesis of V closed is crucial, because Gray's theorem does not hold on open manifolds for instance.

We recall that a choice of a particular contact form α on V gives a unique vector field R α , called Reeb vector field of α, such that α(R α ) = 1 and ι Rα dα = 0 everywhere on V .

Reeb vector fields are particular cases of the following:

TIGHT-OVERTWISTED DICHOTOMY

Definition 2.4. Let ξ be a contact structure on V . A vector field X on V is called contact vector field if, for a (hence every) α contact form defining ξ, there is f :

V → R positive such that L X α = f α.
For this class of vector fields, we have the following correspondence:

Proposition 2.5. Let α be a contact form on V . Then, contact vector fields for ξ = ker α are in 1-1 correspondence with functions H : V → R. More precisely, the correspondence is given by:

• X → H X := α(X);

• H → X H , uniquely defined by the conditions α(X H ) = H and ι

X H dα = dH(R α )α- dH.
With the name contact Hamiltonians, we will hence refer to a function H : V → R, seen as generating a contact vector field X H (after having fixed a contact form α). For a proof of Proposition 2.5, we invite the reader to consult [Gei08, Section 2.3]. As it will be useful for Sections 4.1 and 5.3, we also point out the following link between time-dependent contact Hamiltonians and contact isotopies: Proposition 2.6. Let α be a contact form on V . Then, there is a 1-1 correspondence between time-dependent contact Hamiltonians H t and contact isotopies ϕ t given by:

• H t → ϕ t := ψ t
Xt , where ψ t Xt is the flow at time t of the time dependent contact vector field X t := X Ht (given by Proposition 2.5);

• ϕ t → H t := α(X t )
, where X t is the vector field generating the isotopy ϕ t .

For a proof of Proposition 2.6, the reader can consult for instance [Gei08, Section 2.2].

Tight-overtwisted dichotomy

As already mentioned in the short introduction above, in each odd dimension, contact structures are divided in two different subclasses: tight and overtwisted. The paper [START_REF] Eliashberg | Classification of overtwisted contact structures on 3manifolds[END_REF] introduces this dichotomy in dimension 3, and it is then generalized to all dimensions in [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF].

We point out that the exact definition of overtwistedness given in [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF] is rather technical, and we will not need it explicitly in this thesis. We hence decided to give here some equivalent conditions that we will actually use in the rest of this manuscript, namely in Chapter 4 and Section 9.1. For this, we need to first recall the following definition: Definition 2.7 ( [START_REF] Niederkrüger | The plastikstufe-a generalization of the overtwisted disk to higher dimensions[END_REF]). Let S be a closed smooth manifold of dimension n and λ the standard Liouville form on T * S. Consider also the contact form α ot := cos rdz +r sin rdθ on R 3 , where we use cylindrical coordinates (r, θ, z), and let D be the disk {z = 0, r ≤ π} inside (R 3 , ker α). We call plastikstufe with core S, and denote it PS, the submanifold D×S ⊂ R 3 ×T * S together with the germ of contact structure along it given by ker(α+λ) on R 3 × T * S.

We can now recall the two formulations of overtwistedness in high dimensions that we will need in the following: Theorem 2.8 ( [START_REF] Casals | Geometric criteria for overtwistedness[END_REF][START_REF] Huang | On plastikstufe, bordered Legendrian open book and overtwisted contact structures[END_REF]). Let (V, ξ) be a (2n + 1)-dimensional contact manifold, and α ot := cos rdz +r sin rdϕ on R 3 , where we use cylindrical coordinates (r, ϕ, z). Then, the following are equivalent:

CHAPTER 2. BACKGROUND 1. (V, ξ) is overtwisted (in the sense of [BEM15]);
2. there is a contact embedding (R 3 × C n-1 , ker(α ot + λ std ) → (V, ξ), where λ std is the standard Liouville form n-1 i=1 r 2 i dϕ on C n-1 ; 3. there is a embedding (respecting the germs of contact structures) of a plastikstufe PS into (V, ξ).

We point out that [START_REF] Casals | Geometric criteria for overtwistedness[END_REF] contains the equivalence between points 1, 2 and 3, with the additional hypothesis of PS having spherical core and trivial rotation; then, [START_REF] Huang | On plastikstufe, bordered Legendrian open book and overtwisted contact structures[END_REF] proves that this additional hypothesis on PS is not needed for the above equivalence. In Sections 4.3 and 9.1, we will actually use more specifically the following result, which is used in [START_REF] Casals | Geometric criteria for overtwistedness[END_REF] in order to prove the equivalence between points 1 and 2 above: Theorem 2.9 ([CMP15, Theorem 3.1]). Let (V 2n+1 , ξ = ker α) be overtwisted. Then, (V × C, ker(α + λ std )) is overtwisted too.

The interest in the distinction between overtwisted and tight is the fact, already pointed out above, that the class of overtwisted contact structures satisfy an h-principle with respect to the problem of constructing contact structures. In order to give a specific statement, we need to introduce a formal equivalent of contact structures, which we will need also in Section 4.3: Definition 2.10. An almost contact structure is a couple (ξ, [ω]), where ξ is a hyperplane field on V , ω is a symplectic structure on ξ and [ω] is its conformal class.

We recall that the conformal class of ω is defined by the following relation:

ω 1 ∼ ω 2 if there is a smooth positive function f : V → R such that ω 2 = f ω 1 .
Notice that, given a contact structure ξ on V and a contact form α defining it, the conformal class [dα| ξ ] does not depend on the choice of α defining ξ; it is hence called natural conformal symplectic structure on ξ and denoted by CS ξ . In particular, we have a natural inclusion of the space of contact structures in the space of almost contact structures given by ξ → (ξ, CS ξ ).

As it will be used in the following chapters, we also point out that each almost contact structure (ξ, [ω]) admit a complex structure J (i.e. a J ∈ End(ξ) such that J 2 = -Id) which is tamed by [ω] (i.e.

[ω] is positive on complex lines). Moreover, the space of such J is contractible and, in particular, (ξ, [ω]) have well defined Chern classes; this will be used in Part I.

As anticipated in Chapter 1, overtwisted contact structures satisfy the following hprinciple:

Theorem 2.11 ( [START_REF] Eliashberg | Classification of overtwisted contact structures on 3manifolds[END_REF][START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF]). The natural inclusion of the space of overtwisted contact structures in the space of almost contact structures induces a bijection at the π 0 level.

We point out that [START_REF] Eliashberg | Classification of overtwisted contact structures on 3manifolds[END_REF][START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF] actually contains a more general parametric hprinciple; though, this will not be needed in the following.

Different shades of fillability

Among tight structures, one can further recognize a special subclass: the fillable contact structures.

In the short introduction above, we described the notion of convex boundary of a symplectic manifold, which, from the contact point of view, gives the notion of (strong)

DIFFERENT SHADES OF FILLABILITY

symplectic filling. As it will be the case in this thesis, depending on the particular situation under consideration, it may be more convenient to replace this notion with a stronger or a weaker one. On the stronger side of the spectrum, we have the following: Definition 2.12. A symplectic manifold (W, ω) is called a Liouville manifold if there is an exhaustion W = ∪ +∞ k=0 W k by compact domains and there is a global Liouville vector field Z for ω which is complete and points outwards along the boundary of each W k . Moreover, in the case where W = W k for a certain k, (W, λ := ι Z ω) is called a Liouville (or exact) filling of (V = ∂W, ξ = ker(λ| ∂ W )).

We point out that the fact that λ| M is a positive contact form on M is equivalent to the fact that the Liouville vector field Z of (W, λ) is positively transverse to V = ∂W . Moreover, this notion can also be strengthened further: Definition 2.13. A Weinstein manifold is the data of (W, ω, Z, ψ), where (W, ω) is a symplectic 2n-dimensional manifold, ψ is an exhausting (i.e. proper and bounded from below) Morse function on W and Z is a Liouville vector field for (W, ω) which is gradientlike for ψ. Given a regular value c for ψ, (M := ψ -1 (c), ker(ι Z ω| M )) is said to admit the Weinstein filling (ψ -1 ((-∞, c]), ω, Z, ψ).

The complex counterpart of Weinstein structures are the following: Definition 2.14. A Stein manifold is the data of (W, J, ψ), where (W, J) is a complex manifold and ψ : W → R is exhausting and J-convex.

We recall that d C ψ := dψ •J ∈ Ω 1 (W ), and that ψ is called J-convex if ω ψ := -dd C ψ is a symplectic form on W which is compatible with J, i.e. such that g ψ (., .) := ω ψ (., J.) is a Riemannian metric on W .

As we will never be interested in the integrability of almost complex structures, we will actually only need the following weaker notion: Definition 2.15. An almost Stein manifold is the data of (W, J, ψ), where (W, J) is an almost complex manifold with boundary, ψ : W → R is J-convex, ∂W is a regular level set of ψ and J is ω ψ compatible, where ω ψ := -dd J ψ.

Notice that there is a preferred Liouville vector field Z ψ for ω ψ , given by the symplectic dual of -d J ψ.

In Section 3.3, we will use the following result on Weinstein structures:

Theorem 2.16. Let W be a smooth 2n-dimensional smooth manifold with boundary. Every Weinstein structure (ω, Z, ψ) on W is homotopic, in the space of Weinstein structures with fixed exhausting function ψ, to a Weinstein structure (ω ψ , Z ψ , ψ) coming from an almost Stein structure (J, ψ).

The above theorem, which can in some sense be considered a "folklore" result in the field, is a weaker form of [CE12, Theorem 1.1], which gives a genuine Stein structure.

In the rest of the manuscript we will also need another version of the fillability condition, which is weaker than all those recalled above. Let (V 2n+1 , ξ) be a contact manifold and (W 2n+2 , ω) a symplectic manifold such that V = ∂W . Denote by ω ξ the restriction of ω to ξ and by CS ξ the standard conformal symplectic structure on ξ, i.e. the conformal class of dα| ξ , for an arbitrary 1-form α defining ξ. We point out that CS ξ is well defined because independent of such a choice of α. CHAPTER 2. BACKGROUND Definition 2.17 ( [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF]). We say that (W, ω) weakly fills (V, ξ), or that ω weakly dominates ξ, if ω ξ is symplectic on ξ and ω ξ + CS ξ is a ray of symplectic structures on ξ.

Another way to formulate this condition is the following: ω weakly dominates ξ = ker α if and only if α ∧ (ω ξ + τ dα)

n > 0 for all τ ≥ 0. This will be the formulation we will actually use in the following, as it is more computation-friendly.

Open books in contact geometry

We recall that an open book decomposition on a manifold V is a pair (B, ϕ), where B is a codimension 2 submanifold and ϕ : V \ B → S 1 is a fibration, such that there is a neighborhood B × D 2 of B in V on which ϕ becomes B × D 2 \ {0} → S 1 given by (q, r, θ) → θ, where (r, θ) are polar coordinates on D 2 . Moreover, for each θ ∈ S 1 , the closure ϕ -1 (θ) is called a page of (B, ϕ).

Notice that an open book decomposition (B, ϕ) of V determines a unique element of the mapping class group of the page F , called the monodromy of the open book. Following for instance [START_REF] Giroux | Ideal Liouville Domains -a cool gadget[END_REF], this can be seen as follows. Consider X a spinning vector fields for (B, ϕ), i.e. a vector field X on V such that:

• X = 0 along B and dϕ(X) = 1 on V \ B;

• X lifts to a smooth vector field on the manifold with boundary obtained from V by a real oriented blowup along B.

Then, the restriction of the flow of X at time 1 gives a diffeomorphism of the page F , relative to B. Moreover, as the set of spinning vector fields for (B, ϕ) is an affine space, the isotopy class of this diffeomorphism depends only on (B, ϕ). This mapping class is the monodromy of the open book (B, ϕ).

The link between open book decompositions and contact geometry is captured by the following notion: Definition 2.18 ([Gir02]). Let (V, ξ) be a contact manifold. An open book decomposition (B, ϕ) is said to support the contact structure ξ if the following conditions are satisfied:

1. B is a contact submanifold of (V, ξ); 2. there is a contact form α defining ξ such that, for each θ ∈ S 1 , dα| ϕ -1 (θ) is a positive symplectic form on the fiber ϕ -1 (θ);

3. for each θ ∈ S 1 , the orientation on B induced by ξ coincides with its orientation as boundary of the symplectic manifold (ϕ -1 (θ), dα| ϕ -1 (θ) ).

Such a contact form α is said to be adapted to the open book (B, ϕ) of V .

The above definition is interesting due to the following (deep) result: [START_REF] Giroux | Géométrie de contact: de la dimension trois vers les dimensions supérieures[END_REF]). Each contact structure on a closed manifold is supported by an open book decomposition.

Theorem 2.19 ([
According to [START_REF] Giroux | Géométrie de contact: de la dimension trois vers les dimensions supérieures[END_REF], such a supporting open book is moreover unique up to stabilization in the case of 3-dimensional contact manifolds; as we will need it in Section 9.2, we now give a statement for this uniqueness property. As explained in [START_REF] Giroux | On the stable equivalence of open books in three-manifolds[END_REF], the following result is essentially due to Stallings:

OPEN BOOKS IN CONTACT GEOMETRY

Theorem 2.20 ( [START_REF] Stallings | Constructions of fibred knots and links[END_REF]). Let Σ be a compact surface with boundary in a manifold M and δ 0 a properly embedded arc in Σ. Let also Σ ⊂ M be obtained by plumbing a positive/negative Hopf band to Σ, i.e. if Σ = Σ ∪ A ± where A ± is an annulus in M such that 1. the intersection A ± ∩ Σ is a tubular neighborhood of δ 0 , 2. the core curve of A ± bounds a disk in M \Σ and the linking number of the boundary components is ±1.

If Σ is a page of an open book decomposition (B, ϕ) of M , then Σ is also a page of an open book (B , ϕ ).

Following [START_REF] Giroux | Géométrie de contact: de la dimension trois vers les dimensions supérieures[END_REF], an open book (B , ϕ ) is said to be a stabilization of (B, ϕ) if it is obtained by (B, ϕ) via a finite sequence of plumbings of positive Hopf bands. Then, we have the following:

Theorem 2.21 ([Gir02]
). On a 3-dimensional closed manifold, two open book decompositions support the same contact structure if and only if they have isotopic stabilizations.

We point out that, in this manuscript, and more precisely in Section 9.2, we will actually only need the easier part of this result, i.e. the fact that an open book and all its stabilizations support the same contact structure.

Part I

The topology of the contactomorphism group Chapter 3

On contact mapping classes of infinite order

The aim of this chapter is to give a proof of Théorème I.A; we start by recalling its setting and statement.

Let (F 2n-2 , ω F , Z F , ψ F ) be a Weinstein manifold and consider its stabilization

(F × C, ω F ⊕ ω 0 , Z F + Z 0 , ψ F + |.| 2 C
), where ω 0 = rdr ∧ dϕ and

Z 0 = 1 2 r∂ r if we use coordinates z = re iϕ ∈ C. Suppose that c > min ψ F is a regular value of ψ := ψ F + |.| 2
C and let W be the compact domain ψ -1 ((-∞, c]). We suppose also that there is an almost complex structure J F on F tamed by ω F and such that (T F, J F ) is trivial as complex bundle over F . Consider now the Weinstein manifold (F ×C×R×S 1 , ω , Z , ψ ), where, using coordinates (s, θ) ∈ R × S 1 , ω = ω F + ω 0 + 2ds ∧ dθ, Z = Z + s∂ s and ψ (p, z, s, θ) = ψ(p, z) + s 2 . The preimage (ψ ) -1 (c), which is diffeomorphic to the product of the double DW := W ∪ ∂W W of W and S 1 , is naturally equipped with the contact structure ξ = ker α, where α = ι Z ω | DW ×S 1 . Moreover, the diffeomorphism of F × C × R × S 1 given by (q, z, s, θ) → (q, e iθ z, s, θ) restricts to a well defined diffeomorphism Ψ of DW × S 1 . Théorème I.A. Le difféomorphisme Ψ de DW × S 1 est lissement isotope à un contactomorphisme Ψ c de (DW × S 1 , ξ) tel que, pour tout entier k = 0, sa k-ème itérée n'est pas contacto-isotope à l'identité. This chapter is organized as follows. Section 3.1 describes how, given a Liouville domain W , one can naturally construct an explicit Liouville manifold having DW × S 1 as convex boundary, as well as contactomorphisms of the latter; this will then be used in the case of Weinstein domains in the proof of Théorème I.A. In Section 3.2 we describe a simple invariant, of homotopical nature, for (contact-isotopy classes of) contactomorphisms, introducing the notion of families of Lagrangian bases. This invariant will then be used especially in the case of stably trivial contact structures. Then, in Section 3.3 we prove Théorème I.A recalled above.

Product of doubled Liouville domains and S 1

Let W m be a smooth manifold and f : W → R be a proper and bounded from below function which is also a regular equation of a (cooriented) hypersurface M m-1 ⊂ W , i.e. a smooth proper function transverse to 0 and such that M = f -1 (0) (with coorientation). Denote then by W m the compact submanifold f -1 ((-∞, 0]) of W .

PRODUCT OF DOUBLED LIOUVILLE DOMAINS AND S 1

Definition 3.1. We call f -double of W , and denote it by D f W , the smooth manifold given by {(p, s)

∈ W × R | s 2 + f (p) = 0}.
Notice that this set is indeed a smooth submanifold of W × R because the function W × R → R given by (p, s) → s 2 + f (p) is transverse to 0. Indeed, one can always find a vector field Z on W which is boundary-gradient-like for f , i.e which satisfies df (Z) ≥ 0 everywhere on W and df (Z) > 0 along M = f -1 (0): there is a vector field Z on a neighborhood U of the (cooriented) hypersurface M such that df (Z ) > 0 on U , and we can choose Z to be Z multiplied by a non-negative cutoff function χ supported in U . Then,

d(s 2 + f )(s∂ s + Z) = 2s 2 + df (Z) > 0 along D f W ⊂ W × R, which shows that D f W is a regular hypersurface. Notation 3.2. If f : W → R, we denote by f D : W × R → R the function f D (p, s) = s 2 + f (p); in particular, if f is an equation of the hypersurface M ⊂ W , then f D defines the hypersurface DW ⊂ W × R, as shown above.
In a similar way, if Z is a vector field on W , we denote by

Z D the vector field Z + s∂ s on W × R s ; if Z if boundary-gradient-like for f , then so is Z D for f D .
We have the following uniqueness property of the f -double: Lemma 3.3. Let f 0 , f 1 : W → R be two regular equations for M and Z be a vector field on W which is boundary-gradient-like for both f 0 and f 1 . For each t ∈ [0, 1], denote by f t the function tf 1 + (1 -t)f 0 . Then the flow of the [0, 1] t -parametric vector field

X t := f 1 -f 0 df D t (Z D ) Z D on W × R \ {s = 0, df 0 (Z) = df 1 (Z) = 0}
gives an isotopy which, at time t = 1, restricts to a diffeomorphism from D f1 W to D f0 W .

Lemma 3.3 tells that D f W does not depend on f , up to diffeomorphism. By a slight abuse of notation, we may hence write DW and simply talk about the double of W .

Proof (Lemma 3.3). Notice that if f 0 , f 1 : W → R are two regular equations for M (inducing the same coorientation on it), then so is

f t = tf 1 + (1 -t)f 0 , for each t ∈ [0, 1]. Moreover, if Z is boundary-gradient-like for both f 0 , f 1 , then Z D is also boundary- gradient-like for f D t , for each t ∈ [0, 1]. In particular, the smooth function G : W ×R×[0, 1] → R, given by G(p, s, t) = s 2 +f t (p), is transverse to 0: indeed, dG(Z D ) > 0 along G -1 (0) = t D ft W × {t}. Then, G -1 (0) is a smooth submanifold of W × R × [0, 1], which is moreover contained in W × R \ {s = 0, df 0 (Z) = df 1 (Z) = 0}.
In particular, the (well defined on Im G) vector field -∂ t + X t is tangent to G -1 (0) and the restriction of its flow at time 1 gives a diffeomorphism from

G -1 (0) ∩ W × R × {1} = D f1 W × {1} to G -1 (0) ∩ W × R × {0} = D f0 W × {0}, as wanted.
Let now ( W 2n , λ) be a Liouville manifold and denote by Z its Liouville vector field. Consider also a smooth proper function f : W → R, bounded from below and such that Z is boundary-gradient-like for f ; denote by W the (compact) submanifold f -1 ((-∞, 0]) of W . Notice that (M, η = ker(λ| M )) is a contact manifold and that (W, λ) is a Liouville filling of it.

Consider now the Liouville manifold ( W × R s × S 1 θ , λ + 2sdθ), where R s and S 1 θ denote the manifolds R and S 1 with coordinates s and θ respectively. Notice that the vector

CHAPTER 3. ON CONTACT MAPPING CLASSES OF INFINITE ORDER

field Z D = Z + s∂ s and the function can naturally be seen on

W × R × S 1 . Moreover, Z D is Liouville for λ + 2sdθ and transverse to D f W × S 1 = {f D = 0} ⊂ W × R × S 1 ; so, α f := (λ + 2sdθ)| D f W ×S 1 is a contact form on D f W × S 1 .
In analogy with Notation 3.2, we will also denote the Liouville form λ + 2sdθ on W × R × S 1 by λ D in the following.

Lemma 3.4. Let f 0 , f 1 : W → R be two regular equations for M such that the Liouville vector field Z is boundary-gradient-like for both f 0 , f 1 and, for t ∈ [0, 1], denote by f t the function tf 1 + (1 -t)f 0 . Then, the flow of the [0, 1] t -parametric vector field

X t := f 1 -f 0 df D t (Z D ) Z D on Y := W × R \ {s = 0, df 0 (Z) = df 1 (Z) = 0} induces an isotopy of Y × S 1 which, at time t = 1, restricts to a contactomorphism from (D f1 W × S 1 , ker(α f1 )) to (D f0 W × S 1 , ker(α f0 )).
We may hence drop the f in the notation and just denote it (DW × S 1 , ker α) from now on.

Proof (Lemma 3.4). According to Lemma 3.3, the only thing to show is that the flow

ψ t of X t on Y × S 1 preserves ker(λ D ). An explicit computation shows that L Xt (λ D ) = f1-f0 df D t (Z D ) λ D , which implies that ψ * t λ D = h t λ D for a certain function h t : W × R → R >0 , as wanted.
Remark. In [GS10], Geiges and Stipsicz construct, more generally, contact forms on

(W 1 ∪ M W 2 ) × S 1
, where (W 1 , λ 1 ) and (W 2 , λ 2 ) are Liouville domains with the same (strict) contact boundary (M, α); the contact structure they obtain in the particular case where W 1 = W 2 and λ 1 = λ 2 (and ∂W 1 identified with ∂W 2 via the identity) is the same, up to isotopy, as the contact structure on DW × S 1 that we described above. Even though the construction described here is less general, it has the advantage of involving a natural Liouville filling of the strict contact manifold (DW × S 1 , α), which will be useful in Section 3.3; notice, however, that one cannot always expect a presentation involving a symplectic filling for the construction in [START_REF] Geiges | Contact structures on product fivemanifolds and fibre sums along circles[END_REF]. For instance, in the case

W 1 = D 2 and W 2 = Σ g \ D 2
, where Σ g is a closed surface with genus g = 0, the theory of convex surfaces by Giroux tells that the contact structure on (W 1 ∪ S 1 W 2 ) × S 1 obtained as in [START_REF] Geiges | Contact structures on product fivemanifolds and fibre sums along circles[END_REF] is overtwisted: indeed, it is the unique S 1 -invariant contact structure on Σ g × S 1 such that each Σ g × {pt} is a convex surface with dividing set consisting of a homotopically trivial circle.

We now exhibit an explicit natural way to construct (strict) contactomorphisms of (DW × S 1 , ξ := ker α). Consider an S 1 -family of diffeomorphisms (ϕ θ ) θ∈S 1 of W , each of which preserves both λ and f : W → R; we do not assume that they are the identity on M = ∂W . Take then the diffeomorphism Ψ : DW × S 1 → DW × S 1 induced by the restriction of Ψ :

W × R × S 1 → W × R × S 1 given by Ψ(p, s, θ) = (ϕ θ (p), s, θ); notice that this is well defined because ϕ θ preserves f . Lemma 3.5. The flow ψ t Y of the vector field Y = λ( ∂ϕ θ ∂θ ) 2 df D (Z D ) (2s Z -df (Z) ∂ s ) gives a smooth isotopy Ψ • ψ t Y from Ψ = Ψ • ψ 0 Y to a contactomorphism Ψ c := Ψ • ψ 1 Y of (DW × S 1 , ξ = ker α).

FAMILIES OF LAGRANGIAN BASES

Notice that Y , given above as a section of the tangent bundle of

W × R × S 1 \ {s = 0, df 0 (Z) = df 1 (Z) = 0} is indeed tangent to (the DW factor of) DW × S 1 .
Proof (Lemma 3.5). For notational ease, in the following we denote by X θ the vector field ∂ϕ θ ∂θ and by h θ the function λ(X θ ) defined on W . We start by noticing that we have the identity dh θ = -ι X θ dλ: indeed, ϕ * θ λ = λ for each θ ∈ S 1 , so that L X θ λ = 0, which is equivalent to dh θ = -ι X θ dλ. In particular, an evaluation of the above identity on the Liouville vector field Z tells that dh θ (Z) = h θ .

An explicit computation gives that

Ψ * λ D = λ+(2s + h θ ) dθ, so that Ψ * λ D | T (DW ×S 1 ) = [λ + (2s + h θ ) dθ] | T (DW ×S 1 ) . Denote then, for all t ∈ [0, 1], by λ D t the 1-form λ + (2s + th θ )dθ on W × R × S 1 and by α t the 1-form λ D t | T (DW ×S 1
) on DW × S 1 . We then prove that α t is a contact form for each t ∈ [0, 1]; for this, it's enough to prove that, for each t ∈ [0, 1], λ D t is a Liouville form and that its Liouville vector field

Z D t is transverse to DW × S 1 . We can compute dλ D t = dλ + 2ds ∧ dθ + tdh θ ∧ dθ; then, it is easy to see that (dλ D t ) n+1 = 2(dλ) n ∧ ds ∧ dθ, so that λ D t is indeed a Liouville form on W × R × S 1 . Moreover, its vector field Z t is just Z D : indeed, ι Z D dλ D t = ι Z dλ + tdh θ (Z)dθ + 2sdθ
, which is exactly equal to λ D t , because dh θ (Z) = h θ , as remarked above, and ι Z dλ = λ. In particular, Z t = Z D is transverse to DW × S 1 , as wanted. Now, according to (the proof of) Gray's theorem, the flow of the (a priori timedependent) vector field X t such that α t (X t ) = 0 and ι Xt dα t | ker αt = -αt | ker αt gives an isotopy that pulls back ker α t to ker α 0 . Hence, it's enough to show that the vector field Y in the statement satisfies these two conditions. An explicit computation gives that df D (Y ) = 0 and

λ D t (Y ) = 0, i.e. that Y ∈ ker α t = ker λ D t ∩ T (DW × S 1 ). Moreover, we can compute ι Y dλ D t = h θ 2df D (Z D ) [2sλ + 2ts dh θ (Z) dθ -2df (Z)dθ] (i) = h θ 2df D (Z D ) 2sλ D t -4s 2 dθ -2df (Z)dθ (ii) = 2sh θ 2df D (Z D ) λ D t - d dt λ D t ,
where for (i) we used that dh θ (Z) = h θ and for (ii) we used that df D (Z D ) = 2s 2 + df (Z) and d dt λ D t = h θ dθ. In particular ι Y dα t | ker αt = -αt | ker αt , as wanted.

Families of Lagrangian bases

Let V be a smooth (2n + 1)-manifold and ξ a contact structure on V . Given a compact manifold Y m , we call family of Lagrangian bases of ξ indexed by Y , and we denote it by L, the data of a smooth map γ : Y → V and, for j = 1, . . . , n, of smooth maps

X j : Y → ξ such that the following diagram commutes ξ Y V γ Xj
and such that, for each q ∈ Y , the X 1 (q), . . . , X n (q) are R-linearly independent and generate a Lagrangian subspace of (ξ p , (CS ξ ) p ). Here, CS ξ is the natural conformal symplectic structure on ξ (see the discussion after Definition 2.10); in particular, (CS ξ ) p is a conformal class of symplectic alternating forms on ξ p and, hence, has a well defined class of (isotropic and) Lagrangian subspaces. We point out that if

f : (V, ξ) → (V, ξ) is a contactomorphism, then f * L := (f • γ, df (X 1 ), . . . , df (X n )) is also a Y -family of Lagrangian bases of ξ: indeed, f preserves the conformal symplectic structure CS ξ on ξ. Moreover, if f t : (V, ξ) → (V, ξ) is a contact-isotopy from f 0 = Id to f 1 = f , then (f t ) * L is a path of Y -families of Lagrangian bases of ξ from L to f * L.
In other words, we have the following obstruction to contact-isotopies: Lemma 3.6. Let f : (V, ξ) → (V, ξ) be a contactomorphism. If there is a Y -family of Lagrangian bases L for ξ such that f * L is not homotopic (among families of Lagrangian bases) to L, then f is not contact-isotopic to the identity. Let now J be a complex structure on ξ tamed by CS ξ . If L = (γ, X 1 , . . . , X n ) is a Yfamily of Lagrangian bases for ξ then, for each q ∈ Y , one has ξ γ(q) = X 1 (q), . . . , X n (q) C . Suppose moreover that (ξ, J) is stably trivial, i.e. that there is a k ∈ N and an isomorphism of complex vector bundles Φ :

(ξ, J) ⊕ ε k V ∼ -→ ε n+k V over V ; here, ε V is the trivial complex line bundle V × C → V and ε m
V denotes the direct sum of ε V with itself m times. We point out that the property that (ξ, J) is stably trivial is not dependent on a specific choice of J: indeed, the space of complex structures on ξ which are tamed by CS ξ is contractible, hence (ξ, J) and (ξ, J ) are isomorphic as complex vector bundles if J, J are both tamed by CS ξ . Then, if (e 1 , . . . , e k ) are the sections of ε k V which give, at each point p ∈ V , the canonical bases of the fiber

(ε k V ) p = C k , the image of L stab := (γ, X 1 , . . . , X n , e 1 • γ, . . . , e k • γ) via Φ gives,
pointwisely, a bases of the vector space C n+k given by the fibers of ε n+k V over each point of the image of γ. In particular, considering the linear endomorphism of C n+k obtained by sending the canonical basis to the basis given, pointwisely, by the image of L stab via Φ, we then obtain a smooth map M : Y → GL n+k (C).

In the following, we say that the family L stab is the (e 1 , . . . , e k )-stabilization of L (sometimes omitting the sections (e 1 , . . . , e k ) of ε k V if there is no ambiguity) and denote it more concisely by L ⊕ (e 1 , . . . , e k ). We will also say that the map M is the Y -family of (invertible) matrices associated (via Φ) to L stab .

Given a contactomorphism f : (V, ξ) → (V, ξ), the stabilization

(f * L) stab = (f • γ, df (X 1 ), . . . , df (X n ), e 1 • f • γ, . . . , e k • f • γ) gives, via Φ, another Y -family of invertible matrices, which we denote f * M : Y → GL n+k (C).
As this can also be done parametrically, analogously to Lemma 3.6 above, we obtain: Lemma 3.7. Let (V 2n+1 , ξ) be a contact manifold and J an almost complex structure on ξ such that (ξ, J) is stably trivial, via an isomorphism Φ : (ξ, J) ⊕ ε k V → ε n+k V of complex vector bundles over V . Let also f : (V, ξ) → (V, ξ) be a contactomorphism and L = (γ, X 1 , . . . , X n ) be a Y -family of Lagrangian bases for ξ. If the Y -family of matrices associated via Φ to the (e 1 , . . . , e k )-stabilization (f * L) stab is not homotopic, as map Y → GL n+k (C), to the Y -family of matrices associated via Φ to the (e 1 , . . . , e k )-stabilization L stab , then f is not contact-isotopic to the identity.

Examples of infinite order contact mapping classes

The aim of this section is to prove Théorème I.A, recalled in the beginning of the chapter; in particular, we will use the notations introduced in its statement.

We start by claiming that we can make the following additional assumption: the Weinstein structure (F, ω F , Z F , ψ F ) comes from an almost Stein structure (J F , ψ F ) (i.e.

EXAMPLES OF INFINITE ORDER CONTACT MAPPING CLASSES

ω F is equal to -dd C ψ F and Z F is equal to the vector field ω F -dual to λ F := -d C ψ F ; see Definition 2.15), such that moreover (T F, J F ) is trivial as complex vector bundle. We will indeed need this compatibility between the almost complex structure J F which trivializes T F and the Weinstein structure (ω F , Z F , ψ F ) in the proof of Théorème I.A. Here's how these additional assumptions can be arranged. According to Theorem 2.16, there is an almost Stein structure (J F , ψ F ) on F such that the Weinstein structure (-dd C ψ F , Z F , ψ F ) is Weinstein homotopic, with fixed function ψ F , to the Weinstein structure (ω F , Z F , ψ F ) in the statement of Théorème I.A. Because -dd C ψ F and ω F are homotopic as symplectic structures, (a slight adaptation of the proof of) the contractibility of the space of almost complex structures tamed by a given symplectic form also gives that J F is homotopic to J F ; (T F, J F ) is then isomorphic, as complex bundle, to (T F, J F ) and is, in particular, trivial. Because the Weinstein structure (ω 0 , Z 0 , |.|

2 ) on C already comes from the almost Stein (actually, Stein) structure (i, |.| 2 ), we can moreover apply the Weinstein homotopy only on the F -factor of F × C in order to ensure the same assumption on the manifold F × C. Notice that this Weinstein homotopy do not change W , Ψ and (up to isotopy) the contact structure on DW × S 1 defined in Théorème I.A, because the homotopy is along the F -factor and with ψ F fixed.

With a little abuse of notation, we will hence denote J F again by J F ; let also J := J F ⊕i on F ×C. Notice that J := J F ⊕i can be further extended to

J D on F ×C×R s ×S 1 θ by defining J D (∂ s ) := ∂ θ on T (R s × S 1 θ ). Notice that (J D , ψ D = ψ + s 2 ) is an almost Stein structure on F × C × R × S 1 such that ω D = dλ D , where λ D := -d C ψ D = λ + 2sdθ, and Z D = Z + s∂ s is the Liouville vector field of λ D .
By the hypothesis of Théorème I.A (and the above assumption), there is an isomorphism of complex vector bundles ν :

(T F, J F ) ∼ -→ ε n-1 F over F , where ε n-1 F
is the trivial complex vector bundle (F ×C n+1 , J std ). Moreover, ν naturally extends to a trivialization

µ : T F × C × R × S 1 , J D ∼ -→ ε n+1 F ×C×R×S 1 (3.1)
such that, for each (q, z, s, θ) ∈ F × C × R × S 1 , one has:

• the following diagram commutes

(T q F, J F ) T (q,z,s,θ) F × C × R × S 1 , J D (ε n-1 F ) q = C n-1 (ε n+1 F ) q = C n+1 (ε n+1 F ×C×R×S 1 ) (q,z,s,θ) = C n+1 νq i µ (q,z,s,θ) j

Id

where i and j are the natural inclusions given by

T q F = T q F ⊕ {(0, 0, 0)} ⊂ T (q,z,s,θ) F × C × R × S 1 and C n-1 = C n-1 × {(0, 0)} ⊂ C n+1 ; • µ (q,z,s,θ) (∂ x ) = (0, . . . , 0, 1, 0) ∈ C n+1
, where we use here coordinates (x, y) on the factor

C of F × C × R × S 1 , • µ (q,z,s,θ) (∂ s ) = (0, . . . , 0, 1) ∈ C n+1
, where s is the coordinate on the factor R of

F × C × R × S 1 .

CHAPTER 3. ON CONTACT MAPPING CLASSES OF INFINITE ORDER

Let now a := c-min(ψ F ) 4 > 0 (the exact value of this parameter will intervene later in the proof), and consider a non-decreasing smooth cut-off function χ : R → [-1, 1], equal to 1 on (2a, +∞), equal to -1 on (-∞, -2a), and such that χ(x) = x for x ∈ (-a, a). Then, the function f : Proof. Let f 0 := ψ -c and f 1 := f = χ(ψ -c). According to Lemma 3.4, the flow ψ 1 Xt of the vector field

F × C → R defined by f := χ(ψ -c) is a regular equation of M = f -1 (0) = ψ -1 (c); in particular, Z D = Z + s∂ s on W × R s is transverse to D f W too. Notice also that D f W is essentially a "flattened" version of D ψ-c W , as Figure 3.1 illustrates. W R s D ψ-c W D f W W - s = +1 s = -1 Z D Figure 3.1: D f W and D ψ-c W inside W × R
X t = f1-f0 df D t (Z D ) Z D gives a contactomorphism from (D f0 W × S 1 , ker(α f0 )) to (D f1 W × S 1 , ker(α f1 )).
Hence, in order to prove Lemma 3.8, it's enough to show that the diffeomorphism

ψ 1 Xt • Ψ • (ψ 1 Xt ) -1 of D f1 W is still induced by the diffeomorphism F × C × R × S 1
given by (q, z, s, θ) → (q, e iθ z, s, θ). But this is indeed the case, because the flow ψ 1

Xt fixes the angular component of the C-factor as well as the S 1 -factor of the product F × C × R × S 1 , and hence commutes with (q, z, s, θ) → (q, e iθ z, s, θ).

Let then α, Ψ, Y and Ψ c be obtained as in Section 3.1 from the Liouville manifold (F × C, λ), the regular equation f of M and the family {ϕ θ } θ∈S 1 of diffeomorphisms of F × C which is given by (p, z) → (p, e iθ z). We know from Section 3.1 that, inside the Liouville manifold

(F × C × R s × S 1 θ , λ D = λ + 2sdθ), the preimage of (-∞, 0] via F × C × R s × S 1 θ → R, (p, s, θ) → s 2 + f (p), gives a Liouville filling of (D f W × S 1 θ , α f ).
Moreover, as we are under the hypothesis that the Liouville structure on F comes from an almost Stein structure, the compact manifold {f D ≤ 0} = {s 2 + ψ ≤ c}, together with the almost complex structure induced by the ambient almost Stein manifold

(F ×C×R× S 1 , J D ), is actually an almost Stein filling of (D f W × S 1 θ , α f ). In particular, λ D is equal to -d C ψ D on a neighborhood of {f D = 0} = {s 2 + ψ = c}, which guarantees that the almost complex structure J D on F × C × R × S 1 restricts to a well defined endomorphism of ker α f along D f W × S 1
θ ; this restriction is then automatically tamed by dα f | ker α f .

EXAMPLES OF INFINITE ORDER CONTACT MAPPING CLASSES

Remark. If we relax the condition in Théorème I.A of F being a Weinstein manifold to F being a Liouville manifold, we may not be able to find a J D on F × C × R × S 1 that both restricts to ker α f and splits as J F ⊕ i ⊕ J D | T (R×S 1 ) at the same time; these are both conditions we will need in the following.

Recall now that

Ψ c = Ψ • ψ 1 Y , with ψ t
Y the flow at time t of Y defined in Lemma 3.5. Then, in order to show that, for each k = 0, Ψ k c is not contact isotopic to the identity, we are going to proceed by steps as follows: 3.1); notice that it has nonempty interior by construction of f . We then describe a S 1 -family of Lagrangian bases L for ker(α f ) on W -× S 1 .

1. Let W -:= D f W ∩ {s = -1} ⊂ F × C × R (see Figure
2. We remark that, for all t ≥ 0, ψ t Y (W -× S 1 ) ⊂ W -× S 1 , and we describe the behavior of the restriction of Ψ c , and its iterates, to W -× S 1 . This allows us to describe, for all k ≥ 1, the pushforward

(Ψ k c ) * L of L via the k-th iterate of Ψ c .
3. We describe, for each k ≥ 0, the family of matrices B k : S 1 → GL n+1 (C) associated, via the trivialization µ, to the stabilization

(Ψ k c ) * L ⊕ Z D . We then show that, if k ≥ 1, B k is not homotopically trivial as map S 1 → GL n+1 (C).
According to Lemma 3.7, this proves that, for all k ≥ 1, the k-th iterate of the contactomorphism Ψ c is not contact isotopic to the identity. The space of contactomorphism being a group, this implies the same conclusion for all k = 0.

Step 1 We recall that there is a trivialization ν : (T F, J F ) → (F ×C n-1 ); let (w 1 , . . . , w n-1 ) be the inverse image of the sections (e 1 , . . . , e n-1 ) that give, fiber-wisely, the canonical complex basis for ε n-1 F . Then, we have the following: Lemma 3.9. There are q 0 ∈ F and

x 0 ∈ R >0 ⊂ C such that (q 0 , x 0 , -1) ∈ W -⊂ F × C × R and L := γ, v 1 , . . . , v n-1 , ∂ θ (θ) + 2 x0 ∂ y (x 0
) is an S 1 -family of Lagrangian bases for ker(α f ), where γ : S 1 → W -× S 1 is defined by γ(θ) = (q 0 , x 0 , -1, θ), v j denotes w j (q 0 ) ∈ T q0 F for each j = 1, . . . , n -1, and (x, y) are coordinates on the factor C of

F × C × R × S 1 .
Proof (Lemma 3.9). Let p 0 = (q 0 , x 0 ) ∈ M = ψ -1 (c -3a) ⊂ F × C, where q 0 ∈ F is a point of (global) minimum for ψ F and x 0 ∈ R >0 ⊂ C satisfies ψ F (q 0 ) + x 2 0 = c -3a. Recall that c > min(ψ F ) is the regular value for ψ in the statement of Théorème I.A; also, a = c-min(ψ F ) 4 was introduced in the choice of regular equation f for M and satisfies c -3a > min ψ F , so that an x 0 > 0 as above actually exists. Because W -= DW ∩ {s = -1} ⊂ F × C × R s , this proves the existence of γ as in Lemma 3.9.

We now have to prove that L is an S 1 -family of Lagrangian bases for ker(α f ). Because q 0 is a (global) minimum of ψ F , we have λ D = x 2 0 dϕ-2dθ at the point γ(θ) (here, we use coordinates (r, ϕ) for the factor

C of F × C × R × S 1 ). In particular, ker(α f )| γ(θ) , seen as a sub-bundle of T γ(θ) F × C × R × S 1 = T q0 F ⊕ T x0 C × T -1 R × T θ S 1 , is equal to T q0 F ⊕ Span C ∂ θ (θ) + 2 x0 ∂ y (x 0 ) (recall that ∂ ϕ (x 0 ) = x 0 ∂ y (x 0 )
). This means exactly that L is a family of Lagrangian bases for ker(α f ), as wanted.

Step 2 This step consists in the following two lemmas:

CHAPTER 3. ON CONTACT MAPPING CLASSES OF INFINITE ORDER Lemma 3.10. The contactomorphism Ψ c of (DW × S 1 , ker α f ) satisfies Ψ c (W -× S 1 ) ⊂ W -× S 1 . More precisely, for each k ≥ 0, Ψ k
c has the following form:

Ψ k c : W -× S 1 → W -× S 1 (q, re iϕ , -1, θ) → (Q k (q, r), R k (r)e i(ϕ+kθ) , -1, θ) Lemma 3.11. Let γ, (v 1 , • • • , v n-1
) and L be as in Lemma 3.9 and, for each k ≥ 0, Q k and R k as in Lemma 3.10. Then, for each k ≥ 0, there are a complex basis

(v k 1 , . . . , v k n-1 ) of T Q k (q0,x0) F and a real number s k such that (Ψ k c ) * L is given by Ψ k c • γ, v k 1 , . . . , v k n-1 , ∂ θ (θ) + s k ∂ ϕ (r k e ikθ ) , where r k := R k (x 0 ).
Proof (Lemma 3.10). We give a proof by induction on k. The case k = 0 is trivial; notice that it's actually enough to show that the lemma holds for k = 1, as the inductive step would then become immediate. Indeed, if both Ψ c and Ψ k c can be written in the form given in the statement of Lemma 3.10, it is immediate to check that the same is true for

Ψ k+1 c . Let's then analyze the case k = 1. Recall that Ψ c = Ψ • ψ 1 Y , where ψ t Y : DW × S 1 → DW × S 1 is the flow of Y (given by Lemma 3.5) at time 1. Notice that the function f is constant and s = -1 on W -× S 1 . Then, Y (which is tangent to D f W × S 1 ) restricts to -r 2 2 Z F (q) -r 3 4 ∂ r (re iϕ ) on W -× S 1
; here, we use polar coordinates z = re iϕ on the factor C. In particular, the flow

ψ t Y : DW × S 1 → DW × S 1 of Y at time t ≥ 0 satisfies ψ t Y (W -× S 1 ) ⊂ W -× S 1 : indeed, Y has no component along ∂ s ,
and its flow preserves the scoordinate, hence also

W -× S 1 = (D f W × S 1 ) ∩ {s = -1}. More precisely, at time t = 1, the embedding ψ 1 Y : W -× S 1 → W -× S 1 can be written as ψ 1 Y (q, re iϕ , -1, θ) = (Q(q, r), R(r)e iϕ , -1, θ), for some functions Q : F × C → F and R : C → R, with Q and R both independent of the angular component ϕ on C. Recalling that Ψ : F × C × R × S 1 → F × C × R × S 1
is given by Ψ(q, re iϕ , s, θ) = (q, re i(ϕ+θ) , s, θ), we then obtain an expression for

Ψ c = Ψ • ψ 1
Y which is exactly as in the statement of Lemma 3.10 (with the choices

Q 1 = Q and R 1 = R).
Proof (Lemma 3.11). We are going to describe who v k 1 , . . . , v k n-1 and s k in the statement are. For this, we use the expression for Ψ k c given in Lemma 3.10. Notice that

Ψ k c • γ is given by θ → (Q(q 0 , x 0 ), R k (x 0 )e ikθ , -1, θ) ∈ W -× S 1 ⊂ F ×C×R×S 1 . An explicit computation also gives d Ψ k c •γ(θ) Ψ k c (∂ θ (θ)) = ∂ θ (θ)+k∂ ϕ (r k e ikθ ) and d Ψ k c •γ(θ) Ψ k c (∂ ϕ (x 0 )) = ∂ ϕ (r k e ikθ ), where r k = R k (x 0 ) as in the statement. Then, if we choose s k := k + 2 x 2 0 and v k j := d (q0,x0) G k (v j )
for each j = 1, . . . , n -1, we have that (Ψ k c ) * L can indeed be written as in the statement of Lemma 3.11.

Remark 3.12. The informations in both Lemmas 3.10 and 3.11 could be made much more precise, by computing explicitly the flow ψ 1 Y . For instance, the value of s k given at the end of the proof of Lemma 3.11 is k + 2 x 2 0 . One can easily see that, in order for ∂ θ (θ) + s k ∂ ϕ (r k e ikθ ) to be tangent to ker α f (which it has to be, because image of a tangent vector via the differential of a contactomorphism), one needs the equality k + 2

x 2 0 = 2 r 2 k . An explicit computation of ψ 1
Y would have given us an explicit formula for R k (r) such that r k = R k (x 0 ) satisfies this condition. To improve the readability, we decided not to include these detailed informations, as the content of the lemmas above is actually all we need for Step 3.

Step 3 The main ingredient of the last step is the following:

EXAMPLES OF INFINITE ORDER CONTACT MAPPING CLASSES

Lemma 3.13. The family of matrices B k : S 1 → GL n+1 (C) associated via the trivialization µ (defined in Equation (3.1)) to the stabilization

(Ψ k c ) * L ⊕ Z D = Ψ k c • γ, v k 1 , . . . , v k n-1 , ∂ θ (θ) + s k ∂ ϕ (R k (x 0 )e ikθ ), Z D (Ψ k c • γ(θ))
is given by matrices of the form

B k (θ) =   B 0,k b 0,k b 1,k 0 b 2,k e ikθ b 3,k e ikθ 0 b 4,k b 5,k   , where b 0,k , b 0,k ∈ C n-1 , b 2,k , . . . , b 5,k ∈ C and B 0,k ∈ GL n-1 (C). Proof (Lemma 3.13). Notice that ∂ ϕ (r k e ikθ ) = -r k sin(kθ)∂ x (r k e ikθ ) + r k cos(kθ)∂ y (r k e ikθ ) = r k (-sin(kθ) + J cos(kθ))∂ x (r k e ikθ ) and that ∂ r (r k e ikθ ) = cos(kθ)∂ x (r k e ikθ ) + sin(kθ)∂ y (r k e ikθ ) = (cos(kθ) + J sin(kθ))∂ x (r k e ikθ ).
Then, Lemma 3.13 immediately follows from the expression for (Ψ k c ) * L given in Lemma 3.11 and from 1) . Lemma 3.13 tells in particular that B k is homotopically trivial as map S 1 → GL n+1 (C) if and only if k = 0. Indeed, B 0 is a constant map, and an easy computation tells that

Z D (Ψ k c • γ(θ)) = 1 2 R k (x 0 )∂ r (R k (x 0 )e i(ϕ+kθ) ) -∂ s (-
det(B k (θ)) = b k e ikθ , for a certain b k ∈ C \ {0} (notice that b k = 0 necessarily because B k (θ) ∈ GL n+1 (C)); in particular, θ → det(B k (θ)) is homotopically non-trivial if k ≥ 1.

This concludes

Step 3, hence the proof of Théorème I.A.

Chapter 4

Contact mapping classes on overtwisted manifolds

A general source of non-trivial contact mapping classes

The aim of this section is to prove Théorème I.C, already stated in Chapter 1:

Théorème I.C. On considère une variété lisse W de dimension 2n ≥ 2 et une structure de contact ξ sur la variété V := S 1 × W . On suppose que la première classe de Chern

c 1 (ξ) ∈ H 2 (V ; Z) est toroidale et que, pour tout entier k ≥ 2, la tirée en arrière π * k ξ de ξ via le revêtement à k-feuilles π k : S 1 × W → S 1 × W donné par π k (s, p) = (ks, p) satisfait c 1 (π * k ξ) = k • c 1 (ξ) modulo le sous-module H 2 ator (V ; Z) des classes atoroidales. Alors, la transformation de contact f : (S 1 ×W, π * k ξ) → (S 1 ×W, π * k ξ) définie par f (s, p) = (s + 2π
k , p) est isotope à l'identité lissement mais pas parmi les contactomorphismes.

As each contactomorphism gives in particular an isomorphism of the underlying almost contact structures, Théorème I.C directly follows from the two following lemmas:

Lemma 4.1. Let (S 1 × W 2n , ξ) be a contact manifold. For each natural k ≥ 2, denote by π k : S 1 × W → S 1 × W the k-fold cover π k (s, p) = (ks, p) and by f : (S 1 × W, π * k ξ) → (S 1 × W, π * k ξ) the contactomorphism f (s, p) = (s + 2π k , p). If f is contact-isotopic to the identity, then there is a contactomorphism φ : (S 1 × W, π * kN ξ) ∼ -→ (S 1 × W, π * kN +1 ξ) . Lemma 4.2. Let (V := S 1 × W, ξ), π k and f be as in Lemma 4.1. If moreover c 1 (ξ) is toroidal and c 1 (π * m ξ) = m • c 1 (ξ) mod H 2 ator (V ; Z)
for every natural m ≥ 2, then π * m ξ and π * m+1 ξ are not isomorphic as almost contact structures.

We now prove Lemmas 4.1 and 4.2 above.

Proof (Lemma 4.1). In order to find the desired contactomorphism φ, we use an idea that already appeared in Geiges and Gonzalo Perez [START_REF] Geiges | On the topology of the space of contact structures on torus bundles[END_REF] and in Marinković and Pabiniak [START_REF] Marinković | On displaceability of pre-Lagrangian toric fibers in contact toric manifolds[END_REF], and which consists in cutting off contact Hamiltonians on a particular cover of the manifold we are working with.

A GENERAL SOURCE OF NON-TRIVIAL CONTACT MAPPING CLASSES

By hypothesis, the contactomorphism f :

(S 1 × W, π * k ξ) → (S 1 × W, π * k ξ) defined by f (s, p) = (s + 2π k , p) is contact isotopic to the identity. Call (F r ) r∈[0,1] the isotopy, so that F 0 = Id, F 1 = f and F r is a contactomorphism for all r ∈ [0, 1].
Take now the universal cover R s of the factor S 1 s of the manifold S 1 s × W . Then, pull back π * k ξ to a contact structure η k on the covering R s × W of S 1 s × W and lift the contact isotopy F r to a contact isotopy Φ r of (R s × W, η k ) starting at the identity. Fix a certain contact form β k for η k and denote by H r : R s × W → R the path of contact Hamiltonians β k (Y r ) associated to the contact vector field Y r generating the isotopy Φ r (see the discussions following Proposition 2.5). Now, by compactness of W and [0, 1], there is an

N > 0 such that Φ({0} s ×W ×[0, 1] r ) is contained in (-2 (N -1) π, +∞) s × W .
Consider then an > 0 very small and a smooth function ρ : R → R such that ρ(x) = 0 for x < -2N π + and ρ(x) = 1 for x > -2 (N -1) π -. We can then construct a new contact Hamiltonian: K r (s, p)

:= ρ(s) • H r (s, p), for all (s, p) ∈ R s × W .
We claim that the contact vector field Z r associated to this new Hamiltonian K r can be integrated to a contact isotopy (Ψ r ) r∈[0,1] of (R s × W, η k ) starting at the identity. Indeed, Z r is zero for s < -2N π + and equal to the contact field Y r for s > -2 (N -1) π -, which means in particular that it is integrable outside of a compact set of R s × W (remark that Y r is trivially integrable, because it comes from a contact isotopy); this implies integrability on all R × W . Moreover,

Ψ r | {0}×W = Φ r | {0}×W and Ψ r | {-2N π}×W = Id | {-2N π}×W for all r ∈ [0, 1].
In particular,

Ψ 1 maps [-2N π, 0]×W contactomorphically to [-2N π, 2π k ]×W
, where we consider on the domain and on the codomain the structure η k . Now, by the periodicity of η k , we can identify the two boundary components of [-2N π, 0] × W so that the restriction of η k induces a well defined contact structure on the quotient. More precisely, the quotient contact manifold obtained is (S 1 s × W, π * kN ξ). The analogous procedure for the codomain

[-2N π, 2π k ] × W of Ψ 1 gives as quotient the contact manifold (S 1 s × W, π * kN +1 ξ). Lastly, because Ψ 1 : [-2N π, 0] × W → [-2N π, 2π
k ] × W is the identity on a neighborhood of {-2N π} × W and a lift of the translation f on a neighborhood of {0} × W , it induces on the quotient contact manifolds a well defined contactomorphism

φ : (S 1 s × W, π * kN ξ) ∼ -→ (S 1 s × W, π * kN +1 ξ) .
Proof (Lemma 4.2). Suppose by contradiction that there is an isomorphism of almost contact structures ψ :

(V, π * m ξ) ∼ → V, π * m+1 ξ ; in particular, ψ * c 1 (π * m ξ) = c 1 (π * m+1 ξ) . (4.1)
Now, the submodule H 2 ator (V ; Z) of atoroidal classes is natural, (i.e. it is preserved by pullbacks induced by continuous maps V → V ); in particular, ψ * induces a well defined endomorphism, which is moreover an isomorphism, of the quotient of H 2 (V ; Z) by H 2 ator (V ; Z). We then have

ψ * (π * n ξ) = nψ * c 1 (ξ) mod H 2 ator (V ; Z) for each natural n ≥ 2, so that Equation 4.1 becomes mψ * c 1 (ξ) = (m + 1)c 1 (ξ) mod H 2 ator (V ; Z) . (4.2)
Notice that N := H 2 (V ; Z) H 2 ator (V ; Z) is a finitely generated Z-module without torsion. In particular, the (well defined) divisibility map

d : N \ {0} → N \ {0} n → { k ∈ N | ∃n , n = kn } satisfies d(hn) = hd(n) and d(ψ * n) = d(n), for each n ∈ N \ {0} and h ∈ N \ {0}. Now, because c 1 (ξ)
is toroidal, we can apply d to both the left and right hand sides of Equation 4.2, thus obtaining the wanted contradiction.

Examples from Liouville pairs and half Lutz-Mori twists

In this section we give a proof of Proposition I.D; the main idea is the following. The contact structure η on the manifold V = S 1 ×W in the statement has trivial Chern classes (better, it is trivializable as complex bundle). We then apply a semi-local modification to η and obtain another contact structure ξ; the explicit nature of this modification (as well as the explicit nature of the original contact manifold (V, η)) allows us to compute the first Chern class of ξ, and to show that it satisfies the wanted conditions. More precisely, this section is structured in the following way. We recall in Sections 4.2.1 and 4.2.2, respectively, the notion of half Lutz-Mori twist and the construction of Liouville pairs, both appearing in Massot, Niederkrüger and Wendl [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF]. We then describe in Section 4.2.3 how half Lutz-Mori twists (along contact submanifolds belonging to one of the Liouville pairs constructed in [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF]) affect the Chern classes of the underlying almost contact structure. Finally, Section 4.2.4 contains the proof of Proposition I.D.

The half Lutz-Mori twist

Developing some ideas introduced by Mori in [Mor] in the 5-dimensional case, Massot, Niederkrüger and Wendl introduce in [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF] the notion of Lutz-Mori twist along a manifold belonging to a Liouville pair as a generalization of the well known 3-dimensional Lutz twists. We give here an explicit description of how to perform the half version of the Lutz-Mori twist in particular coordinates in a neighbourhood of the contact submanifold.

Let (V, ξ) be a contact manifold having as a codimension-2 contact submanifold (M, ξ + ) such that α + defining ξ + belongs to a Liouville pair, defined as follows:

Definition 4.3. [MNW13] Let M 2m+1 be an oriented manifold. We call Liouville pair on M a couple of contact forms (α + , α -) such that ±α ± ∧ (dα ± ) m > 0 and such that the form e r α + + e -r α -is a Liouville form on M × R r , i.e. its differential is a symplectic form on M × R.

We point out that the existence of Liouville pairs on closed manifolds is not trivial; at the moment, the only known examples in high dimension are given by the construction in [MNW13, Section 8], which is nonetheless a source of infinitely many non-homeomorphic manifolds with Liouville pairs in each (odd) dimension. In Section 4.2.2 we will recall the properties of this construction which are needed in order to prove Theorem I.C. We now want to find particular coordinates near the submanifold (M, ξ + ). For all ε > 0, denote D 2

ε the (open) disk of radius ε centered at the origin inside R 2 . Consider then a smooth map Ψ :

D 2 ε \ {0} → S 1 × (0, ε), defined by Ψ(r, ϕ) = (ϕ, ψ(r))
, where (r, ϕ) are polar coordinates on D 2 ε and ψ : (0, ε) → (0, ε) is smooth, strictly increasing, equal to r 2 on (0, ε 3 ) and equal to r on ( 2 3 ε, ε). Consider now the 1-form α 0 = 1+cos(s)

2 α + + 1-cos(s) 2 α -+ sin (s) dt on M × S 1 t × (0, ε) s . The fact that (α + , α -) is a Liouville pair implies that α 0 is a contact form; see [MNW13,

EXAMPLES FROM LIOUVILLE PAIRS AND HALF LUTZ-MORI TWISTS

Proposition 9.1] for the details. If Ψ denotes the map

(Id M , Ψ) : M × D 2 ε \ {0} → M × S 1
t × (0, ε) s , then the pull-back (Ψ ) * α 0 can be written as α + + r 2 dϕ + γ, with γ

smooth on M × D 2 ε \ {0} and γ = cos r 2 -1 2 α + + 1 -cos r 2 2 α -+ sin r 2 -r 2 dϕ for r < 3 .
Hence, (Ψ ) * α 0 naturally extends to a (smooth) contact form α on M × D 2 ε , which moreover restricts to α + on M × {0} M . Now, each contact submanifold of codimension 2 having topologically trivial normal bundle also has trivial conformal symplectic normal bundle. Hence, by the contact submanifold neighbourhood theorem (see Geiges [Gei08, Theorem 2.5.15]), each contact manifold (V, ξ) containing (M, ξ + ) as a codimension 2 contact submanifold with trivial normal bundle will also contain, for ε > 0 small enough, the above model M × D 2 ε , ξ = ker (α) as a (codimension 0) contact submanifold, in such a way that (M, ξ + ) coincides with (M × {0}, ξ| M ×{0} ).

We now describe how to modify the contact structure in this particular local coordinates around (M, ξ + ) in order to perform the half twist. Consider another smooth map Φ :

D 2 ε \{0} → S 1 ×(-π, ε), defined by Φ(r, ϕ) = (ϕ, φ(r))
, where (r, ϕ) are again polar coordinates on D 2 ε and φ : (0, ) → (-π, ε) is again smooth, strictly increasing, equal to r on ( 2 3 ε, ε), but this time equal to r 2 -π on (0, ε 3 ). As before, if Φ denotes the map

(Id M , Φ) : M × D 2 ε \ {0} → M × S 1 t × (-π, ε) s , then the contact form (Φ ) * α 0 naturally extends to a contact form α on M × D 2 ε , but this time at M × {0} we have the contact submanifold (M, ξ -= ker (α -)).
We remark though that the contact manifolds

(M × D 2 ε , ξ = ker α) and (M × D 2 ε , ξ = ker α ) coincide on the subset {r ≥ 5 6 ε} of M × D 2 ε . If we denote by D 2 δ the closed disk of radius δ := 11 12 ε centered at the origin inside R 2 , we can thus replace (M × D 2 δ , ξ) with (M × D 2 δ , ξ ) inside (M × D 2 ε , ξ) ⊂ (V, ξ)
; this gives a contact manifold (V, ξ ). Definition 4.4. [MNW13, Remark 9.6] We say that (V, ξ ) is obtained from (V, ξ) by a half Lutz-Mori twist along the contact submanifold (M, ξ + = ker (α + )) belonging to the Liouville pair (α + , α -).

We point out that performing a half Lutz-Mori twist makes the contact manifold overtwisted. Indeed, it is explained in Massot, Niederkrüger and Wendl [MNW13, Remark 9.6] that this half twist always gives a PS-overtwisted manifold, which then is also overtwisted according to [START_REF] Casals | Geometric criteria for overtwistedness[END_REF][START_REF] Huang | On plastikstufe, bordered Legendrian open book and overtwisted contact structures[END_REF].

Construction of Liouville pairs

We recall here the construction in [MNW13, Section 8], leaving the details that are not important for our purposes.

Consider the product manifold R m × R m+1 with the pair of contact structures ξ + , ξ - induced by the following pair of contact forms:

α ± := ±e t1+...+tm dθ 0 + e -t1 dθ 1 + . . . + e -tm dθ m ,
where we use coordinates (t 1 , . . . , t m ) on R m and (θ 0 , . . . , θ m ) on R m+1 . A direct computation shows that (α + , α -) is a Liouville pair on R m × R m+1 .

We now remark that there are two Lie groups acting explicitly on R m × R m+1 by strict contact transformations for both α + and α -. Indeed, the left action of the group R m+1 on R m × R m+1 given by the translations

(ϕ 0 , . . . , ϕ m ) • (t 1 , . . . , t m , θ 0 , . . . , θ m ) := (t 1 , . . . , t m , θ 0 + ϕ 0 , . . . , θ m + ϕ m ) (4.3)
and the left action of R m given by the law (τ 1 , . . . , τ m ) • (t 1 , . . . , t m , θ 0 , . . . , θ m ) := (t 1 + τ 1 , . . . , t m + τ m , e -τ1+...-τm θ 0 , e τ1 θ 1 , . . . , e τm θ m ) (4.4) are Lie group left-actions on R m × R m+1 and they both preserve the contact forms α + and α -. Moreover, these two actions allow us to produce a compact contact manifold from R m × R m+1 . Indeed, there are lattices Λ, Λ of R m and R m+1 respectively, such that the Λ-action on R m × R m+1 induced by the action of R m preserves R m × Λ . This implies that, by first taking the quotient of R m × R m+1 by Λ and then quotienting it by the (well defined by the above property) induced action of Λ, we obtain a compact manifold M .

Finally, this manifold M naturally inherits a Liouville pair, still denoted by (α + , α -), from the Liouville pair on the covering R m × R m+1 , because R m and R m+1 act on R m × R m+1 by strict contactomorphisms for both α + and α -.

We point out that this construction actually gives an infinite number of non homeomorphic manifolds M , hence an infinite number of non isomorphic Liouville pairs, in each odd dimension bigger or equal to 3. Indeed, the existence of the lattices Λ and Λ follows from number theory arguments and the manifold M obtained depends on the choice of a totally real field of real numbers k with finite dimension over Q. Now, for each dimension ≥ 2 over Q, there are infinitely such fields k and the corresponding manifolds are non homeomorphic. See [MNW13, Lemma 8.3] for the details. As far as Proposition I.D is concerned, this means that we have, in each odd dimension 2n + 1 ≥ 5, a contact structure satisfying the hypothesis of Theorem I.C on infinitely many different smooth manifolds T 2 × M 2n-1 ; in dimension 3, we obtain one contact structure on T 2 × M 1 = T 3 . In both cases, Theorem I.C then gives examples of contactomorphisms smoothly isotopic but not contact isotopic to the identity for the countably many contact structures (π * k ξ) k≥2 on each T 2 × M .

Topological effects of the half twists

Using the construction of the previous section, we obtain the following result:

Proposition 4.5. Let (V 2m+3 , ξ) be a contact manifold containing the (M, ξ + ) of Section 4.2.2 as a codimension 2 contact submanifold with trivial normal bundle. Then, if we denote by ξ the contact structure on V obtained by performing a half Lutz-Mori twist along the submanifold (M, ξ + ) (where we consider M with the orientation given by ξ + ), we have the following:

1. for all i = 2, . . . , m + 1, c i (ξ ) -c i (ξ) = 0 in H 2i (V ; Z); 2. c 1 (ξ ) -c 1 (ξ) = -2 PD (j * [M ]) in H 2 (V ; Z), where j : M → V is the inclusion, j * : H 2m+1 (M ; Z) → H 2m+1 (V ; Z)

EXAMPLES FROM LIOUVILLE PAIRS AND HALF LUTZ-MORI TWISTS

half twists. More precisely, we first perform a half twist along a submanifold (M, ξ + ) to obtain ξ ; this changes the core of the tube where we perform the twist from (M, ξ + ) to (M, ξ -). We then perform another half twist, this time along the new core (M, ξ -), to obtain ξ . Hence, applying Proposition 4.5 twice and using the fact that ξ -induces an orientation that is opposite to that induced by ξ + , we get that c i (ξ ) = c i (ξ ) = c i (ξ) for all i = 2, . . . , m + 1 and that c Chern classes are global invariants of complex vector bundles E over a manifold V . In order to prove Proposition 4.5 above, we then have the following problem: it's not clear how local modifications (i.e. over an open set U of V ) of the complex vector bundle E affect its Chern classes. The solution is hence either to use a relative version of Chern classes or to shift to another point of view more local in nature; we adopt here the second strategy. More precisely, following [START_REF] Aguilar | Characteristic classes and transversality[END_REF] we describe in Appendix A how each Chern class of E can be interpreted as the Poincaré dual of (a desingularised version of) the locus of points of V where a "generic" set of sections of E is not linearly independent. We point out that this generalizes the classical fact that the top Chern class of E is the Poincaré dual of the zero locus of a section of E which is transverse to the zero section (see [BT82, Property 20.10.6 and Proposition 12.8]).

1 (ξ ) = c 1 (ξ ) -2 PD (j * [-M ]) = c 1 (ξ) -2 PD (j * [M ]) - 2 PD (j * [-M ]) = c 1 (ξ),
In particular, we will use the following result, which will be proven in Appendix A, in order to prove Proposition 4.5 above: Proposition 4.7. Let E, E be two complex vector bundles of complex rang r over the same smooth oriented manifold V . Suppose also that there exist two open subsets O, U of V , with O compactly contained in U, such that the following are satisfied:

1. There is an isomorphism of vector bundles ψ : Then, we have the following:

E| O c → E | O c over O c := V \ O.

The vector bundles E|

1. c k (E ) = c k (E) in H 2k (V ; Z) for all 2 ≤ k ≤ r; 2. c 1 (E ) -c 1 (E) = PD([Z ]) -PD([Z]) in H 2 (V ; Z).
We are now ready to give a proof of Proposition 4.5:

CHAPTER 4. CONTACT MAPPING CLASSES ON OVERTWISTED MANIFOLDS

Proof (Proposition 4.5). Consider the manifold M and the Liouville pair (α + , α -) constructed in Section 4.2.2 and take a contact manifold (V, ξ) containing (M, ξ + = ker(α + )) as a contact submanifold of dimension 2. Denote by (V, ξ ) the result of a half Lutz-Mori twist on (V, ξ) along (M, ξ + ). According to Section 4.2.1, we have a tubular neighbourhood M × D 2 of M in which we can arrange to have contact forms α, α for ξ and ξ respectively which satisfy the following: if (r, ϕ) are the polar coordinates on D 2 and ψ, φ are as in Section 4.2.1, then

• α = 1+cos(ψ(r)) 2 α + + 1-cos(ψ(r)) 2 α -+ sin (ψ (r)) dϕ , • α = 1+cos(φ(r)) 2 α + + 1-cos(φ(r)) 2 α -+ sin (φ (r)) dϕ .
Now, we have explicit expressions for α + and α -on the cover R m × R m+1 of M , i.e. α ± := ±e t1+...+tm dθ 0 + e -t1 dθ 1 + . . . + e -tm dθ m . Thus, on the cover Let's also consider the following sections:

R m × R m+1 × D 2 of the tubular neighbourhood M × D 2 of M inside V ,
• s m+1 := r∂ r and r m+1 := cos (ψ (r)) ∂ ϕ -sin (ψ (r)) e -m i=1 ti ∂ θ0 for the pullback of ξ,

• s m+1 := r∂ r and r m+1 := cos (φ (r)) ∂ ϕ -sin (φ (r)) e -m i=1 ti ∂ θ0 for the pullback of ξ . These last two couples of sections are R-linearly independent whenever s m+1 and s m+1 are non-zero.

Moreover, for i = 1, . . . , m + 1, s i , r i , s i and r i are invariant under the left-action induced on R m × R m+1 × D 2 by the left-actions of of the Lie groups R m and R m+1 on R m × R m+1 described in Equations 4.3 and 4.4 of Section 4.2.2. Hence, they induce well defined sections of ξ and ξ on M × D 2 , which we will still denote using same notations. We also point out that each section coincide with its "primed version" near M ×D 2 .

We remark now that Span R (s m+1 (p), r m+1 (p)) and Span R s m+1 (p), r m+1 (p) , a priori well defined only for p ∈ M × D 2 \ {0} , actually extend smoothly also over M ×{0}. Indeed, consider the following couples of sections of ξ and ξ respectively:

• S := 1 r (cos (ϕ) s m+1 -sin (ϕ) r m+1 ), R := 1 r (sin (ϕ) s m+1 + cos (ϕ) r m+1 );

• S := 1 r cos (ϕ) s m+1 + sin (ϕ) r m+1 , R := 1 r -sin (ϕ) s m+1 + cos (ϕ) r m+1 .

EXAMPLES FROM LIOUVILLE PAIRS AND HALF LUTZ-MORI TWISTS

These sections, defined on M × D 2 \ {0} , can be smoothly extended to sections on all M × D 2 . For example, the section S can be rewritten near r = 0 as follows:

S = 1 r (cos (ϕ) s m+1 -sin (ϕ) r m+1 ) = cos ϕ ∂ r - sin ϕ cos r 2 r ∂ ϕ + sin ϕ sin r 2 r e -m i=1 ti ∂ θ0 = ∂ x + y 1 -cos r 2 r 2 (-y∂ x + x∂ y ) + y sin r 2 r 2 e -m i=1 ti ∂ θ0 ,
and each coefficient extends smoothly to all M × D 2 . Analogous computations show that also S , R, R extend smoothly to M × D 2 . We will denote these smooth extensions still by S, R, S , R . Moreover, we point out that (s 1 , r 1 , . . . , s m , r m , S, R) are everywhere R-linear independent sections of ξ, which is hence trivialized by them over M × D 2 ; the analogue is true for (s 1 , r 1 , . . . , s m , r m , S , R ). We remark that, unlike the couples (s m+1 , r m+1 ) and (s m+1 , r m+1 ), the (S, R) and (S , R ) do not coincide near the boundary of M × D 2 .

Computing the differentials of α and α thanks to the above explicit expressions (i) and (ii) for their pullbacks, we see that dα(s i , r i ) > 0 and dα (s i , r i ) > 0 for all i = 1, . . . , m and that dα(S, R) > 0 and dα (S , R ) > 0. Then, the identities J(s i ) := r i and J (s i ) := r i , for all i = 1, . . . , m, and the identities J(S) := R, J (S ) := R give two complex structures J, J on ξ and ξ over M × D 2 which are tamed by dα and dα . In particular, the sections s 1 , . . . , s m , S are C-linearly independent on ξ and the sections s 1 , . . . , s m , S are C-linearly independent on ξ .

We point out that J, J satisfy also the identities J(s m+1 ) = r m+1 and J (s m+1 ) = r m+1 . This shows in particular that J and J coincide over a neighbourhood of the boundary of M × D 2 : indeed, each section coincide with its primed version near the boundary of M × D 2 and the span of (s i , r i ) m+1 i=1 and (s i , r i ) m+1 i=1 are respectively ξ and ξ on M × D 2 \ {0} .

We can now extend J and J to complex structures on ξ and ξ over all V , tamed by contact forms that extend α and α , in such a way that they coincide outside M × D 2 . We denote such extensions still with J and J .

We now claim that we are in the hypothesis of Proposition 4.7 if we choose as open set O an arbitrary open set compactly contained in U := M × D 2 and containing the support of the half Lutz-Mori twist.

Indeed, if F, F are the complex span of (s 1 , . . . , s m ), (s 1 , . . . , s m ) and L, L are the complex lines determined by S, S , then the Hypothesis 1, 2a and 2b are trivially satisfied because ξ and ξ coincide outside O and because of the choice of s 1 , . . . , s m and s 1 , . . . , s m .

Let's show that the Hypothesis 2c and 2d are also satisfied in our case. We claim that s m+1 : M × D 2 → ξ and s m+1 : M × D 2 → ξ intersect transversely F ⊂ ξ and F ⊂ ξ in M × {0} and -M × {0} (i.e. M × {0} but with opposite orientation).

Indeed, using the complex trivialization (s 1 , . . . , s m , S) for ξ on U = M × D 2 , we can write s m+1 : U → ξ = U × C m+1 as s m+1 (q) = (q, v 1 (q), . . . , v m+1 (q)), with v i : U → C. More precisely, recalling that JS = R, that S = 1 r (cos (ϕ) s m+1 -sin (ϕ) r m+1 ) and that R = 1 r (sin (ϕ) s m+1 + cos (ϕ) r m+1 ), for each q = (m, x, y) ∈ U = M × D 2 , with m ∈ M , we actually have that v i (q) = 0 for all i = 1, . . . , m and that v m+1 (q) = x + iy, where (x, y) ∈ D 2 are the Cartesian coordinates. In particular,

d (m,0) v m+1 (∂ x ) = ∂ x and d (m,0) v m+1 (∂ y ) = ∂ y , i.e. d (m,0) v m+1 | {0m}⊕T0D 2 : {0 m } ⊕ T 0 D 2 → T 0 C CHAPTER 4. CONTACT MAPPING CLASSES ON OVERTWISTED MANIFOLDS
is an orientation preserving isomorphism of vector spaces. In other words, s m+1 : M × D 2 → ξ intersects transversely F ⊂ ξ in M × {0}, considered as an oriented manifold.

An analogous computation with s m+1 shows that we can write s m+1 : U → ξ = U × C m+1 as s m+1 (q) = q, 0, . . . , 0, v m+1 (q) , with v m+1 (q) = x -iy for each q = (m, x, y) ∈ U = M × D 2 . This gives in particular that d (m,0) v m+1 | {0m}⊕T0D 2 : {0 m } ⊕ T 0 D 2 → T 0 C is an orientation reversing isomorphism of vector spaces hence that s m+1 : M × D 2 → ξ intersect transversely F ⊂ ξ along the oriented submanifold -M × {0}. At this point, Proposition 4.5 follows from Proposition 4.7.

Explicit examples of non-trivial contact mapping classes

We now prove Proposition I.D, of which we recall the statement: Proposition I.D. Soit (M 2n-1 , α + , α -) une des paires de Liouville construites (en nombre infini) dans [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF]. On considère la structure de contact

η = ker 1 + cos (s) 2 α + + 1 -cos (s) 2 α -+ sin (s) dt sur V := T 2 (s,t) × M (ici, la notation T 2 (s,t)
dénote le choix de coordonnées (s, t) sur T 2 ) et soit ξ la structure de contact vrillée obtenue par η via un demi twist de Lutz-Mori le long de {(0, 0)} × M , comme défini dans [START_REF] Massot | Weak and strong fillability of higher dimensional contact manifolds[END_REF].

Alors, c 1 (ξ) ∈ H 2 (V ; Z) est toroidale et, pour tout entier k ≥ 2, on a c 1 (π * k ξ) = k • c 1 (ξ) mod H 2 ator (V ; Z), où π k : T 2 (s,t) × M → T 2 (s,t) × M est donné par π k (s, t, q) = (ks, t, q).
Proof (Proposition I.D). The contact structure η on the manifold T 2 (s,t) × M admits a trivialization as complex vector bundle given by the following sections and choice of J:

1. s i := ∂ ti for i = 1, . . . , m, 2. J(s i ) := e -m j=1 tj cos (s) ∂ θ0 -e ti ∂ θi + sin (s) ∂ t , for i = 1, . . . , m,
where we use locally on M the coordinates (t 1 , . . . , t m , θ 0 , . . . , θ m ) given by the construction in Section 4.2.2. In particular, all the Chern classes of ξ are zero. Hence, applying Proposition 4.5 to the couple (ξ, η) we get the following: if we denote by j : M → T 2 (s,t) × M the inclusion j(p) = (0, 0, p) and by j * :

H 2m+1 (M ; Z) → H 2m+1 (T 2 × M ; Z) the induced map in homology, then c 1 (ξ) = -2 PD (j * [M ]) in H 2 (T 2 × M ; Z).
We now prove that c 1 (ξ) is toroidal. Fix a p ∈ M and consider f :

T 2 → T 2 × M given by f (θ, ϕ) = (θ, ϕ, p), for every (θ, ϕ) ∈ T 2 . Because f is transverse to j(M ), we have f * PD T 2 ×M (j * [M ]) = PD T 2 f -1 (j(M ))
; here, the notation PD X means that we are considering the Poincaré duality on the compact manifold X. Now,

PD T 2 f -1 (j(M )) = PD T 2 ([{(0, 0)}]) generates H 2 (T 2 ; Z) Z; in other words, PD (j * [M ])
is toroidal, and the same is true for c 1 (ξ).

The only thing left to show is that c 1 (π * k ξ) = kc 1 (ξ) mod H 2 ator (V ; Z) for each k ≥ 2. Because η is a trivial complex vector bundle over T 2 × M , the same is true for each π * k η; in particular, each π * k η has trivial Chern classes. Notice that π * k ξ can also be seen as obtained from π * k η by performing a half Lutz-Mori twist along each of the k submanifolds 2lπ m , 0 ×M , where l = 0, . . . , m-1. Then, Proposition 4.5 tells that c 1 (π * k ξ) = kc 1 (ξ); in particular, c 1 (π * k ξ) = kc 1 (ξ) mod H 2 ator (V ; Z) too.

OTHER EXAMPLES OF NON-TRIVIAL CONTACT MAPPING CLASSES

Other examples of non-trivial contact mapping classes

In this section, we show how to obtain examples of (S 1 × W, ξ) as in the hypothesis of Theorem I.C using the existence of adapted open book decompositions due to Giroux [START_REF] Giroux | Géométrie de contact: de la dimension trois vers les dimensions supérieures[END_REF] and the h-principle of Borman, Eliashberg and Murphy [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF].

Examples from adapted open books

We recall the statement of Proposition I.E:

Proposition I.E. Soit (V := S 1 × W 2n , η) une variété de contact vrillée telle que c 1 (η) est toroidale et telle que, pour tout k ≥ 2, la tirée en arrière ξ via le revêtement à k-

feuilles π k : S 1 s × W → S 1 s × W , donné par π k (s, p) = (ks, p), satisfait c 1 (π * k η) = k • c 1 (η) mod H 2
ator (V ; Z). Alors, on a que : i. La première classe de Chern de chaque structure de contact ξ sur V × T 2 obtenue via la construction de Bourgeois [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] sur (V, η) satisfait aussi les conditions ci-dessus, par rapport au revêtement

µ k := (π k , Id) : V × T 2 → V × T 2 .
ii. Soit ν : V × Σ g → V × T 2 induite par un recouvrement Σ g → T 2 qui est ramifié au dessus de deux points (ici, Σ g est une surface fermée de genre g ≥ 2). Alors, la première classe de Chern de chaque revêtement ramifié ξ g de ξ sur V × Σ g satisfait aussi les conditions ci-dessus, par rapport au revêtement µ g k := (π k , Id) : V × Σ g → V × Σ g . De plus, si η est vrillée et g est assez grand, ξ g est aussi vrillé.

In order to give a proof, we need the following lemma which describes the effect of Bourgeois construction [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] and of its branched coverings at the level of almost contact structures (recall Definition 2.10) as well as a sufficient condition for overtwistedness in the case of branched covers: Lemma 4.8. Let (V 2n-1 , η) be a contact manifold, (B, ϕ) an open book decomposition supporting η and α a contact form defining η and adapted to the open book. Then, we have the following:

1. The Bourgeois construction [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] on (V, η) and (B, ϕ, α) gives a contact structure ξ on V × T 2 which is homotopic, as an almost contact structure, to the pair η ⊕ T T 2 , dα ⊕ ω T , where ω T is a volume form on T 2 .

2. Any contact branched covering ξ g of ξ via a branched covering ν : V ×Σ g → V ×T 2 , induced by a covering Σ g → T 2 branched over two points, is homotopic, as an almost contact structure, to (η ⊕ T Σ g , dα ⊕ ω g ), where ω g is a volume form on Σ g .

3. Suppose η is overtwisted. Then, if g is large enough, ξ g is overtwisted too.

Notice that point 1 above has already been pointed out by Lisi, Marinković and Niederkrüger [LMN18, Remark 2.1]. We now prove, in the order, Proposition I.E and Lemma 4.8:

Proof (Proposition I.E). Denote also by p : V × T 2 → V , p g : V × Σ g → V and p g : V × Σ g → Σ g the natural projections. Points 1 and 2 of Lemma 4.8 imply that c 1 (ξ) = p * c 1 (η) and c 1 (ξ g ) = p * g c 1 (η)+p * g c 1 (T Σ g ). Recall now that every continuous map g : T 2 → Σ g has degree 0 (here, we use g ≥ 2); in particular, for each f :

T 2 → V × Σ g , f * p * g c 1 (T Σ g ) = (p g • f ) * c 1 (T Σ g ) = 0 in H 2 (T 2 ; Z), i.e. p * g c 1 (T Σ g ) is atoroidal. We then have that c 1 (ξ) = p * c 1 (η) mod H 2 ator (V × T 2 ; Z) and c 1 (ξ g ) = p * g c 1 (η) mod H 2 ator (V × Σ g ; Z).
In order to show that c 1 (ξ) and c 1 (ξ g ) are toroidal, it's then enough to show that this is true for both p * c 1 (η) and p * g c 1 (η). Because c 1 (η) is toroidal, there is f :

T 2 → V with f * c 1 (η) = 0. Let then h : T 2 → V × T 2 , h(q) = (f (q), * ), and h g : T 2 → V × Σ g , h g (q) = (f (q), * ). As p • h = f and p g • h g = f , we have h * (p * c 1 (η)) = (p • h) * c 1 (η) = f * c 1 (η) = 0 in H 2 (T 2 ; Z) and, similarly, h * g (p * g c 1 (η)) = f * c 1 (η) = 0, i.e. p * c 1 (η) and p * g c 1 (η) are both toroidal. The equalities c 1 (µ * k ξ) = kc 1 (ξ) mod H 2 ator (V × T 2 ; Z) and c 1 ((µ g k ) * ξ g ) = kc 1 (ξ g ) mod H 2 ator (V × Σ g ; Z) follow directly from π k • p = p • µ k , π k • p g = p g • µ g k and c 1 (π * k η) = kc 1 (η) mod H 2
ator (V ; Z). Lastly, if η is overtwisted, point 3 of Lemma 4.8 gives the overtwistedness of ξ g for g large enough, thus concluding the proof.

Proof (Lemma 4.8). We start by proving point 1. The Bourgeois construction [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] on (V, η) and (B, ϕ, α) gives a function Φ = (f, g) : V → R 2 defining the open book (B, ϕ) and such that ξ on V ×T 2 (x,y) is defined by β := α+f dx-gdy. Then, an explicit homotopy of almost contact structures from (ξ, dβ| ξ ) to (η ⊕ T T 2 , dα| η + dx ∧ dy) is given by the [0, 1] t -family of hyperplane fields ξ t given by the kernel of α+(1 -t) (f dx -gdy), together with the symplectic structures given by the restriction of dα+(1-t) [df ∧ dx -dg ∧ dy]+ tdx ∧ dy to ξ t .

As far as point 2 is concerned, as explained in [START_REF] Geiges | Constructions of contact manifolds[END_REF], an explicit contact branched covering ξ g on V × Σ g is given by the kernel of a differential 1-form ν * β + h(r)r 2 dθ; here, (r, θ) are radial coordinates on the D 2 -factor of a neighborhood D 2 × {p, q} of the branching locus {p, q} of the branched covering Σ g → T 2 , > 0 is very small and h = h(r) is a smooth function with support in D 2 × {p, q}, equal to 1 on the branching locus and strictly decreasing in r. As contact branched coverings are unique up to isotopy (see Section 5.2), it's enough to prove that this specific η g is homotopic to the wanted almost contact structure. Now, an explicit computation (analogous to the one in Section 8.5) shows that the wanted homotopy is given by the [0, 1] t -family of hyperplane fields ξ t g defined as the kernel of ν * α + (1 -t) ν * (f dx -gdy) + hr 2 dθ , together with the symplectic structures given by the restriction of ν * dα + (1 -t) ν * (df ∧ dx -dg ∧ dy) + d hr 2 ∧ dθ + tω g to ξ t g . Point 3 will be proven in Section 9.1.2; more precisely, it essentially follows from the following three facts. Firstly, the contact branched covering ξ g can be chosen (up to isotopy) in such a way that it induces on each fiber of V × Σ g → Σ g the original overtwisted contact structure η. Secondly, Niederkrüger and Presas [NP10, page 724] describe how the "size" of a contact neighborhood of each connected component (V, ξ) of the branching set of V × Σ g → V × T 2 is diverging to +∞ as the index g of the branched covering is going to +∞; see also Lemma 9.9. Then, according to Casals, Murphy and Presas [CMP15, Theorem 3.1], topologically trivial contact neighborhoods of overtwisted manifolds in codimension 2 are themselves overtwisted, provided they are sufficiently "large". This concludes the proof of Lemma 4.8.

Examples from the h-principle

We recall the statement of Proposition I.F: Proposition I.F. On considère une variété lisse W de dimension 2n qui est presque complexe, spin et satisfait H 1 (W ; Z) = {0}. Alors, il y a une structure de contact vrillée ξ sur

V := S 1 × W telle que c 1 (ξ) ∈ H 2 (V ; Z) est toroidale et c 1 (π * k ξ) = k • c 1 (ξ) mod H 2 ator (V ; Z), où π k : S 1 s × W → S 1 s × W est le revêtement à k-feuilles π k (s, p) = (ks, p).

OTHER EXAMPLES OF NON-TRIVIAL CONTACT MAPPING CLASSES

The proof is structured as follows. We start from a natural almost contact structure η 0 on V := S 1 × W and we modify it to an almost contact structure η with first Chern class c 1 (η) satisfying the wanted conditions. Then, the h-principle from [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF] tells that η can be deformed to an overtwisted contact structure ξ on V ; the first Chern class of such a ξ will then satisfy the wanted properties too.

Before entering in the details of the proof of Proposition I.F, we state a lemma from algebraic topology, whose proof is postponed: Lemma 4.9. Let (V 2n+1 , η 0 ) be an almost contact manifold. For each class u ∈ H 2 (V ; Z), there is an almost contact structure η u on V such that c 1 (η u ) = c 1 (η 0 ) + 2u.

Proof (Proposition I.F). The hyperplane field η 0 = {0} ⊕ T W on V = S 1 × W is a (coorientable) almost contact structure thanks to the almost complex structure J W on W . Moreover, its first Chern class c 1 (η 0 ) is equal to π * W c 1 (W ), where π W : S 1 × W → W is the projection on the second factor. The hypothesis that W is spin means that the 2nd Stiefel Whitney class

w 2 (W ) ∈ H 2 (W ; Z 2 ) of W is trivial. Because w 2 (W ) is the reduction modulo 2 of c 1 (W ), there is λ ∈ H 2 (W ; Z) such that c 1 (W ) = 2λ. Hence, c 1 (η 0 ) = π * W c 1 (W ) = 2π * W λ.
Consider then a non-trivial c ∈ H 1 (W ; Z), that exists by hypothesis, and let v be a generator of H 1 (S 1 ; Z). Using Kunneth's decomposition theorem, we can see H 1 (S 1 ; Z)⊗ H 1 (W ; Z) as a submodule of H 2 (S 1 × W ; Z). An application of Lemma 4.9 with u = v ⊗ c -π * W λ then gives an almost contact structure η on V with c 1 (η) = 2v ⊗ c. Notice that the map π * k , induced on H 2 (S 1 × W ; Z) by π k , acts as multiplication by k on the submodule H 1 (S 1 ; Z) ⊗ H 1 (W ; Z) of H 2 (S 1 × W ; Z). In particular, the fact that c 1 (η) = 2v ⊗ c implies that c 1 (π * k η) = kc 1 (η) mod H 2 ator (V ; Z). We now claim that c 1 (η) is toroidal. Indeed, according to the universal coefficient theorem and the Hurewicz theorem, H 1 (W ; Z) Hom Z (H 1 (W ; Z); Z) Hom Z (π 1 (W ); Z). In particular, there is γ :

S 1 → W such that γ * c = 0 ∈ H 1 (S 1 ; Z). If we define f = (Id, γ) : T 2 = S 1 × S 1 → S 1 × W , we then have f * c 1 (η) = 2v ⊗ γ * c = 0 in H 1 (S 1 ; Z) ⊗ H 1 (S 1 ; Z) ⊂ H 2 (
T 2 ; Z), i.e. c 1 (η) is toroidal, as wanted.

The h-principle from Borman, Eliashberg and Murphy [START_REF] Strom Borman | Existence and classification of overtwisted contact structures in all dimensions[END_REF] then gives the wanted contact structure ξ as deformation of η.

We now give a proof of the lemma used above:

Proof (Lemma 4.9). Bowden, Crowley and Stipsicz [BCS14, Lemma 2.17.(1)] states that if V is a closed connected manifold of dimension 2n + 1 and ζ is a stable almost complex structure on it, then there is an almost contact structure η on V whose stabilization gives ζ. Recall that a stable almost complex structure on V is the stable isomorphism class of a complex structure on T V ⊕ ε k V , where ε V is the trivial real vector bundle of dimension 1 over V , and the stabilization of η is the stable isomorphism class of the complex structure induced by η on T V ⊕ ε V . In particular, in order to prove Lemma 4.9, it's enough to find a stable almost complex structure

ζ u such that c 1 (ζ u ) = c 1 (η 0 ) + 2u.
The existence of such a ζ u follows, for instance, from [Gei08, Remark 8.1.4], of which we recall here the idea. There is a bijective correspondence, given by the first Chern class, between isomorphism classes of complex line bundles over V and cohomology classes in H 2 (V ; Z). Let then L u be the complex line bundle over V satisfying c 1 (L u ) = u. Consider a direct complement of the dual L * u of L u , i.e. a complex vector bundle E u over V such that there are m ∈ N >0 and an isomorphism ν : L * u ⊕ C E u (ε C V ) m of complex vector bundles over V ; here, ε C V denotes the complexification of ε V . We then claim that the complex vector bundle F u := η 0 ⊕ L u ⊕ E u can be used to define the wanted stable complex structure. The fact that L * u ⊕ C E u is a trivial complex vector bundle implies in particular that

c 1 (E u ) = -c 1 (L * u ) = u; hence, c 1 (F u ) = c 1 (η) + u + u = c 1 (η) + 2u. Now, because L *
u and L u are isomorphic as real vector bundles, ν induces an isomorphism of real vector bundles ν : L u ⊕ R E u ε 2m

V . Moreover, the choice of a vector field X on V transverse to η 0 gives an isomorphism of real vector bundles Ψ : η 0 ⊕ V T V . We then have an isomorphism θ of real vector bundles over V given by the composition

F u = η 0 ⊕ L u ⊕ E u Id ⊕ν η 0 ⊕ ε 2m V = R (η 0 ⊕ ε V ) ⊕ ε 2m-1 V Ψ⊕Id T V ⊕ ε 2m-1 V .
In particular, the pushforward θ * J of the complex structure J on F u via θ gives the wanted stable almost complex structure ζ u on V .

Appendix A

Chern classes as Poincaré duals

Chern classes are global invariants of complex vector bundles E over a manifold V . More precisely, following [START_REF] Aguilar | Characteristic classes and transversality[END_REF] we describe in this appendix how each Chern class of E can be geometrically interpreted as the Poincaré dual of (almost) the locus of points of V where a "generic" set of sections of E is not linearly independent. Consider a complex vector bundle E of complex rank r over an oriented smooth manifold V . Given k sections s 1 , . . . , s k of E, take the homomorphism of vector bundles

h : V × C k → E defined by h(p, u 1 , . . . , u k ) = k j=1 u j s j (p), where V × C k is the trivial complex vector bundle of rank k over V . If τ : Hom C V × C k , E → V ,
is the complex vector bundle over V with fiber over p ∈ V the vector space Hom C C k , E p of C-linear maps from C k to E p , we can reinterpret the map h as a section s h of τ given by s h (p)(w) := h(p, w) for all w ∈ C k . Take now the complex vector bundle π :

Hom C V × C k , E × CP k-1 → V × CP k-1 defined by π(f, d) = (τ (f ), d), for every f ∈ Hom C V × C k , E and d ∈ CP k-1 ,

and consider the section σ

h : V × CP k-1 → Hom C V × C k , E × CP k-1 given by σ h := (s h , Id CP k-1 ). If φ : V × CP k-1 → V and φ : Hom C V × C k , E × CP k-1 → Hom C V × C k , E
are the projections on the first factor, we then have the following commutative diagram:

Hom C V × C k , E × CP k-1 Hom C V × C k , E V × CP k-1 V φ π τ φ σ h s h
Now, in the total space Hom C V × C k , E × CP k-1 of the bundle π we can consider the blown-up non-injectivity locus, i.e. the subset

Σ := (f, d) ∈ Hom C V × C k , E × CP k-1 | d ⊂ ker f .
The adjective blown-up comes from the fact that Σ is a version of the non-injectivity locus

S := f ∈ Hom C V × C k , E | ker (f ) = {0}
where we keep track of the complex lines in the kernel.

Proposition A.1. [ACMFA07, Proposition 4, Proposition 6] Σ is a smooth oriented submanifold of Hom C V × C k , E × CP k-1 , of codimension 2r .
As we will need it in the following, we give a sketch of proof:

Proof (sketch). Let pr :

Hom C V × C k , E × CP k-1 → CP k-1
be the projection on the second factor and γ the tautological line bundle over CP k-1 ; denote then

1 := pr * γ = { (f, d, v) ∈ Hom C V × C k , E × CP k-1 × C k | v ∈ d } .
If φ : V × CP k-1 → V is the projection on the first factor, denote also by 2 the vector bundle π * φ * E over Hom C V × C k , E × CP k-1 , where π :

Hom C V × C k , E × CP k-1 → V × CP k-1 is as above.
Consider then the vector bundle Π :

Hom C ( 1 , 2 ) → Hom C V × C k , E × CP k-1
and take the section Ψ :

Hom C V × C k , E × CP k-1 → Hom C ( 1 , 2 ) of Π defined by Ψ(f, d) = f | d . 1 = pr * γ 2 = π * φ * E Hom C V × C k , E × CP k-1 Hom C ( 1 , 2 ) V × CP k-1 E V π Ψ Π φ σ h
It can be shown that Ψ is transverse to the zero section 0 Π of Π. In particular,

Σ = Ψ -1 (0 Π ) is a smooth submanifold of Hom C V × C k , E × CP k-1 .
Finally, Σ is oriented according to Convention A.2 below, thanks to the fact that Hom C ( 1 , 2 ), Hom C V × C k , E × CP k-1 and 0 Π are naturally oriented: indeed, the first two are complex vector bundles over an oriented base and the third is a section of a vector bundle over an oriented base.

Convention A.2. Let X, Y, Z be oriented manifolds and consider f : X → Y transverse to Z ⊂ Y at p ∈ X . Take a basis (v 1 , . . . , v l ) of T p f -1 (Z), complete it to a positive basis (v 1 , . . . , v l , u 1 , . . . , u n ) of T p X and consider a positive basis (w 1 , . . . , w m ) of T f (p) Z . Then, (v 1 , . . . , v l ) is positive if and only if (w 1 , . . . , w m , d p f (u 1 ) , . . . , d p f (u n )) is a positive basis of T f (p) Y .

Define now the set

Z (h) := σ -1 h (Σ) = (p, d) ∈ V × CP k-1 | d ⊂ ker (h p ) ,
where h p : C k → E p is the C-linear map defined by h p (.) := h(p, .).

Proposition A.3. [ACMFA07, Proposition 5] For a generic choice of vector bundles map h :

V × C k → E , the section σ h : V × CP k-1 → Hom C V × C k , E × CP k-1 is transverse to Σ ⊂ Hom C V × C k , E × CP k-1 . In particular, Z (h) is a closed oriented submanifold of V × CP k-1 of codimension 2r . Theorem A.4. [ACMFA07, Theorem 11] If the section σ h : V ×CP k-1 → Hom C V × C k , E × CP k-1 is transverse to Σ ⊂ Hom C V × C k , E ×CP k-1 , then the Chern class c r-k+1 (E) is equal to the Poincaré dual of φ * [Z (h)],
where φ : V × CP k-1 → V is the projection on the first factor.

Remark A.5. The statement of the theorem shows the advantage of using the blown-up version Σ of S instead of the non-injectivity locus itself. Indeed, while Z (h) is a smooth oriented submanifold for a generic choice of h, hence is has a well defined fundamental class that can be pushed in H * (V ; Z) via φ * , the set s -1 h (S) is only a Whitney stratified submanifold of V (hence not necessarily smooth) for a generic choice of h, and in particular there is no natural way to associate an homology class to s -1 h (S). We point out though that in the complex analytic setting it is possible to construct a cohomology class directly from s -1 h (S) using the theory of currents: this is done, for example, in Griffiths and Harris [GH78, Section 3.3].

As it will be useful later, we remark that there is also a relative version of Proposition A.3. Indeed, we have the following relative transversality result: if M and N are smooth manifolds and f : M → N is a smooth map transverse to a submanifold Z ⊂ N on a neighbourhood of a closed subset C ⊂ M , then f can be C ∞ -perturbed to a map f : M → N everywhere transverse to Z and such that f

| C = f | C .
This can be proven, for example, by introducing a little modification in the proof of Lee [Lee13, Theorem 6.36], where it is shown that, for k ∈ N big enough, there is a parametric family of functions F : M × R k → N that is everywhere transverse to Z and such that F (., 0) = f (.). More precisely, the F appearing in the proof of [Lee13, Theorem 6.36] should be defined in our case as F (p, s) := r (f (p) + χ (p) • e (p) • s), where χ : M → R ≥0 has support C ; the wanted perturbation will then be F s := F (., s) for an s given by the parametric transversality theorem (see Lee [Lee13, Theorem 6.35]). In our setting, using this relative transversality result in the proof of [ACMFA07, Proposition 5] (restated above as Proposition A.3) we can achieve transversality between the map σ h and the submanifold

Σ of Hom C V × C k , E × CP k-1 by C ∞ -perturbing h relative to a closed subset C ⊂ V × CP k-1 near which σ h is already transverse to Σ.
We are then ready to prove Proposition 4.7, whose statement we now recall: Proposition 4.7. Let E, E be two complex vector bundles of complex rang r over the same smooth oriented manifold V . Suppose also that there exist two open subsets O, U of V , with O compactly contained in U , such that the following are satisfied:

1. There is an isomorphism of vector bundles ψ :

E| O c → E | O c over O c := V \ O .
2. The vector bundles E| U and E | U over U admit complex sub-bundles F, L ⊂ E and F , L ⊂ E such that E| U = F ⊕ L and E | U = F ⊕ L and satisfying the following conditions:

(a) ψ • s j = s j over U \ O for all j = 1, . . . , r -1;

(b) L, L have complex rank 1, while F, F have complex rank r -1 and are trivialized by two ordered sets of everywhere C-linearly independent sections (s 1 , . . . , s r-1 ) and s 1 , . . . , s r-1 of E| U and E | U ;

(c) there are two additional sections s r , s r of E| U and E | U respectively, with image contained in L and L and such that s r : U → E| U intersects transversely F and s r : U → E | U intersects transversely F (here F and F are seen here as submanifolds of E| U and E | U );

(d) Z := s -1 r (F ) and Z := (s r ) -1 (F ), which are oriented smooth manifolds of U by Hypothesis 2c above, are actually compactly contained in O .

Then, we have the following:

1. c k (E ) = c k (E) in H 2k (V ; Z) for all 2 ≤ k ≤ r ; 2. c 1 (E ) -c 1 (E) = PD([Z ]) -PD([Z]) in H 2 (V ; Z).
We deduce the above result from Theorem A.4 using the following:

Lemma A.6. Let E be a complex vector bundle of complex rang r over a smooth oriented manifold V with empty boundary. Let s 1 , • • • , s r-1 be C-linearly independent sections of E and denote by F the vector sub-bundle of E generated by them, i.e. the vector bundle with fiber F p = Span C (s 1 (p), • • • , s r-1 (p)) ⊂ E p over a point p ∈ V . Let also L be a complex line sub-bundle of E such that E = F ⊕ L and assume that s r : V → E is an additional section with image contained in L and intersecting transversely F (seen as a submanifold of E ); denote by M the oriented (by Convention A.2) submanifold s -1 r (F ). Then, if h : V × C r → E is defined by h(p, u) = r i=1 u i s i (p) and σ h is obtained from h as above, we have that:

1. σ h : V × CP r-1 → Hom C (V × C r , E) × CP r-1 is transverse to the blown-up non- injectivity locus Σ ⊂ Hom C (V × C r , E) × CP r-1
and, in particular, Z (h) := σ -1 h (Σ) is smooth and naturally oriented; 2. the projection on the first factor φ : V ×CP r → V induces an orientation preserving diffeomorphism φ : Z (h) → M .

Proof (proposition 4.7): Consider another open set V of V , compactly contained in U and containing the closure of O . Take the two complex vector bundle homomorphisms h

V : V × C r → E| V and h V : V × C r → E | V defined by h V (p, u 1 , . . . , u r ) := r i=1 u i s i (p) and h V (p, u 1 , . . . , u r ) := r i=1 u i s i (p)
for all p ∈ V , (u 1 , . . . , u r ) ∈ C r and i = 1, . . . , r . Extend then h V and h V to two vector bundle homomorphisms h : V × C r → E and h : V × C r → E in such a way that ψ (h (p, u)) = h (p, u) for all p ∈ O c , u ∈ C r and i = 1, . . . , r . Such extensions exist because ψ • s i = s i on U \ O for all i = 1, . . . , r by Hypothesis 2a. Given an integer j between 1 and r included, denote respectively by h j and h j the restrictions of h and h to the sub-bundle V × C j of V × C r , where C j is the vector subspace of C r given by the points (u 1 , . . . , u r ) ∈ C r such that u j+1 = . . . = u r = 0. Now, σ hj and σ h j are transverse to the blown-up non-injectivity locus Σ near the closed set O , for all j = 1, . . . , r : indeed, this follows directly from Hypothesis 2b for the case j = 1, . . . , r -1 and from Hypothesis 2c and Lemma A.6 for the case j = r (remark that in Lemma A.6 we do not make compactness assumptions, so we can chose V as base manifold V in the statement of the lemma). Then, using the relative version of the genericity of the transversality condition, we can perturb h j , h j to g j , g j in such a way that g j = h j , g j = h j over O and that σ gj 's and σ g j 's are everywhere transverse to Σ. Moreover, because ψ (h j (p, .)) = h j (p, .) for p ∈ O c , we can also arrange that ψ (g j (p, .)) = g j (p, .) for p ∈ O c : indeed, we can use the same perturbation for h j and h j over O c because they coincide there. Lastly, if we choose the perturbation C 0small, we can arrange to have the submanifolds Z (g j ) and Z g j compactly contained in O × CP j-1 . Now, by construction of the g j 's and the g j 's, if we write

Z (g j ) = Z O (g j ) ∪ Z O c (g j ) and Z g j = Z O g j ∪ Z O c g j , where Z O (g j ) , Z O g j ⊂ O × CP j-1 and Z O c (g j ) , Z O c g j ⊂ O c × CP j-1 , we have that Z O c (g j ) = Z O c g
j for all j = 1, . . . , r and Z O (g j ) = Z O g j = ∅ for j = 1, . . . , r -1. Moreover, if pr j V : V × CP j-1 → V is the projection on the first factor for all j = 1, . . . , r , by Lemma A.6 φ r := pr r V | Z(gr) and φ r := pr r V | Z(g r ) induce orientation preserving diffeomorphisms between Z O (g r ) and Z and between Z O (g r ) and Z respectively.

Moreover, we remark that if

(p, d) ∈ U × CP r-1 is such that σ h (p, d) ∈ Σ then d = [0 : • • • : 0 : 1].
We can hence further restrict to the coordinate chart C r-1 = [z 1 : . . . : z r-1 : 1] ∈ CP r-1 ⊂ CP r-1 containing the point [0 : • • • : 0 : 1] and consider σ h as a map

σ h : U × C r-1 → U × M r (C) × C r-1
(q, z) → (q, M (q) , z)

where z = (z 1 , . . . , z r-1 ) ∈ C r-1 . Now, in order to study the transversality of σ h with respect to Σ, we have to come back at the construction of Σ as preimage of a transverse intersection. We then consider the vector bundles 1 , 2 and the sections Ψ, 0 Π of Π as in the (sketch of the) proof of Proposition A.1 and read them in the given trivialization of E over U and in the chart C r-1 ⊂ CP r-1 . We are then in the following situation:

• 1 , which is globally the product of Hom C (V × C r , E) and the tautological line bundle γ over CP r-1 , becomes the trivial line vector bundle U ×M r (C)×C r-1 ×C over U × M r (C) × C r-1 , and the projection map is just the projection on the first three factors: indeed, γ admits over the coordinate chart C r-1 ⊂ CP r-1 the trivialization C r-1 × C ∼ -→ γ given by (z 1 , . . . , z r-1 , λ) → ([z 1 : . . . : z r-1 : 1] , λz), where z := (z 1 , . . . , z r-1 , 1);

• 2 , defined globally as π * φ * E , becomes the trivial vector bundle

U × M r (C) × C r-1 × C r over U × M r (C) × C r-1
, again via the projection on the first 3 factors;

• the projection Π :

Hom C ( 1 , 2 ) → Hom C (V × C r , E) × CP k-1 becomes locally Π : U × M r (C) × C r-1 × M r,1 (C) → U × M r (C) × C r-1 , (q, A, z, B) → (q, A, z) ;
• the zero section 0 Π of Π is locally the image of the inclusion

U × M r (C) × C r-1 → U × M r (C) × C r-1 × M r,1 (C) (q, A, z) → (q, A, z, 0)
• the section Ψ of Π can be rewritten locally as

Ψ : U × M r (C) × C r-1 → U × M r (C) × C r-1 × M r,1 (
C) and is given by Ψ (q, A, z) = (q, A, z, A • z), where again z := (z 1 , . . . , z r-1 , 1) ∈ C r , if z = (z 1 , . . . , z r-1 ).

Then, using the expression in Equation A.1 for the matrix M (q), we get

Ψ • σ h : U × C r-1 → U × M r (C) × C r-1 × M r,1 (C) (q, z) → q, I r-1 0 0 v(q) , z, z v(q)
Now, because Ψ is transverse to 0 Π and Σ coincides with the oriented preimage Ψ -1 (0 Π ) (see the sketch of proof of Proposition A.1), we actually have that σ h is transverse to Σ at (q, 0) ∈ U × C r-1 (recall that if σ h (q, z) is in Σ then z = 0) if and only if Ψ • σ h is transverse to 0 Π at (q, 0). Moreover, if we have transversality at every point,

Z (h) = σ -1 h (Σ) equals (Ψ • σ h ) -1 (0 Π ) as oriented manifolds.
If we denote by µ :

U × M r (C) × C r-1 × M r,1 (C) → M r,1 (
C) the projection on the last factor, we also get that Ψ • σ h is transverse to 0 Π at (q, 0) ∈ U × C r-1 if and only if µ • Ψ • σ h is transverse to {0} ⊂ M r,1 (C) and that, in case of transversality at every

Part II

Revisiting some known constructions of contact manifolds

Chapter 5

Contact branched coverings and fiber sums

The goal of this chapter is to give definitions of contact branched coverings and contact fiber sums that allow us to naturally obtain uniqueness statements. We will in particular prove Proposition II.A stated in the introduction. We point out that the proofs in this section are mainly a reformulation of those in [START_REF] Geiges | Constructions of contact manifolds[END_REF].

The smooth case

Branched covers

Let W m , W m be smooth manifolds with (possibly empty) boundary.

Definition 5.1. A map π : W → W is a branched covering if each point p ∈ W admits a neighborhood U such that for each connected component U of π -1 (U ) we have the following commutative diagram

D 2 × Ω U D 2 × Ω U ψ π k π | U ψ
where Ω is either R m-2 or R m-3 × [0, +∞) (depending on whether p ∈ W is respectively in the interior or on the boundary of W ), D 2 is the disk of radius 1 and center 0 in R 2 , ψ and ψ are diffeomorphisms, with ψ(0, 0) = p, and π k is the map (z, q) → (z k , q) for all z ∈ D 2 and all q ∈ Ω.

If we denote p = ψ(0, 0) ∈ U , then k is called the branching index at the point p. Remark that it is well-defined, i.e. it does not depend on any choice beside p. Indeed, π| U induces a (unbranched) covering map U \ ψ ({0} × Ω) → U \ ψ ({0} × Ω) and k is its degree, which is independent of the choice of the trivializations.

We denote X m-2 , and call it upstairs branching set the codimension 2 submanifold of W made of the points p ∈ W such that the branching index is > 1, i.e. such that the associated local form π k has a k > 1. We also denote X, and call it downstairs branching set, the codimension 2 submanifold of W given by the image of X via π.

The following properties follow directly from the definition:

THE SMOOTH CASE

(a) X and X are proper submanifolds of W and W respectively, i.e. ∂ X ⊂ ∂ W , ∂X ⊂ ∂W and X ∂ W , X ∂W ;

(b) π induces regular coverings π| W \ X : W \ X → W \ X and π| X : X → X;

(c) if we denote V m-1 := ∂W and V m-1 = ∂ W , then π := π |∂ W is a branched covering π : V → V with branching set M := ∂X ⊂ V ; we then denote M the set of points in V where π has associated local index k > 1.

Fiber sums

Let V, M be two oriented smooth manifolds of dimensions dim V = m + 1 and dim M = m -1, with M connected. We point out that we do not suppose V connected here. Let also j 1 : M → V and j 2 : M → V be two disjoint orientation-preserving embeddings such that there is a fiber-orientation-reversing isomorphism Φ over M between their normal bundles:

N 1 := j * 1 T V (j 1 ) * T M N 2 := j * 2 T V (j 2 ) * T M M Φ ∼
For notational convenience, denote by j : M → V the disjoint union j 1 j 2 .

Definition 5.2. A fiber sum of V along j 1 , j 2 via Φ is the data (W, H, ϕ) of a smooth oriented manifold W m+1 , a cooriented hypersurface H in W and an orientation-preserving diffeomorphism ϕ : V \ j(M ) → W \ H for which there exist (oriented) tubular neighborhoods ρ 1 : N 1 → V of j 1 (M ), ρ 2 : N 2 → V of j 2 (M ) and ρ : N → V of H, where N := T W / T H is the normal bundle of H in W , satisfying the following conditions:

i. ρ 1 and ρ 2 have disjoint images;

ii. for i = 1, 2 and p ∈ M = 0 M ⊂ N i , the differential d p ρ i : T p N i → T ji(p) V , which can be seen as a map d p ρ i :

T p M ⊕ (N i ) p → T ji(p) V if we naturally split T p N i as T p 0 M ⊕ (N i ) p , is such that the composition T ji(p) V pr -→ j * i T ji(p) V (j i ) * T p M = (N i ) p dpρi| (N i ) p -------→ T ji(p) V is exactly Id : T ji(p) V → T ji(p)
V (here, pr is just the natural projection);

iii. if N * 1 , N * 2 and N * denote respectively the bundles N 1 , N 2 and N deprived of their zero sections, ϕ induces a diffeomorphism between the image of ρ 1 ρ 2 in V \ j(M ) and the image of ρ in W \ H, and the composition

f : N * 1 N * 2 ρ1 ρ2 ----→ V \ j(M ) ϕ -→ Im (ϕ • (ρ 1 ρ 2 )) ρ -1 ---→ N *
is a diffeomorphism which is R >0 -equivariant for the natural actions by multiplication of R >0 on N * 1 , N * 2 and N * , and sends N * 1 on the positive part of N * (H is a cooriented hypersurface, hence N is divided by the zero section in two connected components, one positive and one negative thanks to the coorientation); CHAPTER 5. CONTACT BRANCHED COVERINGS AND FIBER SUMS iv. if σ : N → N is the involution given by σ(v) = -v for each v ∈ N , the composition

g : N * 1 f | N * 1 ---→ N σ -→ N f -1 --→ N * 2 coincides with Φ| N * 1 Remark 5.3.
It is a direct consequence of the above definition that if (W, H, ϕ) is a fiber sum of V along j 1 , j 2 via Φ, then we have two diffeomorphisms f 1 and f 2 such that the following diagram is commutative:

N * 1 R >0 H N * 2 R >0 φ f1 f2
where the quotients N * 1 R >0 and N * 2 R >0 are taken with respect to the natural action by multiplication by a positive real on N * 1 and N * 2 respectively and φ is the diffeomorphism induced by Φ.

The following result shows that the above definition is non-empty: Proposition 5.4. Fiber sums of V along j 1 , j 2 via Φ exist.

Proof. If we fix an auxiliary Riemannian metric on V , Φ induces an isomorphism

Φ S : S N 1 → S N 2 v → Φ (v) Φ (v)
(5.1) between the two sub-bundles of N 1 and N 2 made of the vectors of norm 1.

For i = 1, 2, consider an embedding ρ i : N i → V which identifies the normal bundle N i with a regular tubular neighborhood of j i (M ) inside V , in such a way that their images are disjoint. Now, there is a natural isomorphism N * i → S N i × (0, +∞) of oriented bundles over M , for i = 1, 2. We hence obtain embeddings τ i : S N i × (0, +∞) → V thanks to ρ i , for i = 1, 2.

Consider then the set R * := R \ {0} and the embedding

S N 1 × R * Ψ → V \ (j 1 (M ) ∪ j 2 (M )) (v, t) → τ 1 (t • v) for t > 0 τ 2 (-t • Φ S (v)) for t < 0 (5.2)
Let W be the smooth manifold obtained as the quotient of the disjoint union

(V \ j (M )) S N 1 × R by the relation ∼ Ψ defined as follows: p ∼ Ψ (v, t) if p ∈ V \ j (M ), (v, t) ∈ S N 1 × R * and p = Ψ (v, t). Define also by H the hypersurface S N 1 × {0} ⊂ S N 1 × R inside W
and by ϕ the diffeomorhism V \ j(M ) → W \ H given by the natural identification W \ H = V \ j(M ). Then, (W, H, ϕ) is a fiber sum of V along j(M ) via Φ.

THE SMOOTH CASE

The notion of fiber sum in Definition 5.2 also satisfies the following uniqueness property:

Proposition 5.5. If (W, H, ϕ) and (W , H, ϕ) are two fiber sums of V along j 1 , j 2 via Φ, then there is a diffeomorphism Θ : W → W such that:

i. Θ| H : H ∼ -→ H is a coorientation preserving diffeomorphism; ii. θ := ϕ -1 • Θ • ϕ : V \ j(M ) ∼ -→ V \ j(M ) smoothly extends to θ : V ∼ -→ V that
is isotopic to the identity, via an isotopy fixing j(M ) pointwise.

Proposition 5.5 is an immediate consequence of the following: Lemma 5.6. If (W, H, ϕ) and (W , H, ϕ) are two fiber sums of V along j 1 , j 2 via Φ, then there is Θ : W ∼ -→ W , with Θ(H) = H, and an isotopy F t : V ∼ -→ V starting at F 0 = Id V , fixing j(M ) pointwise, and such that the following diagram is commutative:

V \ j(M ) W \ H V \ j(M ) W \ H ϕ F1 Θ ϕ
Proof (Lemma 5.6). Denote by ρ 1 , ρ 2 , ρ, N and by ρ 1 , ρ 2 , ρ, N the embeddings and normal bundle given by Definition 5.2 for (W, H, ϕ) and (W , H, ϕ).

According to the uniqueness theorem for tubular neighborhoods (see for instance [Lan99, Theorem 6.2]), there are vector bundle automorphisms ν 1 , ν 2 , of respectively N 1 , N 2 , and an isotopy F t : V → V starting at F 0 = Id V , fixing j(M ) pointwise and such that

ρ 1 = F 1 • ρ 1 • ν 1 and ρ 2 = F 1 • ρ 2 • ν 2 .
In other words, the following diagram commutes:

N 1 N 2 V N 1 N 2 V ν1 ν2 ρ1 ρ2 F1 ρ 1 ρ 2
Notice moreover that, by condition ii. in Definition 5.2, we can chose F 1 so that ν 1 and ν 2 are actually the identity isomorphisms of N 1 and N 2 respectively. Consider now the composition

Θ : W \ H ϕ -1 --→ V \ j(M ) F1 -→ V \ j(M ) ϕ -→ W \ H .
At this point, it's enough to show that such a Θ extends on all W to a diffeomorphism Θ : W → W as in the statement.

Let X := Im (ϕ • (ρ 1 ρ 2 )), X := Im (ϕ • (ρ 1 ρ 2 )) and µ := ρ -1 • Θ | X • ρ.
We then have the following commutative diagram:

N * 1 N * 2 V \ j (M ) X N * N * 1 N * 2 V \ j (M ) X N * Id ρ1 ρ2 ϕ F1 Θ | X ρ -1 µ ρ 1 ρ 2 ϕ ρ -1
Now, by Definition 5.2, the upper and the lower rows give two R >0 -equivariant diffeomorphisms. Hence, µ is R >0 -equivariant too. Because N and N are vector spaces of real dimension 1 over H and H, this means that µ actually extends to a vector bundle isomorphism µ : N → N .

By the commutativity of the above diagram, this implies that Θ can be extended to a diffeomorphism Θ : W ∼ -→ W , as wanted.

Branched coverings of contact manifolds

Suppose π : V 2n+1 → V 2n+1 is a branched covering map of manifolds without boundary, branched along the codimension 2 submanifold M 2n-1 ⊂ V . We denote, like in the previous section, M 2n-1 the locus of points of V with branching index > 1. Be careful, though, that here V , V , M , M play, respectively, the roles of W , W , X, X of Section 5.1, and they all have no boundary.

In this section we also suppose that we are dealing with (V, η) and (M, ξ) contact manifolds, where ξ is the sub-bundle η ∩ T M of T V | M .

The pullback π * η is a well defined hyperplane field on V , because if we fix a contact form α for η then π * α, which defines π * η, is nowhere vanishing. Though, it is not a contact form, because at each point p of M we have π * (α ∧ dα n ) | p = 0. We point out that, nonetheless, the restriction of π * η to M is a honest contact structure on M . We then want to show that π * η gives a "natural" way to construct contact structures on V .

We start by considering a more general setting. Let Y 2n+1 be a smooth manifold, Z 2n-1 a codimension-2 submanifold and η an hyperplane field on Y . Definition 5.7. We say that η is adjusted to Z if it is a contact structure on the complement of Z and η ∩ T Z is a contact structure on Z. If that's the case, we also call contactization of η a contact structure ξ such that there is a smooth path {η s } s∈[0,1] of hyperplane fields, all adjusted to Z, which starts at η 0 = η, ends at η 1 = ξ and such that η s is a contact structure for all s ∈ (0, 1].

We then have the following general result: Proposition 5.8. Let η be an hyperplane field on Y adjusted to Z. Contactizations of η exist and are all isotopic.

We recall from [ET98, Section 1.1.6] that a confoliation on a manifold V is an hyperplane field ζ = ker α on V that admits a complex structure J : ζ → ζ tamed by dα| ζ , i.e. such that dα(X, JX) ≥ 0 for all vector fields X tangent to ζ. In our situation, if η is an hyperplane field on Y adjusted to a codimension 2 submanifold Z, then η is a confoliation. Indeed, we have the following: Fact 5.9. Let (η n ) n∈N be a sequence of contact structures on a manifold Y 2n+1 which C 1 -converges to a hyperplane field η on Y . Then, η = ker α admits a complex structure J tamed by dα| η .

From now on, we will hence talk directly about confoliations adjusted to a certain codimension 2 submanifold.

Proof (Fact 5.9). A first idea could be to take, for each n ∈ N, a complex structure J n on η n = ker α n tamed by dα n | ηn (which exists because η n is a contact structure) and to define J as "the limit" of the sequence (J n ) n∈N . However, such a limit does not necessarily exist for any choice of J n . The solution is to assure the orthogonality of each of the J n with respect to an auxiliary riemannian metric g using the polar decomposition of matrices, as follows.

Let ξ = ker β be a contact structure on Y and fix an auxiliary riemannian metric g on Y . Then, by the contact condition, dβ induces an isomorphism dβ : ξ → ξ * , where ξ * is the vector bundle dual of ξ over M . Now, g also induces an isomorphism g : ξ * → ξ. We can hence consider A : ξ → ξ given by A := g • dβ; in other words, dβ(X, Y ) = g(AX, Y )

BRANCHED COVERINGS OF CONTACT MANIFOLDS

for each couple of sections (X, Y ) of ξ. Consider then the (left) polar decomposition RJ of A, i.e. the unique couple of vector bundle-isomorphisms R, J : ξ → ξ with R positive definite and J orthogonal with respect to g. More explicitly, R is the positive definite square root of the symmetric and positive definite AA * , where A * is the g-dual of A, and J is the g-orthogonal endomorphism of ξ given by R -1 A. The positive definiteness of R implies that J is dβ| ξ -tamed. Moreover, by definition, A is skew-symmetric, i.e. A * = -A. This and the fact that J is g-orthogonal imply that J 2 = -Id, i.e. that J is a complex structure on ξ.

This being said, let's go back to the setting of Fact 5.9. Once fixed a sequence of 1-forms (α n ) n∈N with α n defining η n and that C 1 -converges to a 1-form α defining η, consider an auxiliary riemannian metric g on M . Then, according to what we just said, we can define a sequence (J n ) n∈N such that, for all n ∈ N, J n : η n → η n is a complex structure on η n and is orthogonal with respect to g. Then, after obtaining from each J n an orthogonal B n : T M → T M that sends the vector of norm 1 positively orthogonal to η n to itself, by compactness of the space of orthogonal linear maps T M → T M , we get that there is a sequence of naturals (n k ) k∈N diverging to +∞ such that B n k converges (in C 1 -topology) to a limit B : T M → T M . Now, we recall that η n C 1 --→ η, that J 2 n = -Id | ηn and that dα n (X, JX) > 0 for each X section of η n . Hence, B restricts to a well defined J : η → η such that J 2 = -Id | η and dα(X, JX) ≥ 0 for all X section of η, as wanted.

Proposition 5.8 is a consequence of the following lemma, which deals with the more general situation of any number of parameters: Lemma 5.10. Given K a compact set and (η k ) k∈K a smooth K-family of confoliations on V adjusted to M , there is a smooth family of confoliations

(η s k ) s∈[0,1], k∈K such that (η s k ) s∈[0,1] is contactization of η k , for each k ∈ K. Moreover, if η k is contact for all k in a closed subset H ⊂ K, then η s
k can be chosen so that η s k = η k for all k ∈ H and s ∈ [0, 1].

In the above statement, by a smooth K-family of hyperplane fields we mean the following. Let X be a closed smooth manifold and K ⊂ X a compact subset. Then, we say that (η k ) k∈K is a smooth K-family if there is an open set U ⊂ X containing K and a differential form α ∈ Ω 1 (V × U ) such, that for all k ∈ K, η k = ker (i * k α), where

i k : V → V × U is the inclusion i k (p) = (p, k).
Proof (Proposition 5.8). The existence of contactizations follows directly from Lemma 5.10 with K a point. Given two contactizations ξ, ξ of η, we have by definition two associated paths of adjusted confoliations η t , η t , with t ∈ [0, 1], such that η 0 = η 0 = η, η 1 = ξ, η 1 = ξ and η t , η t contact for t ∈ (0, 1]. Then, the path

t → η t := η 1-2t if t ∈ [0, 1 /2] η 2t-1 if t ∈ [ 1 /2, 1] (5.3)
is a continuous path of adjusted confoliations from η 0 = ξ to η 1 = ξ . Moreover, up to perturbing it smoothly at t = 1 /2, we can suppose that η t is smooth in t. Then, applying Lemma 5.10 to η t , with K = [0, 1] and H = {0, 1}, we get a family ( η s t ) s∈[0,1], t∈[0,1] of adjusted confoliations such that η s 0 = ξ, η s 1 = ξ for all s ∈ [0, 1] and such that η s t is contact for s > 0. The subfamily η 1 t is then a path of contact structures from ξ to ξ , and it can be turned into an isotopy by Gray's theorem.

CHAPTER 5. CONTACT BRANCHED COVERINGS AND FIBER SUMS

Proof (Lemma 5.10). This proof follows almost step by step the construction and the computations made in [Gei97b, Section 2].

Because the contact condition is an open condition in the space of 1-forms (with the C 1 -topology), there is an open subset U of K which contains H and such that ξ k is contact for all k ∈ U . We consider then a smooth cut-off function ρ : K → [0, 1], equal to 0 on H and equal to 1 on the complement of U .

Take now an auxiliary Riemannian metric on V and consider the circle bundle S (N M ) given by the vectors of norm 1 in the normal bundle N M of M inside V . Let γ be a connection form on S (N M ), i.e. a nowhere vanishing 1-form defining an hyperplane field which is transversal to the fibers of the fibration S (N M ) → M . Then, thanks to the natural retraction R 2 \ {0} → S 1 , γ can also be seen as a 1-form on N M \ M . Moreover, the form r 2 γ, where r is the radial coordinate in N M \ M , smoothly extends over the zero section M to all N M .

We consider then a non-increasing cut-off smooth function g = g(r) which is 1 near r = 0 and vanishes for r > 1 and we identify N M with a neighborhood of M inside V . If α k is a smooth K-family of 1-forms defining ξ k , set

α s k := α k + s ρ(k)g(r)r 2 γ .
Here is a positive real constant which will be chosen very small later. Suppose, without loss of generality, that ≤ 1. Remark that ξ s k := α s k is a well defined hyperplane field. Moreover, it is adjusted to M , for all values of s, k.

We now want to show that, for an small enough, ξ s k is actually a contact structure on V for all s > 0, k ∈ K.

We can compute

α s k ∧ (dα s k ) n = α k ∧ (dα k ) n + + ns [rg (r) + 2g (r)] ρ (k) α k ∧ (dα k ) n-1 ∧ rdr ∧ γ + + s r 2 g (r) ρ (k) h vol
where vol is the Riemannian volume form on V and h is a function of

p ∈ V , k ∈ K, s ∈ [0, 1], ∈ R >0 and is polynomial in . Define now P k , Q k : V → R by the identities α k ∧(dα k ) n = P k vol and n [rg (r) + 2g (r)] α k ∧ (dα k ) n-1 ∧ rdr ∧ γ = Q k vol. Also, define R k ( ) := r 2 g (r) h( , k). Then, α s k ∧ (dα s k ) n = {P k + s ρ (k) [Q k + R k ( )]} • vol . Now, Q k > 0 and R k ( ) = 0 along M , for all k ∈ K and ∈ [0, 1] (remark we allow here = 0). Hence, by compactness of M and [0, 1], there is an open neighborhood O of M inside V such that Q k + R k ( ) > 0 on O for all ∈ [0, 1].
P k is independent of , s and is non-negative everywhere on V for all k. Moreover, P k is strictly positive on the complement of O for all k ∈ K, and even on all

V if k ∈ U ⊂ K (remember ξ k is contact if k ∈ U ). Then, P k + s ρ (k) [Q k + R k ( )] > 0 on O, for all k ∈ K and all ∈ (0, 1].
Lastly, for very small, P k dominates s ρ(k

) [Q k + R k ( )]
wherever it is strictly positive, because the latter is bounded above in norm (recall we are working with ≤ 1). Hence, by compactness of V \ O,

P k + s ρ (k) [Q k + R k ( )]
is also positive on the complement of O for all k ∈ K, for > 0 small enough.

FIBER SUMS OF CONTACT MANIFOLDS

Coming back to the specific case of branched coverings, the hyperplane field π * η on V is adjusted to M (hence is in particular a confoliation).

We can thus redefine branched coverings in contact topology as follows:

Definition 5.11. We say that a contact structure on V is a contact branched covering of η if it is a contactization of π * η and it is invariant under all the diffeomorphisms of V lifting the identity of V .

We point out that, by definition of contactization, if η is a contact branched covering of η, the upstairs branching locus M is naturally a contact submanifold in ( V , η). Then, Proposition 5.8 easily implies the following: Proposition 5.12. Let V → V be a smooth branched covering and η a contact structure on V . Then, contact branched coverings of η on V exist and are all isotopic (among contact branched coverings).

We point out that, in order to deduce this result from Proposition 5.8, the contactization in the statement Proposition 5.8 has to be invariant under deck transformations of π, as requested in Definition 5.11, and the isotopy has to be among invariant contactizations. From the explicit formula in the proof of Lemma 5.10 above, it's clear that both these conditions can be easily arranged.

Let's now recall and prove Proposition II.A, stated in the introduction:

Proposition II.A. Soit (V 2n-1 , η) une variété de contact et π : V → V un revêtement ramifié avec lieu de ramification (en bas) M . On suppose que η ∩ T M est une structure de contact sur M . Alors :

1. il y a une famille, indexée par [0, 1], de distributions d'hyperplans η t sur V telle que η 0 = π * η et η t est une structure de contact pour tout t ∈ (0, 1];

2. si η t et η t sont comme dans le point 1, alors η r est isotope à η s pour tout r, s ∈ (0, 1].

De plus, dans le point 1, η t peut être choisie invariant par les automorphismes (locales) du revêtement π, pour tout t ∈ (0, 1]; de façon analogue, l'isotopie du point 2 peut être choisie parmi les structures de contact invariant par les automorphismes (locales) de revêtement, si η t et η t le sont aussi.

Proof (Proposition II.A). This is a simple consequence of Gray's theorem and the fact that contact branched coverings exist and are unique up to isotopy. Indeed, the [0, 1]families of hyperplane fields in points 1 and 2 in the statement are automatically adjusted to the upstairs branching locus for small parameters t ≥ 0.

Fiber sums of contact manifolds

Let's start here with some general definitions and results. Let V 2n+1 be a smooth manifold and (M 2n-1 , ξ) be a contact manifold. Consider now an oriented circle bundle π : S → M and suppose that there is a smooth embedding S → V .

Definition 5.13. A smooth hyperplane field η on V is adjusted to S if η is a contact structure away from S and η ∩ T S = π * ξ as bundles over S. If that's the case, we call contactization of η a smooth path of hyperplane fields (η s ) s∈[0,1] such that η 0 = η and such that η s is a contact structure for all s ∈ (0, 1].

FIBER SUMS OF CONTACT MANIFOLDS

Definition 5.16. We call confoliated fiber sum of (V, η) along the contact embeddings j 1 , j 2 via Φ the data (W, H, ϕ, ζ) of a smooth fiber sum (W, H, ϕ) of V along j 1 , j 2 via Φ and a smooth hyperplane field ζ on W that coincides with ϕ * η on W \ H and is adjusted to H.

Here, according to Remark 5.3, H can naturally be seen as the total space of the circle bundle S N 1 → M . Hence, ζ is adjusted to H is in the sense of Definition 5.13. In particular, η is a confoliation on V , which explains the nomenclature "confoliated" fiber sum.

Remark 5.17. If such a confoliation ζ exists, it is uniquely determined by ϕ * η, because

H is a hypersurface in W , hence W \ H is dense in W .
The following result guarantees that the above definition is non-empty: Proposition 5.18. Confoliated fiber sums of (V, η) along contact embeddings j 1 , j 2 via Φ exist.

Proof (Proposition 5.18). Let α be a contact form defining ξ on M . Denote also by γ 2 a connection form on the bundle S N 2 and γ 1 the connection -Φ * S γ 2 on S N 1 , where Φ S is as in Equation (5.1).

Lemma 5.19. There are > 0 and embeddings ρ 1 : N 1 → V and ρ 2 : N 2 → V with disjoint images and such that ρ * 1 (η) = ker π * 1 α + arctan r 2 γ 1 and ρ * 2 (η) = ker π * 2 α + arctan r 2 γ 2 , where π i : N i → M is the projection of the vector bundle N i onto its base space M . Proof (Lemma 5.19). This follows directly from the standard neighborhood theorem for contact submanifolds [Gei08, Theorem 2.5.15], together with the fact that a pair of codimension 2 embeddings of the same contact submanifold have isomorphic conformal symplectic normal bundles if and only if they have isomorphic smooth normal bundles.

We now call τ i : S N i × (0, +∞) → V the embedding induced by ρ i , for i = 1, 2, and we define and embedding Ψ as follows:

S N 1 × R * Ψ → V \ j (M ) (v, t) →    τ 1 tan (t 3 ) • v for t > 0 τ 2 tan (-t 3 ) • Φ S (v) for t < 0 (5.4)
We point out that Ψ is an embedding that preserves the orientation (recall N 1 is oriented, hence S N 1 too).

Using Lemma 5.19 and the fact that

Φ * S γ 2 = -γ 1 , a direct computation gives that Ψ * η = ker π * 1 α + t 3 γ 1 on S N 1 × R * . Now, the 1-form γ := π *
1 α + t 3 γ 1 on the domain of Ψ naturally extends to a smooth differential form on S N 1 × R, which we will still denote by γ. Moreover, this extension is clearly positively contact away from the submanifold S N 1 × {0} and defines a smooth confoliation on all S N 1 × R * .

Then, denote by W the smooth manifold obtained as the quotient of the disjoint union

(V \ j (M )) (S N 1 × R) by the relation ∼ Ψ defined as follows: p ∼ Ψ (v, t) if p ∈ V \ j (M ), (v, t) ∈ S N 1 × R * and p = Ψ (v, t).
Let also H be the hypersurface S N 1 × {0} ⊂ S N 1 × R inside W , and ϕ the diffeomorhism V \ j(M ) → W \ H given by the identification W \ H = V \ j(M ).

WEAK FILLINGS OF CONTACT BRANCHED COVERINGS

U 1 U 2 N 1 N 2 V U 1 U 2 N 1 N 2 V Id H0=i R0=ρ1 ρ2 F1 H1 R1=ρ 1 ρ 2
Now, consider the composition

Θ : W \ H ϕ -1 --→ V \ j(M ) F1 -→ V \ j(M ) ϕ -→ W \ H .
It's enough to show that such a Θ extends on all W to a diffeomorphism Θ : W → W as in the statement.

We have the following commutative diagram:

U * 1 U * 2 N * 1 N * 2 V \ j (M ) X ρ -1 (X) U * 1 U * 2 N * 1 N * 2 V \ j (M ) X ρ -1 X Id H0=i ρ1 ρ2 ϕ F1| V \j(M ) Θ | X ρ -1 µ H1 ρ 1 ρ 2 ϕ ρ -1
where

X := Im (ϕ • (ρ 1 ρ 2 ) • i), X := Im (ϕ • (ρ 1 ρ 2 ) • H 1 ) and µ := ρ -1 • Θ | X • ρ is defined only on the subset ρ -1 (X) of N * and has values in ρ -1 X ⊂ N * . Now, by Definition 5.2, the compositions ρ -1 • ϕ • (ρ 1 ρ 2 ) and ρ -1 • ϕ • (ρ 1 ρ 2 ) are R >0 -equivariant maps.
Moreover, we can see from the explicit proof of [Gei08, Theorem 2.6.12], that for i ∈ {1, 2} and all

p ∈ M ⊂ U i , d p H 1 : T p (N i ) → T p (N i ) is of the form Id TpM ⊕L p : T p M ⊕ N p → T p M ⊕ N p ,
where L : N → N is an isomorphism of vector bundles over M . Here, we used the natural splitting

T p (N i ) = T p M ⊕ N p for p ∈ M ⊂ N i .
By commutativity of the above diagrams, this implies that µ smoothly extends over H, in such a way that µ(H) = H.

Hence, Θ extends to a diffeomorphism Θ : W → W such that Θ (H) = H, which concludes the proof of Lemma 5.21.

Once showed that we have a natural (and unique, in the sense of Proposition 5.20) confoliation ζ adjusted to the hypersurface H on smooth fiber sum (W, H, ϕ), we can now define the contact fiber sum in terms of deformations of this confoliation: Definition 5.22. A contact fiber sum on the confoliated fiber sum (W, H, ϕ, ζ) is a contactization of ζ.

Analogously to Section 5.2, the existence and uniqueness up to isotopy of the contact fiber sum is a direct consequence of Proposition 5.14.

Weak fillings of contact branched coverings

Let now π : W 2n+2 → W 2n+2 be a branched covering of even dimensional manifolds with non-empty boundaries V 2n+1 = ∂ W and V 2n+1 = ∂W . We also use the same notations as in Section 5.1. In particular, we denote by X 2n the upstairs branching set, by X the downstairs branch set, by M, M the boundaries of X, X respectively and by π the restriction π| V : V → V .

Here's a more detailed version of Théorème II.B from Chapter 1: CHAPTER 5. CONTACT BRANCHED COVERINGS AND FIBER SUMS Theorem 5.23. Suppose we are in the following situation:

(a) η is a contact structure on V and ξ := η ∩ T M is contact on M ;

(b) η on V is a contact branched covering of (V, η);

(c) ω on W weakly dominates η on V ;

(d) X is a symplectic submanifold of (W, ω) and it weakly fills (M, ξ).

Then, W admits a symplectic form ω that weakly dominates η on V .

Notice that, because π |

M : M → M is a (unbranched) covering map, ξ := π | M * ξ = η ∩ T M is a contact structure on M .
Proof. Consider the normal bundle of X inside W and see it as a neighborhood U of X.

Similarly for a neighborhood M of M in V . In particular, we have a norm function on U and M, and we can denote by U r , M r the set of vectors of norm less than r.

Fix now an arbitrary smooth function f : W → R ≥0 , compactly supported in U 1 , depending only on r, non-increasing in it, and equal to 1 on a neighborhood of X. We also denote by g its restriction to V = ∂ W . We point out that in particular f (r) = 0, hence g (r) = 0, for r = 0.

Let now δ be a connection 1-form on the circle bundle S U given by the vectors of norm 1 in U. Denote also by γ the restriction of δ to the sub-bundle S M given by the vectors of norm 1 in M; then, γ is in particular a connection form on S M.

The explicit formula in the proof of Lemma 5.10 then shows that, up to isotopy, we can assume the contact branched covering η to be the kernel of α := π * α + g(r)r 2 γ, for each smaller than or equal to a certain constant 0 > 0.

As far as the symplectic structure on W is concerned, consider the closed 2-form ω := π * ω + d f (r)r 2 δ on W , where > 0.

Lemma 5.24. There is 1 > 0 such that ω is symplectic for all 0 < < 1 .

Proof. We compute

ω = π * ω + (2f + rf ) rdr ∧ δ + f r 2 dδ ,
which gives, once fixed a volume form vol on W ,

ω n+1 = π * ω + (2f + rf ) rdr ∧ δ + f r 2 dδ n+1 = π * ω n+1 + (n + 1) (2f + rf ) π * ω n ∧ rdr ∧ δ + r 2 f hvol ,
where h is a smooth function depending on p ∈ W and on > 0. Using the facts that π * ω is symplectic away from X and that the restriction of ω to X is symplectic on X, we can now conclude, with arguments analogous to those in the proof of Lemma 5.10, that ω n+1 > 0 for small enough.

We then want to show that ω weakly dominates η = ker( α ), provided that > 0 is small enough (and in particular such that < := min ( 0 , 1 )). By (the discussion following) Definition 2.17, we need to check that α ∧ ( ω ,V + τ d α )

n > 0 , ∀τ ≥ 0 , CHAPTER 5. CONTACT BRANCHED COVERINGS AND FIBER SUMS Lemma 5.28. Consider a smooth manifold S and a continuous function p : S×R ≥0 → R such that, for each s ∈ S, p s : R ≥0 → R defined by p s (τ ) := p(s, τ ) is polynomial in τ . Suppose there is s 0 ∈ S and a neighborhood U of s 0 such that for all s ∈ U the followings are satisfied:

1. deg τ (p s0 ) ≥ deg τ (p s );
2. the leading coefficient of p s0 is positive.

Then, there is a neighborhood O of s 0 contained in U such that, for all s ∈ O, the minimum m s of p s exists and, moreover, depends continuously on s.

In particular, if moreover m s0 > 0, then m s > 0 for s sufficiently near to s 0 .

Proof (Lemma 5.28). Hypothesis 1 tells us that p s is of fixed degree n := deg τ (p s0 ) for all s ∈ U . In particular, the leading coefficient a s n of p s converges to the positive leading coefficient a s0 n of p s0 for s → s 0 . Hence, there is a neighborhood O ⊂ U of s 0 such that, for all s ∈ O, a s n > a s 0 n /2 > 0. Then, for s ∈ O, p s (τ ) → +∞ for τ → +∞. Hence, by the continuity of p s in τ , p s admits a global minimum m s for s ∈ O.

Notice that, up to shrinking O, we can suppose moreover that it has compact closure.

For the dependence of the global minimum m s on the variable s, we recall the following fact: if X, Y are topological spaces with Y compact, f : X × Y → R is continuous and f x (.) := f (x, .) : Y → R admits global minimum m x for all x ∈ X, then m is continuous in x ∈ X. Indeed, for each sequence x n in X converging to x, there is a subsequence

x n k such that m(x n k ) → m(x), because, if y n ∈ Y denotes a sequence such that m(x n ) = f (x n , y n ), by compactness of Y there is a subsequence y n k converging to a certain y ∈ Y , hence m(x n k ) = f (x n k , y n k ) C 0 --→ f (x, y), by continuity of f , and f (x, y) is exactly the global minimum m(x) of f x .
Going back to the case of Lemma 5.28, in order to apply this discussion we need to show that, for each s ∈ O, the τ s realizing the identity p(s, τ s ) = m s can be chosen in a compact subset K of R ≥0 which is independent of s ∈ O.

To prove this, we write p s (τ ) = a s n τ n + i<n a s i τ i and we compute, for s ∈ O,

p s (τ ) (a) ≥ a s n τ n -M n-1 i=1 τ i (b) ≥ a s0 n 2 τ n -M n-1 i=1 τ i (c) ≥ 2a s0 0 (d) ≥ a s 0 ,
where

M := max s∈O,i<n (|a s i |) (recall that O is compact).
Here, (a) comes from the triangular inequality and the fact that a s n > 0 for s ∈ O, (b) comes from the fact that a s n > a s 0 n /2 for s ∈ O, (c) is true for all τ ∈ [τ , +∞), for a certain τ > 0 depending only on s 0 , and (d) is true up to shrinking O, because a s 0 converges to a s0 0 for s → s 0 . The above inequality tells that p s (τ ) > a s 0 = p s (0) for all s ∈ O and τ ∈ [τ , +∞). In other words, a τ s realizing m s = p s (τ s ) has to be in the compact subset K := [0, τ ], for all s ∈ O. This is exactly what we needed to prove in order to show that m s is continuous in s ∈ O, according to the above discussion.

The last part of the statement is now obvious.

Proof (Lemma 5.26). We would like to use Lemma 5.28, with S := V × [0, ) and P := P 1 + P 2 : S × R ≥0 → R, i.e. P q, (τ ) is given by [P 1 (τ ) + P 2 (τ, )] (q) for (q, ) ∈ S = Chapter 6

Open books and contact vector fields

We describe here a reinterpretation of adapted open book decompositions in terms of contact vector fields:

Théorème II.E. Sur une variété de contact (M 2n-1 , ξ), chaque paire de champs de vecteurs de contact X, Y , telle que [X, Y ] est partout transverse à ξ, donne un livre ouvert explicite de M qui porte ξ. Vice versa, un livre ouvert qui porte ξ donne une paire X, Y comme ci dessus.

A part of this result (i.e. how to go from adapted open books to contact vector fields) has been stated by Giroux during the Yashafest in June 2007 and the AIM workshop in May 2012 (see [Gir12, Claim on page 19]).

The whole chapter is devoted to the proof of Théorème II.E, already stated in Section 1.2. More precisely, in Section 6.1 we describe how to recover the data of an open book decomposition adapted to a certain contact structure ξ from the data of two contact vector fields with Lie bracket everywhere transverse to ξ. In Section 6.2, we show that it is also possible to recover such a couple of contact vector fields from an adapted open book, as claimed in [START_REF] Giroux | The existence problem, contact geometry in high dimensions[END_REF].

From contact vector fields to open books

Proposition 6.1. Let (M 2n-1 , ξ) be a closed contact manifold. Suppose X, Y are two contact vector fields with Lie bracket [X, Y ] everywhere negatively transverse to ξ. Then, if we denote X θ := cos θ •X +sin θ •Y and Y θ := X θ+π/2 for θ ∈ S 1 , we have the following: (a) The set Σ θ := {X θ ∈ ξ} is a non-empty regular hypersurface, which is moreover ξ-convex.

(b) For θ = θ mod π, Σ θ and Σ θ intersect transversely along a non-empty contact submanifold K of M (independent of the couple (θ, θ )).

(c) For each θ ∈ S 1 , consider the set

F θ := { p ∈ Σ θ | Y θ (p) is positively transverse to ξ p } ,
and define ϕ :

M \ K → S 1 as ϕ(p) := θ if p ∈ F -θ . Then, (K, ϕ
) is an open book decomposition of M and is adapted to ξ.

(ii) dα is a positive symplectic form on the fibers of ϕ| M \N .

This concludes the proof of point (c).

We now proceed to prove the lemmas used in the above proof.

Proof (Lemma 6.7). Clearly, φ -1 (0) = Σ 0 ∩ Σ π/2 = K as subsets of M . Moreover, we can compute dφ(X) = d (α (X)) (X)∂ x -d (α (Y )) (X)∂ y along K. Now, by Lemma 6.6 and Corollary 6.5, d (α (X)) (X) = 0 and

d (α (Y )) (X) = α ([X, Y ]) along K, hence dφ(X) = -α ([X, Y ]) ∂ y . Similarly, we can compute dφ(Y ) = -α ([X, Y ]) ∂ x along K.
In other words, φ is transverse to the origin of R 2 and the oriented couple (Y, X) gives the positive coorientation of φ -1 (0), hence (Y θ , X θ ) too.

To study ϕ -1 (θ), we do the following. Suppose ϕ(p) = θ and write φ(p) ∈ R 2 in polar coordinates as φ(p) • (cos θ, sin θ). Then, we can compute

α X -θ-π/2 = α (X) sin θ + α (Y ) cos θ = φ 1 (p) sin θ -φ 2 cos θ = φ(p) • (cos θ sin θ -sin θ cos θ) = 0 ,
i.e. we have that p ∈ Σ -θ-π/2 . Hence, to show that p ∈ F -θ-π/2 , we need to check that Y -θ-π/2 is positively transverse to ξ at p, i.e. that α Y -θ-π/2 | p > 0. This follows from the following computation:

α Y -θ-π/2 | p = α (X) | p cos θ -α (Y ) | p sin θ = φ 1 (p) cos θ + φ 2 sin θ = φ (p) cos 2 θ + sin 2 θ = φ (p) > 0 .
We now check that ϕ -1 (θ) is positively cooriented by Y -θ-π/2 . For this, we need to check that dϕ Y -θ-π/2 | p is positive. We can compute

φ dϕ Y -θ-π/2 | p = (cos θ dφ 2 -sin θ dφ 1 ) Y -θ-π/2 | p = [-cos θ d (α (Y )) -sin θ d (α (X))] Y -θ-π/2 | p = d α X -θ-π/2 Y -θ-π/2 | p ( * ) = -α ([X, Y ]) | p > 0 ,
where ( * ) comes from Lemma 6.4. This concludes the proof of Lemma 6.7.

Proof (Lemma 6.8). Let α be a contact form for ξ as in the statement. We notice that Hypothesis (iii) actually means that there is a very small > 0 such that dα is a symplectic form on each fiber of the restriction of

ϕ to M \ K × D 2 1-, where D 2 1-is the disk of radius 1 -in R 2 .
The aim is to find a function f : M → R >0 such that f α satisfies the conditions of being compatible with the open book decomposition (K, ϕ). In other words, we want that f α induces a positive contact form on K (which is trivially satisfied because it induces the same contact structure as α) and such that d (f α) is a positive volume form on the pages. We search such a function f of the following form: f is a smooth function that depends only on the radius coordinate r on D 2 inside N , non-increasing in r, which is equal to 1 on M \ K × D 2 1-/2 and equal to 1 + e -kr 2 on K × D 2 1-, where k > 0 is a

FROM CONTACT VECTOR FIELDS TO OPEN BOOKS

constant yet to determine. We can then compute

dϕ ∧ d (f α) n-1 = dϕ ∧ ( df ∧ α + f dα ) n-1 = f n-1 dϕ ∧ dα n-1 + (n -1)f n-2 dϕ ∧ ∂f ∂r dr ∧ α ∧ dα n-2 = f n-2 f dϕ ∧ dα n-1 -(n -1) ∂f ∂r dr ∧ dϕ ∧ α ∧ dα n-2 . Now, on M \ K × D 2 1-/2 we have that f α = α, hence dϕ ∧ d(f α) n-1 > 0 as wanted.
We then need to control the situation in K × D 2 1-/2 . Let's start by analyzing what happens on K × D 2 1-. Here, ∂f ∂r = -2kre -kr 2 , so that

f dϕ ∧ dα n-1 -(n -1) ∂f ∂r dr ∧ dϕ ∧ α ∧ dα n-2 = = e -kr 2 dϕ ∧ dα n-1 + 2 (n -1) krdr ∧ dϕ ∧ α ∧ dα n-2 .
By Hypothesis (ii), the form rdr ∧ dϕ ∧ α ∧ dα n-2 is strictly positive on N , hence on K × D 2 1-/2 , and dϕ ∧ dα n-1 is bounded above in norm, even if we don't know its exact sign. This means that for k > 0 big enough, the second form will dominate the first, i.e. their sum will still be positive.

It then remains to study the sign on the open set

K × D 2 1-/2 \ D 2 1-.
Here, the situation is easy because dϕ ∧ dα n-1 is strictly positive and -∂f ∂r dr ∧ dϕ ∧ α ∧ dα n-2 is non-negative (remember f is a non-increasing function of r in this set), so their sum is also strictly positive.

Proof (Lemma 6.9). We start with point (i). We can explicitly evaluate the differential dΨ at points of the form (p, 0, 0). On K ×{0}, we simply have that dΨ(∂ x ) = Y , dΨ(∂ y ) = X and that dΨ(V ) = V for all vector fields V which are tangent to K × {0}. This shows that Ψ is a local diffeomorphism at each point (p, 0, 0). Hence, by compactness, Ψ is also a diffeomophism from K × D 2 δ onto its image, provided δ is small enough. We now prove point (ii). For θ ∈ S 1 , consider the function H θ : K × [0, δ) → M defined as H θ (p, r) = Ψ(p, r cos θ, r sin θ). We then have to show that ϕ(H θ (p, r)) = θ. Noticing that Y -θ = sin θ•X +cos θ•Y , we can rewrite more explicitly H θ (p, r) = ψ r Y -θ (p), i.e. H θ (., r) is the flow of Y -θ at time r. By Lemma 6.6, Y -θ = -X -θ-π/2 is tangent to Σ -θ-π/2 and entering in F -θ-π/2 ; in particular, for r > 0 we have ψ r Y -θ (p) ∈ F -θ-π/2 . Now, by Lemma 6.7, ϕ -1 (θ) = F -θ-π 2 , which implies ϕ(H θ (p, r)) = θ, as wanted. We can now finish with point (iii). Because the contact condition is open, up to shrinking δ, it is enough to prove that K 0 = Ψ(K × {0}) is a positive contact submanifold. This actually follows from general results from [START_REF] Giroux | Convexité en topologie de contact[END_REF]: indeed, X θ defines the characteristic foliation of Σ θ , and K is transverse to it. For completeness' sake, we prove it here also with an explicit computation. We showed in point (b) of Proposition 6.1 that Y is transverse to K. Due to the symmetry of the situation, the same is true for the contact vector field X. If we consider an arbitrary point p ∈ K and denote (K i ) i=1,...,2n-3 a local base of T K in a neighborhood N of p in K, we then know that

ι Y ι X α ∧ dα n-1 (K 1 , . . . , K 2n-3 ) = α ∧ dα n-1 (X, Y, K 1 , . . . , K 2n-3 ) = 0 , CHAPTER 6. OPEN BOOKS AND CONTACT VECTOR FIELDS because (X, Y, K 1 , . . . , K 2n-3 ) is a basis for T M over N . We can also compute ι Y ι X α ∧ dα n-1 (i) = -(n -1) ι Y α ∧ (ι X dα) ∧ dα n-2 (ii) = (n -1) dα (X, Y ) α ∧ dα n-2 + -(n -1)(n -2)α ∧ dα (X, .) ∧ dα (Y , .) ∧ dα n-3 (6.4)
where equalities (i) and (ii) come from the graded Leibniz rule for the interior product (i.e. the formula ι Z (µ ∧ ν) = ι Z µ ∧ ν + (-1) deg µ µ ∧ ι Z ν for all differential forms µ, ν and vector fields Z) and from the fact that α (X) = 0 and α (Y ) = 0 along K. Now, dα (X, Y ) = α ([X, Y ]) = 0 along K by Corollary 6.5, so if we manage to prove that the last term in the second line of Equation (6.4) is zero when evaluated on (K i ) i , we will get that α ∧ dα n-2 (K 1 , . . . , K 2n-3 ) = 0 too, which is exactly what we wanted. From Equation (6.2), with θ = 0, we deduce that, for a certain µ ∈ Ω 1 (M ),

dα ( X , . ) ∧ dα ( Y , . ) = d (α (X)) ∧ d (α (Y )) + α ∧ µ .
Replacing this in the last summand of the second right hand side of Equation (6.4), we get

α ∧ dα ( X , . ) ∧ dα ( Y , . ) ∧ dα n-3 = α ∧ d (α (X)) ∧ d (α (Y )) ∧ dα n-3 .
Now, evaluating this on (K 1 , . . . , K 2n-3 ) gives zero, because K i ∈ T K and both d (α (X)) p and d (α (Y )) p are zero on T p K = T p Σ 0 ∩ T p Σ π/2 , for every p ∈ K. Then, by Corollary 6.5, the orientation induced on K by the contact structure ξ on M is the one such that the ordered couple of vector fields (Y, X) induces a positive orientation of the normal bundle of K in M . But, according to the computation of dφ in the proof of Lemma 6.7, this is exactly the case also for Ψ(K × {0}) (oriented as image of K = φ -1 (0)). This means that K 0 is a positive contact submanifold, as wanted.

Proof (Lemma 6.10). We search a function f such that α := f α satisfies dϕ ∧ d α n-1 > 0 on M \ Int(N ). We can compute

dϕ ∧ d α n-1 = f n-1 dϕ ∧ dα n-1 + (n -1)f n-2 dϕ ∧ df ∧ α ∧ dα n-2 = f n-2 f dϕ ∧ dα n-1 -(n -1) df ∧ dϕ ∧ α ∧ dα n-2 .
Let now > 0 be such that { φ < 2 } ⊂ N and chose f to be a smooth function, depending only on φ and non-increasing in it, equal to 1/ on the set { φ < } and equal to 1/ φ on the set M \ { φ < 2 }.

Let's now analyze dϕ ∧ d α on N c . Here, we have f = 1 / φ and df = -d φ / φ 2 , so that

φ n+1 dϕ ∧ d α n-1 = φ 2 dϕ ∧ dα n-1 + (n -1) φ d φ ∧ dϕ ∧ α ∧ dα n-2 .
Moreover, recalling that φ = (α (X) , -α (Y )), we also have that φ

2 dϕ = φ 1 dφ 2 - φ 2 dφ 1 = -α (X) d (α (Y ))+α (Y ) d (α (X)) and φ d φ ∧dϕ = dφ 1 ∧dφ 2 = -d (α (X))∧ d (α (Y )), so that φ n+1 dϕ ∧ d α n-1 = [-α (X) d (α (Y )) + α (Y ) d (α (X))] ∧ dα n-1 -(n -1) d (α (X)) ∧ d (α (Y )) ∧ α ∧ dα n-2 . (6.5)

FROM OPEN BOOKS TO CONTACT VECTOR FIELDS

Notice now that α ∧ d (α (Y )) ∧ dα n-1 = 0 on M , because dim M = 2n -1. Hence, ι X α ∧ d (α (Y )) ∧ dα n-1 = 0 too. In other words, according to the graded Leibniz rule for the interior product,

α(X) • d (α (Y )) ∧ dα n-1 -d (α (Y )) (X) • α ∧ dα n-1 + (n -1)α ∧ d (α (Y )) ∧ dα(X, .) ∧ dα n-2 = 0 , i.e. α(X) • d (α (Y )) ∧ dα n-1 = d (α (Y )) (X) • α ∧ dα n-1 -(n -1)α ∧ d (α (Y )) ∧ dα(X, .) ∧ dα n-2 . (6.6)
Exchanging the roles of X and Y in the above computations, we also get

α(Y ) • d (α (X)) ∧ dα n-1 = d (α (X)) (Y ) • α ∧ dα n-1 -(n -1)α ∧ d (α (X)) ∧ dα(Y, .) ∧ dα n-2 . (6.7)
Now, recall that X, Y are contact vector fields for ξ, i.e. there are functions f, g :

M → R such that d (α (X)) = f α -dα(X, .) , d (α (Y )) = gα -dα(Y, .
) . (6.8)

Then, Equations (6.5) to (6.8), together with the fact that α ∧ α = 0, tell that

φ n+1 dϕ ∧ d α n-1 = -d (α (Y )) (X) • α ∧ dα n-1 + d (α (X)) (Y ) • α ∧ dα n-1 + (n -1)α ∧ dα(X, .) ∧ dα(Y, .) ∧ dα n-2 .
(6.9) Now, again for dimensional reasons, we have dα n = 0 on M , so that ι X ι Y dα n = 0 too on M . This gives (n -1)dα(X, .) ∧ dα(Y, .) ∧ dα n-2 = dα(X, Y ) • dα n-1 , so that Equation (6.9) finally becomes

φ n+1 dϕ ∧ d α n-1 = -d (α (Y )) (X) • α ∧ dα n-1 + d (α (X)) (Y ) • α ∧ dα n-1 + dα(X, Y ) • α ∧ dα n-1 = -α ([X, Y ]) • α ∧ dα n-1 , because dα(X, Y ) = d (α (Y )) (X) -d (α (X)) (Y ) -α ([X, Y ]
) according to the exterior derivative formula. Now, [X, Y ] is negatively transverse to ξ by hypothesis, so that the above equation implies that d α is symplectic on the fibers of ϕ| M \N , as wanted. Consider the standard open book decomposition OB of R 2 , i.e. the one with the origin as binding and with radii starting from the origin as pages.

From open books to contact vector fields

FROM OPEN BOOKS TO CONTACT VECTOR FIELDS

Define now χ : R ≥0 → R ≥0 as follows: χ(x) is non-increasing in x, is equal to x for x < /2 and equal to 1 for x ≥ . Denote then f := χ( φ ) / φ : M → R >0 and define φ := f φ : M → R 2 ; then, φ defines (B, ϕ) too. Let also ρ := φ and θ := φ /ρ : M \ B → S 1 . Notice that θ = ϕ.

Then, we claim that

Ω := nρ 2 dθ ∧ dα n-1 + n(n -1)ρdρ ∧ dθ ∧ α ∧ dα n-2
is a volume form on M . Indeed, the first term is non-negative everywhere and positive away from B, because dα is symplectic on the fibers of θ = ϕ, and the second term is positive along B and non-negative everywhere, by construction of f . This being said, we now denote by X, Y the contact vector fields associated, respectively, to the contact Hamiltonians φ 1 , -φ 2 via the contact form α given in the statement.

Because ρdθ = φ 1 dφ 2 -φ 2 dφ 1 and ρdρ ∧ dθ = dφ 1 ∧ dφ 2 , we can then write

Ω = n [-α (X) d (α (Y )) + α (Y ) d (α (X))] ∧ dα n-1 -n(n -1)d (α (X)) ∧ d (α (Y )) ∧ α ∧ dα n-2 .
Notice then that the right hand side is exactly the same (up to a factor n) as the one of Equation (6.5) in the proof of Lemma 6.10. Hence, the exact same computations made in that proof tell us that

Ω = -nα ([X, Y ]) • α ∧ dα n-1 .
This shows that [X, Y ] is negatively transverse to ξ, because Ω is a volume form on M and α is a positive contact form.

Chapter 7

Lutz' study and Bourgeois' construction of invariant contact manifolds

The aim of this chapter is twofold: we want to reinterpret a part of Lutz' study from [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF] using the language of adapted open books introduced in [Gir02] by Giroux, and we want to describe the links between Lutz' work and the Bourgeois construction from [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF]. In order to do this, we proceed as follows.

Section 7.1 describes the study from [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF] of T 2 -invariant contact structures on the total space of a principal T 2 -bundle π : V 2n+1 → M 2n-1 . In this context, we recall how Lutz recovers an open book decomposition (B, ϕ) and a differential 1-form β on the base manifold M such that its restriction to B is a contact form and, in the case of a flat principal bundle π : V → M , its differential is symplectic on the fibers of ϕ. In this part, we recall (and complete the missing details) of the proofs from [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF]. Finally, we conclude Section 7.1 by showing, using a lemma on adapted open book decompositions due to Giroux (already stated as Lemma 6.8), that if the induced β happens to be contact on M (which is not always the case), this open book (B, ϕ) supports the contact structure ξ = ker β, at least in the case of flat bundles.

In Section 7.2 we recall the construction presented in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF], where, starting from a contact manifold (M, ξ) and an open book decomposition on M supporting ξ, Bourgeois gives an explicit T 2 -invariant contact structure on the particular flat principal T 2 -bundle M × T 2 . We will also show that this can be interpreted as a converse of the construction by Lutz recalled in Section 7.1.

Lutz's study revisited

Let π : V → M be a principal T 2 -bundle and η be a contact structure on V invariant under the T 2 -action χ : T 2 × V → V on V . Assume that V, M are oriented, that the T 2 -action preserves the orientation and that η is positive on V . We then want to give a detailed statement that summarizes some of Lutz' results from [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF] in this context their reinterpretation in terms of adapted open books, as introduced by Giroux in [START_REF] Giroux | Géométrie de contact: de la dimension trois vers les dimensions supérieures[END_REF]. In order to do this, we first need to introduce some notations.

For each g ∈ T 2 , denote by χ g : V → V the diffeomorphism given by the action of g on the total space V . If we denote by t the Lie algebra of the Lie group T 2 , we can then 7.1. LUTZ'S STUDY REVISITED 0, because φ(p) = 0, and dα(Z, .) |p = 0 by Lemma 7.3 and becausesin θ d p φ 1 + cos θ d p φ 2 = 0. By the contact condition, this implies Z p = 0, giving a contradiction.

Take now p ∈ M \B, with φ 1 (p) = 0 let's say. Then, to show that d p ϕ : T p (M \ B) → T ϕ(p) S 1 has rank 1, it's enough to show that the differential of

Ψ 2 := φ 2 ρ : M \ B ϕ -→ S 1 πy| S 1 ---→ R
has rank 1 at p. Here, π y : R 2 → R is the projection on the y-axis.

Because dρ = 1 2ρ d ρ 2 = 1 ρ (φ 1 dφ 1 + φ 2 dφ 2 ) ,
we can write

dΨ 2 = dφ 2 ρ - φ 2 ρ 2 dρ = dφ 2 ρ - φ 2 ρ 3 (φ 1 dφ 1 + φ 2 dφ 2 ) . (7.1)
For notational convenience, let ρ, Ψ 2 denote respectively the compositions ρ • π and Ψ 2 • π.

Suppose now by contradiction that dΨ 2 = 0 and consider the vector field

Z := Ψ 2 φ 1 X 1 + φ 2 X 2 ρ -X 2 defined on V \ B.
Then, an easy computation shows that α(Z) = 0 on all V \ B. Moreover, Equation (7.1) and the hypothesis d p Ψ 2 = 0 imply that dα (Z, .) |p = 0. By the contact condition, we get Z(p) = 0. In particular, the coefficient multiplying X 1 (p) in Z(p) has to be zero, i.e. Ψ 2 (p) φ 1 (p) = 0. Because φ 1 (p) is non-zero by assumption, Ψ 2 (p) has to be zero. But this implies Z(p) = -X 2 (p), which contradicts the fact that Z(p) = 0. This concludes the proof of the fact that (B, ϕ) is an open book decomposition of M .

We then proceed to the proof of points b. and c. of Theorem 7.2. We start by showing that, thanks to the T 2 -invariant contact form α and a choice of connection form ω defining F, we can induce an explicit 1-form β on M whose kernel is exactly π * (η ∩ F). This is also explained in [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF]Section I.3.6].

We recall that the connection form ω ∈ Ω 1 (V, t) defining F can be written as as ω = ω 1 ⊗ e 1 + ω 2 ⊗ e 2 , with ω 1 , ω 2 ∈ Ω 1 (V ) such that ker ω 1 ∩ ker ω 2 = F and (e 1 , e 2 ) the basis of the vector space t = R 2 associated to the choice of coordinates (θ 1 , θ 2 ).

Moreover, the fact that F is a foliation gives, by Frobenius' theorem, the flatness of ω, i.e. tells that its curvature is zero. We recall that the curvature form Ω of ω is the 2-form on M with values in t defined by dω + 1 2 [ω, ω]. More explicitly, for each couple of vector fields U, V on M and each p ∈ M , Ω p (U p , V p ) := dω q (X q , Y q ) + 1 2 [ω q (X q ), ω q (Y q )], where X, Y are lifts of U, V horizontal for ker ω = F and q ∈ π -1 (p). This is indeed well defined because the right-hand side is independent of the choice of X, Y, q as above. Notice moreover that [ω, ω] = 0, because T 2 is abelian, so that Ω = dω. Hence, in our case, the flatness of ω means that dω = dω 1 ⊗ e 1 + dω 2 ⊗ e 2 = 0. Now, the 1-form γ := α -φ 1 ω 1 -φ 2 ω 2 on V is T 2 -invariant and such that γ(X 1 ) = γ(X 2 ) = 0. Hence, there is a 1-form β on M such that γ = π * β, i.e. such that α = φ 1 ω 1 + φ 2 ω 2 + π * β.

Then, we have the following result, whose proof is postponed:

Lemma 7.4 ([Lut79, Proposition (page 296)]). α is a positive contact form on V if and only if the two following conditions are satisfied: INVARIANT CONTACT MANIFOLDS i. β induces a negative contact form on B;

ii. for each θ ∈ S 1 , d β ρ + φ1 ρ Ω 1 + φ2 ρ Ω 2 induces a negative symplectic form on ϕ -1 (θ).

This tells in particular that, if there is a flat connection ω for π : V → M , the β obtained on M via ω induces a contact structure on B and the differential of β /ρ is symplectic on the fibers of ϕ.

Let's suppose, for the moment, that β defines a contact structure on all M . Then, we want to show point c., i.e. that ξ = ker β is supported by the open book (B, ϕ), where B is B with orientation inversed and ϕ is ϕ composed with the orientation reversing diffeomorphism S 1 → S 1 given by the restriction of the complex conjugation to S 1 ⊂ C. We recall from what said after the statement of Theorem 7.2 that the particular choice of reversing diffeomorphism S 1 → S 1 is not important, hence we can choose an arbitrary one.

In order to prove this, we need again Lemma 6.8, already used in Section 6.1. We recall here the statement, adapting the notations to this situation: Lemma 6.8 (Giroux). Let (M 2n-1 , ξ) be a contact manifold. Suppose there are an open book decomposition (K, θ) of M (in particular, B is oriented as the boundary of each fiber of θ), a tubular neighborhood N = K × D 2 of K (here D 2 is the open unit disk in R 2 ) and a contact form γ defining ξ such that:

(i) θ restricted to N \K is the angular coordinate of the projection on the second factor N = K × D 2 → D 2 ;

(ii) ξ induces a positive contact structure on each submanifold K z := K × {z} of N (notice each K z is oriented because K is);

(iii) dγ induces a positive symplectic form on each fiber of θ |M \N .

Then, the open book decomposition (K, θ) supports the contact structure ξ.

Choose now an > 0 very small and replace β by β := χ β, with χ : M → R a positive function equal to 1 /ρ on the complement of {ρ ≥ 2 } ⊂ M and equal to 1 / on B. Then, according to Lemma 7.4, (B, ϕ), N = {ρ < 2 } and β satisfy the hypothesis of Lemma 6.8, provided that is small enough. Hence, we get that (B, ϕ) supports ξ, as wanted.

The only thing left in order to finish the proof of Theorem 7.2 is then to show point b., i.e. that ξ = π * η ∩ F is a contact structure on M if and only if (F, η ∩ F) is a codimension 2 contact foliation on V .

This actually follows from the fact that

β ∧ dβ n-1 > 0 on M ⇐⇒ π * β ∧ dβ n-1 F > 0 ⇐⇒ α ∧ dα n-1 F > 0 ,
where we used the facts that the differential d p π is an isomorphism between F p and T π(p) M and that the connection ω is zero on F. This concludes the proof of Theorem 7.2.

We now prove Lemma 7.4 used above.

Proof (Lemma 7.4). The form α is a positive contact form on V if and only if ι X2 ι X1 α ∧ dα n induces a positive section of the vector bundle Λ 2n-1 (F * ) over V , where F * is the dual of F. Indeed, F is a complement of the vector sub-bundle spanned by X 1 and X 2 in T V .

LUTZ'S STUDY REVISITED

Now, because ι X2 ι X1 α ∧ dα n is T 2 -invariant and becomes zero after contraction with X 1 or X 2 , it can be written as pullback via π of a differential form δ of maximal degree on M . Hence, ι X2 ι X1 α ∧ dα n is a positive section of Λ 2n-1 (F * ) if and only if δ is a volume form on M . Then, what we need to do is find this form δ.

We start by computing

ι X2 ι X1 α ∧ dα n (a) = ι X2 φ 1 dα n -nα ∧ ι X1 dα ∧ dα n-1 (b) =n φ 1 ι X2 dα ∧ dα n-1 -n φ 2 ι X1 dα ∧ dα n-1 + ndα (X 1 , X 2 ) α ∧ dα n-1 -n(n -1)α ∧ ι X1 dα ∧ ι X2 dα ∧ dα n-2 (c) = -ndα n-1 ∧ π * (φ 1 dφ 2 -φ 2 dφ 1 ) -n(n -1)α ∧ dα n-2 ∧ π * (dφ 1 ∧ dφ 2 ) (d) = -ndα n-1 ∧ π * ρ 2 dϕ -n(n -1)α ∧ dα n-2 ∧ π * (ρdρ ∧ dϕ) . (7.2)
Here, the equalities (a) and (b) come from the graded Leibniz rule for the interior product (i.e. the formula ι Z (µ ∧ ν) = ι Z µ ∧ ν + (-1) deg µ µ ∧ ι Z ν for all differential forms µ, ν and vector fields Z) and from the facts that ι X1 α = φ 1 and ι X2 α = φ 2 . Also, the equality (c) comes from Lemma 7.3 and the (d) follows from the fact that φ 1 dφ 2 -φ 2 dφ 1 = ρ 2 dϕ and dφ 1 ∧ dφ 2 = ρdρ ∧ dϕ by definition of ρ and ϕ. Now, along B = π -1 (B) we have that

ι X2 ι X1 α ∧ dα n | B (a) = -n(n -1)α ∧ dα n-2 ∧ π * (ρdρ ∧ dϕ) | B (b) = -n(n -1) π| B * β ∧ dβ n-2 ∧ ρdρ ∧ dϕ| B (7.3)
Here, the equality (a) comes from the fact that ρ 2 dϕ = 0 on B = φ -1 (0) (by definition) and the (b) from the fact that, for well chosen differential forms µ 1 , µ 2 , ν 1 , ν 2 , we can write

α ∧ dα n-2 = π * β ∧ dβ n-2 + φ 1 µ 1 + φ 2 µ 2 + d φ 1 ∧ ν 1 + d φ 2 ∧ ν 2 ,
so that the only non-zero contribution along B after wedging with the form π * (dφ 1 ∧ dφ 2 ) is given by the term π * β ∧ dβ n-1 .

We now analyze the situation on the complement of B. Let ϕ := ϕ • π and recall that we already introduced the notation ρ = ρ • π. Then, away from B, we compute

ρ d α ρ n-1 ∧ d ϕ = ρ -d ρ ∧ α + ρdα ρ 2 n-1 ∧ d ϕ = ρ ρ n-1 dα n-1 -(n -1) ρ n-2 d ρ ∧ α ∧ dα n-2 ρ 2n-2 ∧ d ϕ = 1 ρ n ρ 2 d ϕ ∧ dα n-1 + (n -1) α ∧ dα n-2 ρ ∧ (d ρ ∧ d ϕ) .
Hence, from Equation (7.2) we get

ρ d α ρ n-1 ∧ d ϕ = - 1 n ρ n (ι X2 ι X1 α ∧ dα n ) . (7.4) INVARIANT CONTACT MANIFOLDS
Now, we point out that on the horizontal space F of the connection ω we have

α = π * β , dα = φ 1 dω 1 + φ 2 dω 2 + π * dβ. (7.5)
Indeed, the terms ω 1 , ω 2 in α and d φ 1 ∧ ω 1 and d φ 2 ∧ ω 2 in dα are zero on F.

Then, on F we also have

ρ d α ρ n-1 ∧ d ϕ = ρ -d ρ ∧ α + ρdα ρ 2 n-1 ∧ d ϕ (a) = ρ -d ρ ∧ π * β + ρπ * dβ ρ 2 + φ 1 ρ dω 1 + φ 2 ρ dω 2 n-1 ∧ d ϕ (b) = π * ρ d β ρ + φ 1 ρ Ω 1 + φ 2 ρ Ω 2 n-1 ∧ dϕ . (7.6)
Then, Equation (7.3) together with Equations (7.4) and (7.6) tell us that ι X2 ι X1 α ∧ dα n is a positive section of Λ 2n-1 (F * ) over V if and only if

• the form β ∧ dβ n-2 ∧ ρdρ ∧ dϕ on B is a negative volume form along B, • the form ρ d β ρ + φ1 ρ Ω 1 + φ2 ρ Ω 2 n-1
∧ dϕ is a negative volume form away from B, i.e. if and only if • β induces a negative contact form on B,

• for each θ ∈ S 1 , d β ρ + φ1 ρ Ω 1 + φ2 ρ Ω 2 induces a negative symplectic form on ϕ -1 (θ).

This concludes the proof of Lemma 7.4.

The Bourgeois construction

Bourgeois gives in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] a construction that goes in the opposite direction with respect to that in [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF] recalled above. Indeed, using the notion of open book decompositions for contact manifolds (M 2n-1 , ξ) from [START_REF] Giroux | Géométrie de contact: de la dimension trois vers les dimensions supérieures[END_REF], he constructs explicit contact structures on the total space of the principal T 2 -bundle π : M × T 2 → M . More precisely, he proves the following:

Theorem 7.5 (Bourgeois). Let (M 2n-1 , ξ) be a contact manifold and (B, ϕ) an open book decomposition of M supporting ξ.

a. There is a smooth map φ = (φ 1 , φ 2 ) : M → R 2 defining the open book (B, ϕ) and such that γ ∧ dγ n-2 ∧ dφ 1 ∧ dφ 2 ≥ 0 on M , where γ is any contact form defining ξ.

b. If φ is as in point a., then for any choice of coordinates (θ 1 , θ 2 ) coordinates on T 2 and for any contact form β defining ξ and adapted to the open book (B, ϕ), the 1-form α :

= β + φ 1 dθ 1 -φ 2 dθ 2 is a contact form on M × T 2 .
all isotopic by Gray's theorem. Notice that α 0 = β defines the hyperplane field ξ ⊕ T T 2 , which is not a contact structure on M × T 2 . Nonetheless, in Chapter 8 we will call it contact fiber bundle on M ×T 2 and show that it plays an important role in understanding the properties of the construction in Theorem 7.5.

We are now interested in studying in detail the relations between the Bourgeois construction and Lutz' one, already recalled in Section 7.1.

First of all, we point out that the application of Lutz' procedure after the Bourgeois construction gives back the original data. More precisely, if we start from (M 2n-1 , ξ) and (B, ϕ) supporting ξ, we can apply Theorem 7.5 to obtain the contact manifold (M × T 2 , η := ker α), Vice versa, we now study the other possible composition, i.e. the application of the Bourgeois construction after the Lutz' one. We then have the following: Proposition 7.7. Let η be a T 2 -invariant contact structure on the principal bundle π : M × T 2 → M such that the couple (F, F ∩ η), where F := T M ⊕ {0}, is a contact foliation on M × T 2 . Let (B, ϕ) and ξ be, respectively, the open book and the contact structure on M given by Theorem 7.2. Then, for any choice of coordinates (θ 1 , θ 2 ) on T 2 , there is a couple (β, φ) as in the statement of Theorem 7.5 such that η = ker(α),

with α = β + φ 1 dθ 1 -φ 2 dθ 2 . Then, if ω ∈ Ω 1 (M × T 2 , t)
where α = β + φ 1 dθ 1 -φ 2 dθ 2 on M × T 2 , with φ = (φ 1 , φ 2 ).
In other words, provided the T 2 -invariant contact structure η we start with satisfies the fact that the push-forward of η ∩(T M × {0}) via dπ is a contact structure ξ on M , we can apply the Bourgeois construction after the Lutz' one to get back the initial structure η.

Proof (Proposition 7.7). According to Theorem 7.5, it's enough to show that there are a contact form β defining ξ, a neighborhood N = B × D 2 of the binding B in M and a map φ = (φ 1 , φ 2 ) : M → R 2 such that i. β is adapted to (B, ϕ),

ii. φ defines (B, ϕ) and β ∧ dβ n-2 ∧ dφ 1 ∧ dφ 2 ≥ 0 on M , iii. η = ker (β + φ 1 dθ 1 -φ 2 dθ 2 ).

As already done in the proof of point a. of Theorem 7.5, we point out that for the point ii. above it's enough to find a map φ = (φ 1 , φ 2 ) and a neighborhood N = B × D 2 of B in M such that φ defines (B, ϕ), φ is constant on M \ N and each B × {pt} in N = B × D 2 is a fiber of φ and a contact submanifold of (M, ξ).

Let's start by writing η = ker (α ), with α = β + φ 1 dx -φ 2 dy, where β is a contact form on M and φ := (φ 1 , φ 2 ) : M → R 2 defines (B, ϕ). This can be done as in the proof of Theorem 7.2, using here the natural flat connection dθ 1 ⊗ e 1 + dθ 2 ⊗ e 2 on π : M × T 2 → M .

Consider now a small normal neighborhood N = B × D 2 of B in M such that each B × {pt} is a fiber of φ and a contact submanifold of (M, ξ). Such N exists because φ defines the open book (B, ϕ) adapted to ξ.

Then, we claim that there are a contact form β for ξ on M and a map φ := (φ 1 , φ 2 ) : M → R 2 such that α := β + φ 1 dx -φ 2 dy still defines η, dβ is symplectic on the fibers of ϕ| M \N , and each B × {pt} in N is a fiber of φ . Indeed, let > 0 be such that { φ < 2 } ⊂ N and chose f to be a smooth function, Chapter 8

Bourgeois' construction revisited

In Section 7.2, we presented the construction in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] in terms of invariant structures on the principal T 2 -bundle M × T 2 → M . Though, the following "orthogonal" point of view can also be adopted: the examples by Bourgeois are contact fiber bundle structures on the fiber bundle M × T 2 → T 2 . This change of perspective is motivated by [START_REF] Presas | A class of non-fillable contact structures[END_REF][START_REF] Van Koert | Every contact manifolds can be given a nonfillable contact structure[END_REF][START_REF] Niederkrüger | Some remarks on the size of tubular neighborhoods in contact topology and fillability[END_REF], where the contact fiber bundle structure of the examples from [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] was used to give other important constructions of high dimensional contact manifolds. Actually, the contact fiber bundle structure, together with the existence of a flat connection, is what remains in the higher genus case, for example taking contact branched coverings of the constructions in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF], as already proposed in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF]Corollary 3].

The aim of this chapter is then to reinterpret and generalize the construction from Bourgeois using the notion of contact fiber bundles, as introduced in [START_REF] Lerman | Contact fiber bundles[END_REF], under the hypothesis of the existence of a flat connection.

More precisely, in Section 8.1 we recall the definitions and the main properties of contact fiber bundles. Then, we use this notion to generalize the construction from [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] recalled in Section 7.2. In particular, in Section 8.2 we take a general fibration admitting a flat contact connection and we consider on it two non-trivial subclasses of all its contact connections. The first subclass, described in Section 8.3, is characterized in terms of deformations to the flat contact connection, in a flavor similar to the notion of contactizations introduced in Definitions 5.11 and 5.22. The second subclass, included in the first, is a direct generalization of the examples from [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] in the setting of contact fiber bundles and is presented in Section 8.4, where Proposition II.D from the introduction is also proven. Lastly, in Section 8.5 we study the stability of the first class under the operation of contact branched covering.

Generalities

Let B 2m , M 2n-1 and V 2m+2n-1 be smooth manifolds and π : V → B a smooth fiber bundle with fiber M . Denote by M b the fiber of π over b ∈ B.

Suppose now that V, B are oriented, and give to each fiber M b the natural orientation as preimage π -1 (b).

CONTACT DEFORMATIONS OF FLAT CONTACT BUNDLES

that define a contact structure on the total space V . To describe this subclass in terms of the potential, we have to introduce another two tensors. The first one is the exterior derivative d 0 A ∈ Ω 2 (Σ, X cont f ib (V )) of A, which is defined as follows: for all p ∈ Σ and u, v

∈ T p Σ, d 0 A(u, v) := [ X, A Y ] -[ Y , A X ] -A [X,Y ]
, where X, Y are vector fields tangent to Σ defined near p and such that X(p) = u and Y (p) = v. We remark that this is indeed well defined because the quantity on the right hand side depends only on u and v: this can easily be checked using the tensoriality criterion. The second tensor which we will need is the Lie bracket of A with itself,

[A, A] ∈ Ω 2 (Σ, X cont f ib (V )), given by [A, A](u, v) := [A X , A Y ],
where u, v, X, Y are as in the definition of d 0 A above. Again, the right hand side depends only on u, v, hence this is a well defined tensor.

We can now characterize the potentials that define contact structures:

Proposition 8.4. On a flat contact fiber bundle (V, Σ, M, {ξ p } p∈Σ , H 0 ) with Σ surface, a contact connection H with vector potential A gives a contact structure η on the total space if and only if, for all p in Σ and all oriented basis (u, v) of T p Σ, the vector field

d 0 A(u, v) + [A u , A v ] on M b is a negative contact vector field for (M p , ξ p ).
Proof. This is a direct consequence of Proposition 8.3 and of the following computation: for all X, Y vector fields on Σ such that X(p) = u and

Y (p) = v, [u # , v # ] -[u, v] # = [ X + A X , Y + A Y ] -[X, Y ] + A [X,Y ] = [ X, A Y ] -[ Y , A X ] -A [X,Y ] + [A X , A Y ] + [ X, Y ] -[X, Y ] ( * ) = d 0 A(u, v) + [A, A](u, v) , where (

Contact deformations of flat contact bundles

Fix for all this section a flat contact fiber bundle (V, Σ, M, η 0 ).

Definition 8.5. We say that a contact fiber bundle η defining a contact structure on the total space of the fiber bundle (V, Σ, M ) is a contact deformation of the flat bundle (V, Σ, M, η 0 ) if there is a smooth family of contact fiber bundles (η s ) s∈[0,1] starting at η 0 , ending at η 1 := η and satisfying the followings:

1. for all p ∈ Σ and all s ∈ [0, 1], η s defines the same ξ p on the fiber M p ; 2. η s defines a contact structure on V for all s > 0.

Thanks to Hypothesis 1, we could also rephrase the above notion in terms of a path of contact connections H s interpolating between H 0 and H.

We point out that this definition is "non-empty", i.e. given a flat contact fiber bundle (V, Σ, M, η 0 ), not all the contact fiber bundles for the same underlying fibration structure (V, Σ, M ) and inducing the same contact structures ξ p on each fiber M p , are contact deformations of η 0 . For instance consider the contact fiber bundle structure on T 3 = S 1 × T 3 which is given by the kernel η of α = dθ + cos(θ)dxsin(θ)dy, where θ ∈ S 1 and (x, y) are coordinates on T 2 . This contact fiber bundle structure is a contact deformation of the flat contact fiber bundle structure given by η 0 = ker (dθ): the deformation is given by α t := dθ + t cos(θ)dx -t sin(θ)dy, with t ∈ [0, 1]. We point out that, by [Gir99, Lemma 10], η admits prelagrangian tori only in the isotopy class of {pt} × T 2 . Take now a diffeomorphism ψ of T 3 sending (θ, x, y) to (θ + x, x, y). Then, ψ * η is still transverse to the S 1 factor, hence it still is a contact fiber bundle on the chosen fibration, and obviously it still defines a contact structure on the total space. However, it has prelagrangian tori in an isotopy class which is different from that of the prelagrangian tori of η. According to [Vog16, Proposition 9.9], this implies that φ * η cannot be a contact deformation of η 0 = T T 2 ⊕ {0} ⊂ T T 2 × S 1 .

We also remark that, even though the above definition is of a very similar flavor to Definitions 5.11 and 5.22, the objects they define behave differently. For instance, there is no uniqueness up to isotopy for contact deformations. Indeed, if we take again the fiber bundle T 3 = T 2 × S 1 → T 2 where we see the fibers as contact manifolds S 1 , ker (dθ) , then the flat contact bundle defined by η 0 = ker(dθ) actually admits as contact deformations every contact structure on T 3 defined by α n := dθ + cos(nθ)dxsin(nθ)dy. Though, these are not isotopic one to the other as contact fiber bundles defining contact structures on the total space, because they are not even isomorphic as contact structures on T 3 due to different Giroux torsion (see [START_REF] Giroux | Une infinité de structures de contact tendues sur une infinité de variétés[END_REF]).

Bourgeois' contact structures

The aim here is to use what we defined in the previous sections to generalize the construction by Bourgeois recalled in Section 7.2. Let's start by reformulating it with this new terminology.

We start from the trivial fiber bundle M × T 2 → T 2 with fixed contact fiber (M, ξ = ker(α)), and we consider the flat contact fiber bundle structure ξ ⊕ T T 2 on the total space of the fibration. Once fixed an open book decomposition (B, ϕ) adapted to ξ on M and a particular contact form β with differential symplectic on the pages of ϕ, consider a function φ = (φ 1 , φ 2 ) : M → R 2 as in the statement of Theorem 7.5. Now take the contact vector fields X and Y on (M, ξ) associated, respectively, to the contact hamiltonians φ 1 and -φ 2 via the contact form β, and define A := -X ⊗ dx -Y ⊗ dy, where (x, y) is a choice of coordinates for T 2 . A direct computation shows that the contact fiber bundle associated to this potential is exactly the kernel of the contact form α = β + φ 1 dθ 1 -φ 2 dθ 2 given by Theorem 7.5. We also remark that, because X and Y are independent from the point of T 2 in the product M × T 2 , the 2-form d 0 A is zero everywhere. Because α is a contact form, Proposition 8.4 then tells us that [A, A] takes values in the space of negative contact vector fields of the fibers (M, ξ). In particular, [X, Y ] is a negative contact vector field.

We can then generalize the construction by Bourgeois via the following:

Definition 8.6. Let (V, Σ, M, η 0 ) be a flat contact fiber bundle with associated connection H 0 . We call Bourgeois contact structure each contact structure on the total space V given by a contact fiber bundle structure η on V → Σ with potential A such that d 0 A = 0.

We remark that the curvature R of the connection H determined by η 0 and A (i.e. the contact connection of η) is just d 0 A + [A, A]. In particular, this curvature has two terms which behave differently under rescaling A → A, for > 0: the term d 0 A is rescaling linearly in , whereas [A, A] is rescaling quadratically in it. Then, if we denote by R the curvature associated to the connection H of potential A with respect to η 0 , space V , which is also transverse to the fibers of π : V → T 2 . Moreover, because of our particular choice of ρ : π 1 (T 2 ) → SO(n), each leaf L of F 0 intersects every fiber twice. Now, the connection T T 2 on the trivial bundle p : M × T 2 → T 2 gives a foliation F 1 with leaves {pt} × T 2 , which only intersects each fiber once. In particular, no isomorphism Ψ of fiber bundles (equipped with connections) over T 2 can exist between (V, T 2 , M, H 0 ) and (M × T 2 , T 2 , M, {0} ⊕ T T 2 ): indeed it should send F 0 to F 1 , which is not possible because their leaves intersects the fibers (of respectively π and p) a different number of times.

We point out that the motivation behind Definition 8.6 doesn't only consist in Proposition 8.7 above. Indeed, we now show that the condition d 0 A = 0 above, while being general enough to be satisfied by a class of contact structures strictly larger than those given by the construction in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF], is also strong enough to ensure some nice properties from the points of view of contact deformations, weak fillability and adapted open book decompositions.

We start by noticing that each Bourgeois contact structure η is in particular a contact deformation of the underlying flat contact bundle η 0 . Indeed, we have the natural path of contact bundle structures (η t ) t∈[0,1] that is given by the potential A t := t • A with respect to H 0 , where A is the potential of η. This has the wanted starting point and ending point and gives a contact structure η t for t > 0, according to Proposition 8.4, because

d 0 A t = t • d 0 A is zero and [A t , A t ] = t 2 [A, A] is negatively transverse to ξ = η ∩ T M for t > 0.
This property is a direct generalization of the fact that the examples in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] are actually all contact deformations of the trivial flat contact bundle on M × T 2 (see Remark 7.6).

The study of weak fillability of Bourgeois contact structures is postponed to Section 9.1.1, where Proposition 9.1 states that if (M, ξ) is weakly fillable then a Bourgeois contact structure η on the flat contact bundle (M × T 2 , T 2 , M, ξ ⊕ T T 2 ) is weakly fillable too (see also [MNW13, Example 1.1] and [LMN18, Theorem A.a], that both deal with the particular case of the contact structures obtained as in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF]). This stability of weak fillability is also true in a more general case, as stated in Proposition 9.5.

As far as adapted open book decompositions are concerned, we have the following property: given a Bourgeois contact structure η on the flat contact bundle (V, Σ, M, η 0 ), we can "naturally" associate to each point b of Σ an open book decomposition of the fiber M b supporting the contact structure ξ b . In order to give a precise statement, let's introduce some notations. Consider a smooth contact bundle η on X → Y , where X is not assumed to be closed. Denote by Λ the space of maps Φ : X → R 2 such that, for each y ∈ Y : i. the restriction φ y :

= Φ| π -1 (y) : π -1 (y) → R 2 is transverse to {0} ⊂ R 2 , ii. the map φy φy : π -1 (y) \ φ -1 y (0) → S 1 is a fibration, iii. (φ -1 y (0), φy φy ),
which is an open book decomposition of π -1 (y) according to points i., ii. above and to Lemma 6.7, is moreover adapted to the contact structure η ∩ T π -1 (y) .

Notice that this space Λ comes endowed with a natural C ∞ -topology induced by that on the space of functions X → R 2 in which it is contained. Consider then the quotient Λ / ∼ of Λ by the relation ∼ defined as follows: Φ 1 , Φ 2 ∈ Λ are equivalent via ∼ if there is a positive function f : X → R such that Φ 2 = f Φ 1 . Notice that Λ / ∼ inherits a natural topology as quotient of the topological space Λ. We then call smooth Y -family of open 8.4. BOURGEOIS' CONTACT STRUCTURES books in X (adjusted to η) each element of Λ / ∼. Remark also that if we have a contact bundle η on a smooth fiber bundle π : X → Y and f : Z → Y is a smooth map, we can define the pullback contact bundle f * η on the pullback bundle

f * X := {(z, x) ∈ Z × X| f (z) = π(x)} X pr X as the vector sub-bundle {W ∈ T (f * X)| d(pr X )(W ) ∈ η} of T (f * X)
, where pr X : Z × X → X is the projections on the second factor. This is indeed a contact bundle on f * X → Z because its trace on each fiber

(pr Z ) -1 (z) ∩ f * X = {z} × π -1 f (z) of f * X → Z is exactly {0} ⊕ η f (z)
; here, pr Z : Z × X → Z is the projections on the first factor. We now go back to the specific case of Bourgeois contact structure η on a flat contact bundle (V, Σ, M, η 0 ). Denote, for all b ∈ Σ, (M b , ξ b ) the contact fiber over b, i.e. M b := π -1 (b) and ξ b := η 0 ∩ T M b , where π : V → M is the given fiber bundle. We then say that a couple (K, ϕ) is a fiber adapted open book if there is a point b ∈ Σ such that (K, ϕ) is an open book decomposition of M b supporting ξ b . Denote lastly by pr : F Σ → Σ the frame tangent bundle of Σ, i.e. the (principal) bundle over Σ with fiber over b ∈ Σ given by the set of all oriented basis of T b Σ. Then, we can state the following result: Proposition 8.9. Given a Bourgeois contact structure η on the flat contact bundle (V, Σ, M, η 0 ), there is a map

(f t ) t∈[0,1] of the fiber M b , starting at φ 0 = Id, such that K 1 = f 1 (K 0 ), ϕ 1 = ϕ 0 • f -1 1 and f t (K 0 ) , ϕ 0 • f -1 t is

BOURGEOIS' CONTACT STRUCTURES

We lastly point out another property of Bourgeois contact structures in the case of trivial flat contact bundle: on the flat contact bundle (M × T 2 , T 2 , M, ξ ⊕ T T 2 ), where ξ is contact on M , Bourgeois contact structures can also be studied from the point of view of the natural T 2 -action on the principal bundle, in the spirit of Chapter 7. Let's give some precise statements.

We start by noticing that the potential A of a contact bundle η, with respect to the natural flat connection {0} ⊕ T T 2 ⊂ T M × T 2 on π : M × T 2 → T 2 , can actually be seen as a 1-form defined on T 2 and with values in the contact vector fields of (M, ξ), thanks to the canonical identification of each fiber of π with M . Then, we have a natural inclusion map i :

   T 2 -invariant contact bundles on (M × T 2 , T 2 , M, ξ ⊕ T T 2 ), defining a contact structure on M × T 2    → Bourgeois contact structures .
Indeed, a direct computation shows that each T 2 -invariant contact bundle η has potential A invariant under the T 2 -action, which hence satisfies d 0 A = 0.

The other way around, we have the following:

Proposition 8.11. Let η be a Bourgeois contact structure on the flat contact bundle (M × T 2 , T 2 , M, ξ ⊕ T T 2 ) and denote by A its potential. The average A of A, taken under the natural T 2 -action, is the potential of a T 2 -invariant Bourgeois contact structure η on

(M × T 2 , T 2 , M, ξ ⊕ T T 2 ).
In particular, taking the average of the potential gives a well defined map

F : Bourgeois contact structures →    T 2 -invariant contact bundles on (M × T 2 , T 2 , M, ξ ⊕ T T 2 ), defining a contact structure on M × T 2    , which satisfies F • i = Id.
Proof (Proposition 8.11). It is clear that the average A satisfies d 0 A = 0, because it is invariant under the T 2 -action. By Proposition 8.4, what we need to show is then that [A, A] is with values in the negative contact vector fields for (M, ξ). Let's start by analyzing this condition more explicitly. Write A = -X ⊗ dx -Y ⊗ dy, with (x, y) coordinates on T 2 = ( R /2πZ) 2 and X, Y a T 2 -family of vector fields on M parametrized smoothly by (x, y). Then, we know that [X, Y ] is negatively transverse to ξ everywhere on M , for all (x, y) ∈ T 2 , and we want to show that their averages X, Y are such that [X, Y ] is also negatively transverse to ξ everywhere on M . Notice that, if Z, W are T 2 -parametric vector fields on M , it is not true in general that the T 2 -average of [Z, W ] is equal to the Lie bracket of the averages of Z and W . This being said, what we want to show here is that this is actually true for X, Y , thanks to the additional condition d 0 A = 0. Now, X, Y can be seen as smooth functions from T 2 to the space of vector field on M , which has a natural structure of vector space over R. As such, they both admit a complex Fourier series expansion

X = m,n∈Z e i(mx+ny) X m,n and Y = h,k∈Z e i(hx+ky) Y h,k , (8.1)
where, for all m, n, h, k ∈ Z, X m,n , Y h,k are complex vector fields on M , i.e. sections of the complexified tangent bundle T M ⊗ R C → M . Because X, Y are actually real, we have the following condition on the coefficients:

X m,n = X -m,-n and Y h,k = Y -h,-k , for all m, n, h, k ∈ Z, (8.2)
where X m,n denotes here the conjugated of X m,n and similarly for Y h,k . What's more, the condition d 0 A = 0 also gives some information on the Fourier coefficients. We have indeed the following: Claim 8.12.

d 0 A = 0 if and only if -∂ ∂x Y + ∂ ∂y X = 0
Proof (Claim 8.12). We can explicitly compute

d 0 A (∂ x , ∂ y ) = ∂ x , A ∂y -∂ y , A ∂x -A [∂x,∂y] (i) = -∂ x , Y + ∂ y , X (ii) = - ∂ ∂x Y + ∂ ∂y X ,
where (i) comes from the fact that ∂ x and ∂ y commute and (ii) follows from the expression in coordinates of the Lie bracket.

A straightforward computation shows that Claim 8.12 is equivalent to the following condition:

mY m,n = nX m,n for all m, n ∈ Z . ( 8.3)
Notice now that the averages of X and Y are, respectively, X 0,0 and Y 0,0 , which are then in particular real vector fields on M . To avoid confusion with the conjugation, we will hence drop the notation X and Y for the averages and just denote them by X 0,0 and Y 0,0 instead. Let [. , .] C be the Lie bracket induced on the complex vector space of the sections of T M ⊗ C → M by the Lie bracket [. , .] on the space of tangent vector fields on M . We can then compute:

[X, Y ] =   m,n∈Z e i(mx+ny) X m,n , h,k∈Z e i(hx+ky) Y h,k   C (a) = m,n∈Z h,k∈Z e i[(m+h)x+(n+k)y] [X m,n , Y h,k ] C (b) = r,s∈Z e i(rx+sy)   m,n∈Z [X m,n , Y r-m,s-n ] C   ,
where the equality (a) comes from the fact that the Lie bracket is C-bilinear and is taken on each fiber M × {pt} of M × T 2 → T 2 (where the exponentials are constant), and the equality (b) comes from replacing r = m + h and s = n + k.

The above computation shows that [X, Y ] has Fourier coefficients

[X, Y ] r,s = m,n∈Z [X m,n , Y r-m,s-n ] C (8.4)
for r, s ∈ Z.

BRANCHED COVERINGS AS CONTACT DEFORMATIONS

In particular, its average is given by

[X, Y ] 0,0 = m,n∈Z [X m,n , Y -m,-n ] C (a) = [X 0,0 , Y 0,0 ] + m,n∈Z\{0} [X m,n , Y -m,-n ] C (b) = [X 0,0 , Y 0,0 ] + m,n∈Z\{0} m n Y m,n , Y m,n C (c) = [X 0,0 , Y 0,0 ] -2i m,n∈Z\{0} m n [ Y m,n , Y m,n ] (d) = [X 0,0 , Y 0,0 ] ,
where Because [X, Y ] is negatively transverse to ξ everywhere on M for all (x, y) ∈ T 2 , its average [X, Y ] 0,0 = [X 0,0 , Y 0,0 ] is also negatively transverse to ξ everywhere on M . Now, notice that the average A of A is exactly A = -X 0,0 ⊗ dx -Y 0,0 ⊗ dy. Then, the fact that [X 0,0 , Y 0,0 ] is everywhere negatively transverse to ξ means that [A, A] is a 1-form on T 2 with values in the negative contact vector fields for (M, ξ). This concludes the proof of Proposition 8.11. Remark 8.13. In analogy with the case of Bourgeois contact structures, we could have also considered, on a flat contact fiber bundle (V, Σ, M, η 0 ), the class of contact structures η on V given by a contact fiber bundle structures having potential A with [A, A] = 0.

For such an η, Proposition 8.4 tells us that d 0 A is with values in the negative contact vector fields of the fibers. Such a condition, though, is not compatible with the fact that the surface Σ is closed. Indeed, by explicit computations (analogous to those in the proof of Lemma 9.2 in the following) it can be proved that this condition on d 0 A implies the existence of an exact volume form on Σ. Now, the latter can't exist if Σ is closed, according to Stoke's theorem.

Moreover, even if we allow Σ to have boundary, we do not recover all the informations on the fiber that we have with a Bourgeois contact structure. More precisely, we can't recover in general an (isotopy class of ) open book decomposition supporting the contact structure on the fiber. For instance, consider on the flat contact bundle (M ×Σ, Σ, M, ξ M ⊕T Σ) the contact fiber bundle structure η = ker (α + λ), with ξ M = ker α and dλ symplectic on Σ (that hence has non-empty boundary). Then, an explicit computation shows that A = -R α ⊗ λ, where R α is the Reeb vector field of α. In particular, [A, A] = 0 and d 0 A = -R α ⊗ dλ, and we do not have any way to recover an (isotopy class of ) open book decomposition on M from A.

Branched coverings as contact deformations

We show in this section that the class of contact fiber bundles that are contact deformations of a flat contact fiber bundle is stable under the operation of contact branched CHAPTER 8. BOURGEOIS' CONSTRUCTION REVISITED coverings. More precisely we have the following: Proposition 8.14. Let (V, Σ, M, η 0 ) be a flat contact fiber bundle and p : Σ → Σ a branched covering map that lifts to a branched covering map p : V → V . Consider now the pull-back flat contact fiber bundle ( V , Σ, M, η 0 ) induced by p, i.e. η 0 := p * η 0 . If η is a contact deformation of η 0 , then there is a contact branched covering η of η to V that is a contact deformation of η 0 .

Proof. This follows essentially from the explicit formula for the contact branched covering in the proof of Lemma 5.10.

More precisely, take a smooth family of 1-forms (α t ) t∈[0,1] on V defining the family η t which, by definition of contact deformation, interpolates between η = ker(α 1 ) and η 0 = ker(α 0 ) in such a way that η t is contact for t > 0 and for all t the fibers of the fibration π : V → Σ are contact submanifolds, with induced contact structure independent of t.

According to the proof of Lemma 5.10, we can chose η on V to be the kernel of α = p * α 1 + g(r)r 2 dθ, with the same notations as in that proof, using the particular choice of closed form γ = dθ as connection on the trivial unit normal bundle of M in V . Define now ( α t ) t∈[0,1] by α t = p * α t + t g(r)r 2 dθ. We remark that ker( α 1 ) = η and that ker( α 0 ) = η 0 .

We then want to show that this path of 1-forms α t is the path giving η as contact deformation of η 0 . Now, α t gives on each fiber a contact structure independent of t, hence the only thing left to show is that α t actually defines a contact structure for t > 0. We can explicitly compute α t ∧ d α n t = C n+1 p * (α t ∧ dα n t ) + + C n t (rg (r) + 2g (r)) p * α t ∧ dα n-1 t ∧ rdr ∧ dθ + C n t g(r) r 2 dθ ∧ p * dα n t .

Notice that p * α t ∧ dα n-1 t ∧ rdr ∧ dθ = p * α 1 ∧ dα n-1 1 ∧ rdr ∧ dθ , because α t and α 1 induce the same contact form on each fiber. In particular, p * α t ∧ dα n-1 t ∧ rdr ∧ dθ is bounded below by a positive volume form independent of t.

We can now use the same argument as in the proof of Lemma 5.10 to conclude that the whole sum is positive everywhere for every t > 0.

VIRTUALLY OVERTWISTED CONTACT STRUCTURES IN HIGH DIMENSIONS

-φ 2 are the contact Hamiltonians associated, respectively, to -(A ) ∂x = -A ∂x and -(A ) ∂y = -A ∂y . Hence, up to isotopy we have that η = ker(α ), with > 0 that will be chosen very small in the following.

According to the weak fillability hypothesis for M (and the discussion after Definition 2.17), we have β ∧ (ω M + τ dβ) n-1 > 0 for all τ ≥ 0 on M , where ω M denotes the restriction of ω to M = ∂X. We want to verify that, for > 0 small enough, we also have α ∧ (ω M + ω T 2 + τ dα ) n > 0 for all τ ≥ 0 on M × T 2 . Lemma 9.2. Let Ω be an arbitrary volume form on M × T 2 . We then have

α ∧ (ω M + ω T 2 + τ dα ) n = nβ ∧ (ω M + τ dβ) n-1 ∧ ω T 2 + + 2 τ n α 1 ∧ dα n 1 + 2 h Ω ,
where h is independent of and polynomial in τ , with deg τ (h) ≤ n -1.

The proof of this lemma is postponed.

Denote now f and g the functions defined by the equalities f Ω = nβ∧(ω M +τ dβ) n-1 ∧ ω T 2 , g Ω = τ n α 1 ∧ dα n 1 . Then, what we want to show is that f + 2 (g + h) > 0 on all M × T 2 .

We remark that for each p ∈ M × T 2 , f (p), g(p) and h(p) are polynomials in τ , by explicit computation in the case of f and g, and by Lemma 9.2 in the case of h. Moreover, we have the following properties: for each p ∈ M × T 2 , (a) f (p) > 0, because (X, ω) weakly fills (M, ξ);

(b) g(p) > 0, because α 1 is a contact form for the Bourgeois contact structure η;

(c) h(p) has degree in τ strictly less than g(p), by Lemma 9.2. We now use the following lemma, whose proof is easy (and omitted): Lemma 9.3. Let P 1 , P 2 ∈ R [τ ] of degree n, with P 1 (τ ) > 0 ∀ τ ≥ 0 and with P 2 with positive leading coefficient. Then ∃ 0 > 0 such that ∀ 0 < < 0 , P 1 + 2 P 2 > 0 on R ≥0 . Now, for each p ∈ M × T 2 , if we define P 1 = f (p) and P 2 = g(p) + h(p), Lemma 9.3 gives an p > 0 such that f (p) + p (g + h)(p) > 0. Then, the compactness of M × T 2 guarantees that there is > 0 independent of p such that f + (g + h) > 0, as wanted.

We now prove Lemma 9.2 used above: Proof (Lemma 9.2). We can compute dα = dβ + dφ 1 ∧ dθ 1 -dφ 2 ∧ dθ 2 -∂φ 1 ∂θ 2 + ∂φ 2 ∂θ 1 dθ 1 ∧ dθ 2 . (9.1) Claim 9.4. d 0 A = 0 if and only if ∂φ1 ∂θ2 + ∂φ2 ∂θ1 = 0 .

Proof (Claim 9.4). We have A = -X ⊗ dθ 1 -Y ⊗ dθ 2 , with X, Y the contact vector fields on (M, ξ) with contact Hamiltonians φ 1 , -φ 2 via β, respectively. By Claim 8.12, we have that d 0 A = 0 if and only if -∂ ∂θ1 Y + ∂ ∂θ2 X. Now, because -∂ ∂θ1 Y + ∂ ∂θ2 X is a contact vector field on each fiber (M, ξ), it is zero if and only if its contact Hamiltonian via β is zero, i.e. if and only if

0 = - ∂ ∂θ 1 β (Y ) + ∂ ∂θ 2 β (X) = ∂ ∂θ 1 φ 2 + ∂ ∂θ 2 φ 1 .
Because η is a Bourgeois contact structure, Equation (9.1) becomes just

dα = dβ + dφ 1 ∧ dθ 1 -dφ 2 ∧ dθ 2 .
For dimensional reasons, we now get

ω | T M + ω T 2 +τ dα ) n = = n ω | T M + τ dβ n-1 ∧ (ω T 2 + τ dφ 1 ∧ dθ 1 -τ dφ 2 ∧ dθ 2 ) + + τ 2 2 n(n -1) ω | T M + τ dβ n-2 ∧ dφ 1 ∧ dφ 2 ∧ dθ 1 ∧ dθ 2 .
Hence, we compute

α ∧ (ω | T M + ω T 2 + τ dα ) n = = n(β + φ 1 dθ 1 -φ 2 dθ 2 ) ∧ (ω M + τ dβ) n-1 ∧ ∧ (ω T 2 + τ dφ 1 ∧ dθ 1 -τ dφ 2 ∧ dθ 2 )+ + τ 2 2 n(n -1) (β + φ 1 dθ 1 -φ 2 dθ 2 ) ∧ ω | T M + τ dβ n-2 ∧ ∧ dφ 1 ∧ dφ 2 ∧ dθ 1 ∧ dθ 2 = = nβ ∧ (ω M + τ dβ) n-1 ∧ ω T 2
+ nτ 2 (φ 1 dφ 2 -φ 2 dφ 1 ) ∧ (ω M + τ dβ) n-1 ∧ dθ 1 ∧ dθ 2 + τ 2 2 n(n -1)β ∧ ω | T M + τ dβ n-2 ∧ dφ 1 ∧ dφ 2 ∧ dθ 1 ∧ dθ 2 . (9.2) Now, a similar explicit computation (using again Claim 9.4) shows that α 1 ∧ dα n 1 = n (φ 1 dφ 2 -φ 2 dφ 1 ) ∧ dβ n-1 ∧ dθ 1 ∧ dθ 2 + + n(n -1)β ∧ dβ n-2 ∧ dφ 1 ∧ dφ 2 ∧ dθ 1 ∧ dθ 2 , so that the second and third term in the right hand side of the last equality in Equation (9.2) give 2 τ n α 1 ∧ dα n-1 1 + 2 h Ω, where h is as in the statement. This conclude the proof of Lemma 9.2.

Even if we will not use it in the following, we remark that the local nature of the condition d 0 A = 0 and of all the computations in the above proof actually gives the following more general result: Proposition 9.5. Let (M 2n-1 , ξ) be a contact manifold weakly filled by (X 2n , ω). Suppose that a representation ρ of π 1 (Σ g ) in the group of symplectomorphisms of (X, ω) gives, by restriction to the boundary, a representation ρ of π 1 (Σ g ) in the group of contactomorphisms of (M, ξ). Let also η be a Bourgeois contact structure on the flat contact bundle (M × ρ Σ g , Σ g , M, η 0 ) (as constructed in Section 8.2). Then, there is a symplectic form Ω on X × ρ Σ g that weakly fills η on M × ρ Σ g . More precisely, if R 2 → Σ g denotes the universal covering map, Ω can be chosen to be the symplectic form on X × ρ Σ g induced by ω + ω g R 2 on X × R 2 and where ω g R 2 is a symplectic form on R 2 invariant by the action of π 1 (Σ g ) on R 2 by deck transformations.

Consider now η := (p, Id) * η on M × T 2 and ζ g := (p, Id) * η g on M × Σ g . Notice that the restriction of ζ g to the upstairs branching locus of (Id, π) : M × Σ g → M × T 2 is exactly ξ.

We claim moreover that (M × Σ g , ζ g ) is a branched contact covering of (M × T 2 , η). Indeed, we can easily see that ζ g is a contact deformation of the confoliation (Id, π) * η on M ×Σ g as follows. If (η t g ) t∈[0,1] is a path of confoliations adapted to the upstairs branching locus of (Id, π) : M × Σ g → M × T 2 starting at η 0 g = (Id, π) * η, ending at η 1 g = η g and such that η t g is contact for t ∈ (0, 1], then (p, Id) * η t g is the path of confoliations on M ×Σ g which shows that ζ g is a contact deformation of (Id, π) * η.

We now use the following result, whose proof is postponed:

Lemma 9.9 ([NP10]). For k ∈ N >1 , let π k : V k → V 2n+1 be a branched covering map of branching index k. Suppose that all π k 's have same downstairs branching locus M and that the upstairs branching locus M k of π k is connected (in particular, π k | M k induces a diffeomorphism between M k and M ). Suppose also that there is a tubular neighborhood N := M × D 2 (where D is the 2-disk centered at 0 and of radius 1) of the downstairs branching locus M over which all the π k 's are trivialized at the same time, i.e. such that π k : M × D 2 → M × D 2 is just (p, z) → (p, z k ).

Let now η be a contact structure on V inducing a contact structure ξ on M and consider a contact branched covering η k of η on the total space of V k . Then, there is > 0 such that, for all k ≥ 2, the upstairs branching locus ( M k , ξ k = ker π * k α) π k (M, ξ = ker α) has a neighborhood of the form M × D 2 √ k , ker α + r 2 dϕ inside ( V k , η k ) (here, by D 2 r we denote the open disk centered in 0 and of radius r inside R 2 ).

In our situation, we have a sequence of branched coverings M × Σ g of M × T 2 , together with contact branched coverings ζ g of η, as in the hypothesis of Lemma 9.9. Then, a direct application of such lemma tells that each of the fibers (M , ξ = ker(α)) that belong to the (upstairs) branching set has a contact neighborhood of the form

M × D 2
Rg , ker α + r 2 dθ , with R g → +∞ for g → +∞. Because ξ on M is overtwisted, this implies, according to [CMP15, Theorem 3.1], that if g is big enough then the upstairs branching set has an overtwisted neighborhood, so that (M × Σ g , ζ) is also overtwisted. In other words, we just proved that, for g big enough, (M × Σ g , η g ) has a finite cover which is overtwisted.

Let's now prove Lemma 9.9 used above: Proof (Lemma 9.9). This result is a rephrasing of the discussion made by Niederkrüger and Presas in [NP10, page 724], using our definition of contact branched coverings. Here's a detailed proof.

We start by remarking that, by the uniqueness of the branched contact coverings up to isotopy (Proposition 5.8), it is enough to show that the result is true for a particular choice of contact branched coverings η k 's. Indeed, if η k contains a contact neighborhood of the upstairs branching locus (M, ξ = ker α) as in the statement, so does any contact structure isotopic to it via an isotopy fixing M . Suppose for the moment that we are in the case where η = ker α + r 2 dϕ on a subneighborhood M × D The neighborhood N = M × D 2 induces (via the choice of a basis of T 0 D 2 ) a normal framing of M inside V . Now, the conformal symplectic normal bundle of the downstairs branching locus (M, ξ) in (V, η) is trivial as symplectic bundle (because M has a trivial normal bundle inside V ) and admits moreover a trivialization (as a symplectic bundle) that induces a normal framing of M inside V which is homotopic to that induced by N . The contact neighborhood theorem [Gei08, Theorem 2.5.15] gives then a contact neighborhood M × D 2 δ , ker α + r 2 dϕ that induces (via the choice of a basis of T 0 D 2 δ ) a normal framing of M inside V that is also homotopic to that induced by N . Hence, by the uniqueness (up to isotopy) of smooth tubular neighborhoods inducing a fixed homotopy class of normal framing (see for instance [Lan99, Theorem 6.2]), there is an isotopy of V that fixes M pointwise and puts η in the wanted normal form ker(α+r 2 dϕ) on a sub-neighborhood M × D 2 of N = M × D 2 , for a certain 0 < < δ; notice that this does not depend on k (indeed, it depends only on N , which does not depend on k). In other words, we have a smooth family of contact structures (η t ) t∈[0,1] , starting at η 0 = η, ending at a certain η 1 that has the wanted normal form on M ×D 2 , and such that each η t restricts to η = ker α on M = M × {0}. Then, by Lemma 5.10 (with K = [0, 1]), we have a smooth family of contactizations η k,t of π * k η t . Now, the above discussion shows that, for each k ≥ 2, the upstairs branching locus ( M k , ker(π * k α))

π k = (M, ξ) has a neighborhood M × D 2
√ k , ker α + r 2 ϕ inside ( V k , η k,1 ). Then, because η k,0 and η k,1 are isotopic, the same is true inside ( V k , η k,0 ), as wanted.

We point out that taking g = 1 in the statement of Proposition 9.8, i.e. working directly on M × T 2 without taking a branched covering, is in general not enough to ensure the same conclusion. For instance, this follows from Section 9.2, where we will show that for each contact manifold (M 3 , ξ), with π 1 (M ) = {1}, there is an open book decomposition of M supporting ξ such that the construction in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF] yields a hypertight contact form α on M × T 2 . In particular, even if (M, ξ) is virtually overtwisted, with (M , ξ) an overtwisted finite cover, the pullback α of α to M × T 2 will still define a tight contact structure η = ker α. Indeed, if by contradiction η is overtwisted, according to [START_REF] Casals | Geometric criteria for overtwistedness[END_REF] and [START_REF] Albers | On the Weinstein conjecture in higher dimensions[END_REF], α admits a contractible Reeb orbit in M × T 2 , which then projects to a contractible Reeb orbit of α in M × T 2 , contradicting the hypertightness of α.

Notice also that we preferred to take a very big g in Proposition 9.8 in order not to enter too much in technical details and to keep the construction simple, but actually g = 2 is already enough: Observation 9.10 (Niederkrüger). If (M, ξ) is overtwisted, the contact manifold (M × Σ g , η g ) is overtwisted already for g = 2.

Proof (sketch). Take an arc γ on T 2 going from one (downstairs) branching point of the cover Σ 2 → T 2 to the other, and such that it is radial in the local model (given by Definition 5.1) around the two branching points, in such a way that its double cover δ in Σ 2 is a smooth closed curve. Denote also p ∈ Σ 2 one of the two upstairs branching points, and see δ as a loop based at p.

Then, the monodromy of the contact fiber bundle M × Σ 2 → Σ 2 over δ is trivial. Indeed, as the proof of Lemma 5.10 shows, the contact branched covering η 2 of a Bourgeois contact structure (M × T 2 , T 2 , M, η = ker β) can be chosen to be defined by a form β on M × Σ 2 which is invariant under deck transformations of the branched covering π : M × Σ 2 → M × T 2 and C ∞ -close to π * β. Then, it can be shown that the monodromy of (M × Σ 2 , Σ 2 , M, η 2 ) over δ is obtained as the concatenation of the monodromy f γ of M × T 2 , T 2 , M, η = ker (β) over γ plus a C ∞ -little perturbation h, and the monodromy (f γ )

-1 over -γ plus the inverse h -1 of the same perturbation.

Then, using the techniques from [START_REF] Presas | A class of non-fillable contact structures[END_REF], we can find an embedded plastikstufe inside M × δ ⊂ M × T 2 . In practice, this PS is obtained by "moving around" an overtwisted disk in M ×{p} M via the monodromy of (M × Σ 2 , Σ 2 , M, η 2 ) along δ. This procedure actually gives an embedded PS because the monodromy along the loop δ is the identity as map M × {p} → M × {p}. At this point, [START_REF] Huang | On plastikstufe, bordered Legendrian open book and overtwisted contact structures[END_REF] tells us that each PS-overtwisted manifold is also overtwisted.

Bourgeois construction and Reeb dynamics

The main aim of this section is to give a proof of Théorème II.G stated in Chapter 1. Let's recall the statement:

Théorème II.G. Chaque 3-variété de contact fermée (M, ξ) avec π 1 (M ) = {1} peut être plongée avec fibré normal trivial dans une 5-variété de contact (V 5 , η) fermée (hyper)tendue.

In order to give a proof, in Section 9.2.1 we consider, starting from a contact manifold (M 2n-1 , ξ) and an open book (B, ϕ) adapted to ξ, a Bourgeois contact structure η on the flat contact bundle (M × T 2 , T 2 , M, ξ ⊕ T T 2 ), with fibers (M, ξ), which admits a contact form α with very controlled Reeb vector field. This η is actually one of the examples described in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF]. We then show that the Reeb dynamics of α on M × T 2 is strictly related to the Reeb dynamics on the binding B of the open book (B, ϕ). This will give a criterion for the existence of closed contractible Reeb orbits of α on M × T 2 .

Then, we show in Section 9.2.2 how to deduce Théorème II.G as a corollary of this study in the case of 3-dimensional M . Now, by the hypothesis that π 1 (M ) is not trivial, at least one of the generators a i , b i of π 1 (M, p) is non-trivial, let's say b g . Make then a plumbing of a positive Hopf band, as defined in Theorem 2.20, along a properly embedded arc δ 0 which is in the same class as β g in π 1 (Σ, ∂Σ), as shown in Figure 9.2. We then obtain an open book decomposition (K , ϕ ) for M , which still supports the original contact structure, according to Theorem 2.21. Notice that the page Σ is obtained, as embedded surface, from Σ and δ 0 by plumbing a positive Hopf band to Σ. We recall that this means that Σ = Σ ∪ A where A is an annulus in M such that 1. the intersection A ∩ Σ is a tubular neighborhood of δ 0 ,

Bourgeois contact structures and contractible Reeb orbits

Figure 1

 1 Figure 1.1: Découpage, en rouge, sur le tore {θ 0 } × T 2 (t,s) .

  Figure 3.1: D f W and D ψ-c W inside W × R s and the vector field Z D , transverse to both; W -:= D f W ∩ {s = -1} that will appear in Step 1 is also represented. As we would like to prove Théorème I.A using the equation f instead of ψ -c, we need the following: Lemma 3.8. If the conclusion of Théorème I.A holds with the special choice of equation f for D f W ⊂ F × C × R, then it holds also for D ψ-c W defined by ψ -c (i.e. as in the statement of Théorème I.A).

  is the induced map and PD(α) denotes the Poincaré dual of the homology class α ∈ H * (V ; Z). Remark 4.6. This result is not in contradiction with Massot, Niederkrüger and Wendl [MNW13, Theorem 9.5], where the authors prove that the contact structures before and after a full Lutz-Mori twist (as defined in [MNW13, Section 9.1]) are homotopic through almost contact structures, hence have the same Chern classes. Indeed, the result ξ of a full Lutz-Mori twist can be interpreted as a couple of successive

  as we expected from [MNW13, Theorem 9.5].

  U and E | U over U admit complex sub-bundles F, L ⊂ E and F , L ⊂ E such that E| U = F ⊕ L and E | U = F ⊕ L and satisfying the following conditions: (a) ψ • s j = s j over U \ O for all j = 1, . . . , r -1; (b) L, L have complex rank 1, while F, F have complex rank r -1 and are trivialized by two ordered sets of everywhere C-linearly independent sections (s 1 , . . . , s r-1 ) and s 1 , . . . , s r-1 of E| U and E | U ; (c) there are two additional sections s r , s r of E| U and E | U respectively, with image contained in L and L and such that s r : U → E| U intersects transversely F and s r : U → E | U intersects transversely F (here F and F are seen here as submanifolds of E| U and E | U ); (d) Z := s -1 r (F ) and Z := (s r ) -1 (F ), which are oriented smooth manifolds of U by Hypothesis 2c above, are actually compactly contained in O.

  we can write in a more explicit form (i) α = e m i=1 ti cos (ψ (r)) dθ 0 + m i=1 e -ti dθ i + sin (ψ (r)) dϕ , (ii) α = e m i=1 ti cos (φ (r)) dθ 0 + m i=1 e -ti dθ i + sin (φ (r)) dϕ . Take now the following 2m R-linearly independent sections of the pullback of ξ and ξ to R m × R m+1 × D 2 : for i = 1, . . . , m, • s i := ∂ ti and r i := e -m j=1 tj cos (ψ (r)) ∂ θ0 -e ti ∂ θi + sin (ψ (r)) ∂ ϕ for the pullback of ξ, • s i := ∂ ti and r i := e -m j=1 tj cos (φ (r)) ∂ θ0 -e ti ∂ θi + sin (φ (r)) ∂ ϕ for the pullback of ξ .

  We start by recalling another point of view on open book decompositions in terms of functions with values in R 2 . This is presented, for instance, in [Lut79, Page 289] and [Gir17, Page 11].

  denotes the natural flat connection form dθ 1 ⊗ e 1 + dθ 2 ⊗ e 2 on M × T 2 , the construction of open book preceding the statement of Theorem 7.2 gives back the original open book (B, ϕ) and contact structure ξ on M .

  Definition 8.1. [Ler04] A contact fiber bundle is a cooriented hyperplane field η on V such that for each fiber M b of π the intersection ξ b := η ∩ T M b is a positive contact structure on M b .

  * ) comes from the definition of d 0 A, [A, A] and from the fact that H 0 flat means [ X, Y ] = [X, Y ] for all X, Y vector fields on Σ.

Ψ

  η : F Σ → {fiber adapted open book} satisfying the following properties: i. Ψ η sends, for all b ∈ Σ, each positive basis of T b Σ to an open book decomposition of M b adapted to ξ b ; ii. for each smooth path γ : [0, 1] → F Σ, the composition Ψ η • γ : [0, 1] → {fiber adapted open books} describes a smooth [0, 1]-family of open books in γ * pr * V adjusted to γ * pr * η. (γ * pr * V, γ * pr * η) (pr * V, pr * η) From the above result, we can deduce a more precise version of Proposition II.D stated in Section 1.2: Corollary 8.10. The map Ψ η in Proposition 8.9 induces a well defined ψ η : Σ → {fiber adapted open books} / ∼ , where (K 0 , ϕ 0 ) ∼ (K 1 , ϕ 1 ) if they are both adapted open books on a same fiber (M b , ξ b ) and there is an isotopy

  an open book of M b adapted to ξ b . In other words, η uniquely determines an isotopy class of adapted open book decompositions for each fiber (M b , ξ b ) of (V, Σ, M, η 0 ). Moreover, if η = ker α is the Bourgeois contact structure on (M ×T 2 , T 2 , M, ξ⊕T T 2 ) given by Theorem 7.5 starting from an open book (B, ϕ) for (M, ξ), then the corresponding map ψ η sends each b ∈ T 2 to an isotopy class of adapted open books on (M b , ξ b ) that (via the natural identification (M b , ξ b ) (M, ξ) given by the projection M ×T 2 → M ) corresponds to the isotopy class of the original open book (B, ϕ) on (M, ξ).

  2 of N = M × D 2 . Then, π * k η = ker α + kr 2k dϕ on a neighborhood M × D 2 k √ of the upstairs branching locus M = M inside V . Let now f k : (0, k √ ) → (0, ) be a smooth strictly increasing function such that f k (r) = r near 0 and f k (r) = r k near k √ . Then, we can construct a contact branched covering η k of η by replacing π * k η = ker α + kr 2k dϕ on the neighborhood M × D 2 k √ with 111 9.1. VIRTUALLY OVERTWISTED CONTACT STRUCTURES IN HIGH DIMENSIONS ker α + kf 2 k (r)dϕ . Notice that we have the following contactomorphism:M × D 2 k √ , ker α + kf 2 k (r)dϕ ∼ -→ M × D 2 √ k , ker α + r 2 dϕ (p, r, ϕ) → (p, √ kf k (r), ϕ) This means that ( M k , ker(π * k α)) π k (M, ξ) has a contact neighborhood of the form M × D 2 √ k , ker α + r 2 ϕ inside ( V k , η k), as wanted. It only remains to show that we can reduce to the case where η has the wanted form ker α + r 2 dϕ on a sub-neighborhood M × D 2 of N = M × D 2 , for a certain > 0 independent of k.

Figure 9 . 1 :

 91 Figure 9.1: Curves on a page Σ of (K, ϕ) which give the generators of the presentation in Equation (9.3).

Figure 9 . 2 :

 92 Figure 9.2: Stabilization arc δ 0 . Only a part of the surface is shown.

  Bourgeois et on démontre la Proposition II.D. Le Chapitre 9 contient ensuite deux applications des outils développés du Chapitre 6 jusqu'au 8. Plus précisement, elle contient l'étude de la remplissabilité faible des structures de contact de Bourgeois, i.e. les preuves de la Proposition II.C et du Théorème II.F, ainsi qu'une analyse de la dynamique du champ de Reeb des formes de contact [Bou02], i.e. les preuves du Théorème II.G et du Corollaire II.H.

	Chapter 2

  Y m,n and Y m,n denote respectively the real and imaginary part of Y m,n . Moreover, (a) comes from the fact that X m,n is zero if m = 0, n = 0 and Y -m,-n is zero if n = 0, m = 0 by Equation (8.3), (b) comes from Equations (8.2) and (8.3), (c) comes from the C-bilinearity of [. , .] C and the anti-symmetry of [. , .] and, finally, (d) comes from the fact that [X, Y ] 0,0 is a (real) tangent vector field, because average of [X, Y ], hence has zero imaginary part.

Acknowledgements

By Theorem A.4 and the identities above, we have that for all j = 1, . . . , r -1 Proof (lemma A.6). Because the transversality and the orientation preserving conditions are local, we can restrict our attention to an open set U on which there is an everywhere non-zero section s of L| U .

In particular, the r -tuple of sections (s 1 , . . . , s r-1 , s) trivializes E| U , i.e. the map U × C r → E| U given by (q, w) → r-1 i=1 w i s i (q) + w r s(q) is an isomorphism of complex vector bundles.

In this local trivialization, we can rewrite s r as s r : U → U × C r q → (q, 0, . . . , 0 r-1 times , v(q)) for a certain v : U → C. Also, the fact that s r is transverse to F at a certain point q ∈ U means that v : U → C is transverse to {0} ⊂ C at q . Moreover, if we denote by ν : U ×C r → C r the projection on the second factor, in the open set U the submanifold M = s -1 r (F ) (oriented according to Convention A.2) is actually equal to the oriented manifold (ν • s r )

-1 (0); in other words, remarking that ν •s r = (0, . . . , 0, v), we have that M ∩U = v -1 (0) as oriented manifolds.

Now, let's rewrite σ h using the chosen trivialization of E over U . Firstly, the map h becomes h : U × C r → U × C r , h(q, w) = (q, M (q) • w), where • denotes the matrix product, M : U → M r (C) is with values in the space M r (C) of square matrices r×r with complex coefficients and is defined by

with I r-1 the identity matrix of dimension (r -1) × (r -1). In other words, s h : U → U × M r (C) is given by s h (q) = (q, M (q)).

point, Z (h) ∩ U × C r-1 = (µ • Ψ • σ h ) -1 (0) as oriented manifolds. In other words, using the fact that

we get that σ h is transverse to Σ at (q, 0) ∈ U ×C r-1 if and only if v : U → C is transverse to {0} ⊂ C at q and that, if there is transversality everywhere, Z (h) ∩ U × C r-1 = v -1 (0) × {0} ⊂ U × C r-1 as oriented manifolds.

This concludes the proof of lemma A.6, because v is transverse to 0 ∈ C (as said in the beginning), hence σ h is transverse to Σ over U × C r-1 , and φ : V × CP r-1 → V clearly induces an orientation preserving diffeomorphism

53

As in Section 5.2, we have the following: Proposition 5.14. Let η be a hyperplane field adjusted to S. Contactizations of η exist and are all isotopic.

We point out that exactly as remarked in Fact 5.9 for the case of hyperplane fields adapted to a codimension 2 submanifold, η is automatically a confoliation on V . We will hence talk directly of confoliation adjusted to S in the following.

In order to prove Proposition 5.14, analogously as in the proof of Proposition 5.8, we use the following: Lemma 5.15. Given K a compact set and (η k ) k∈K a smooth family of confoliations on V adjusted to S, there is a smooth family of adjusted confoliations (η s k ) s∈[0,1], k∈K such that, for all k ∈ K, η 0 k = η k and η s k is contact for s ∈ (0, 1]. Moreover, if η k is contact for all k in a closed subset H ⊂ K, then the η s k can be chosen so that η s k = η k for all k ∈ H and s ∈ [0, 1].

Proof (Lemma 5.15). Take an open subset U of K which contains H and such that ξ k is contact for all k ∈ U . Consider a smooth cut-off function ρ : K → [0, 1], equal to 0 exactly on H and equal to 1 on the complement of U . Now take an auxiliary Riemannian metric on V and consider a connection form γ on the circle bundle π : S → M . The hypothesis of S being an oriented circle bundle (together with the fact that M, V are oriented by ξ, η respectively) tells that S is orientable as smooth manifold, and that its normal bundle N S inside V is trivial. In other words, S has a neighborhood of the form S × R inside V . Denote by r a coordinate on the R factor.

Take now a non-increasing cut-off smooth function g = g(r) which is 1 near r = 0 and vanishes for r > 1. Consider then a smooth K-family of 1-forms α k defining ξ k and let α s k := α k + s ρ(k)g(r)rγ .

Here is a positive real constant which will be chosen very small later. We can in particular set once and for all < 1. We claim that α s k defines a contact structure ξ s k on V for all s > 0. Indeed,

where vol is the Riemannian volume form on V and h is a function of p ∈ V , k ∈ K, s ∈ [0, 1], ∈ R >0 , and is polynomial in . At this point, the same arguments as in the proof of Lemma 5.10 show that, for small enough, the above (2n + 1)-form is a volume form on all V .

Let's now consider the case of fiber sums. Exactly as we did in the case of branched coverings, once showed that there is a natural (in the sense of Definition 5.2) notion of smooth fiber sum (W, H, ϕ), we want to define on it a natural confoliation adapted to H.

Let (V 2n+1 , η), (M 2n-1 , ξ) be two contact manifolds and consider two contact embeddings j 1 : (M, ξ) → (V, η) and j 2 : (M, ξ) → (V, η) such that there is a fiber-orientationreversing isomorphism Φ : N 1 → N 2 of vector bundles over M between the normal bundle N 1 of j 1 (M ) and the normal bundle N 2 of j 2 (M ). Again, let j := j 1 j 2 for notational convenience.

CHAPTER 5. CONTACT BRANCHED COVERINGS AND FIBER SUMS

Finally, let ζ be the smooth hyperplane (and confoliation) induced by η and ker γ respectively on the subsets V \ j (M ) and S N 1 × R * of W .

Then, (W, H, ϕ, ζ) is a confoliated fiber sum of V along j(M ) via Φ.

We also have the following uniqueness property:

Proposition 5.20. If (W, H, ϕ, ζ) and (W , H, ϕ, ζ) are two confoliated fiber sums of V along the contact embeddings j 1 , j 2 via Φ, then there is a diffeomorphism Θ : W → W such that:

We deduce Proposition 5.20 from the following: Lemma 5.21. If (W, H, ϕ) and (W , H, ϕ) are two fiber sums of V along j 1 j 2 via Φ, then there is Θ :

Proof (Proposition 5.20). This follows directly from Lemma 5.21, once noticed that the commutativity of the square in the lemma tells that Θ| W \H * (ϕ * η) = ϕ * η and that, according to Remark 5.17, ζ and ζ are determined, respectively, by ϕ * η and ϕ * η.

Proof (Lemma 5.21). As in the proof of Lemma 5.6, denote by ρ 1 ,ρ 2 ,ρ,N and by ρ 1 ,ρ 2 ,ρ,N the embeddings and normal bundle given by Definition 5.2 for respectively (W, H, ϕ) and (W , H, ϕ).

Again according to the uniqueness theorem for tubular neighborhoods (see [Lan99, Theorem 6.2]), there is an isotopy

and denote R t := ρ t 1 ρ t 2 . Then, as explained in the proof of [Gei08, Theorem 2.6.12], we can find a family of embeddings

such a H t corresponds in the proof of [Gei08, Theorem 2.6.12] to the ϕ t obtained from

Then, using a cut-off of contact Hamiltonians, we can then obtain a contact isotopy F t : (V, η) → (V, η) such that, for smaller neighborhoods U 1 , U 2 of the zero section of N 1 , N 2 compactly contained in N 1 , N 2 , the following diagram is commutative:

WEAK FILLINGS OF CONTACT BRANCHED COVERINGS

where ω ,V denotes the pullback of ω via the inclusion V → W , i.e.

Once fixed a volume form vol on V , using the fact that

where h is a smooth function of p ∈ V , and τ , which is moreover polynomial in and in τ , with deg τ h ≤ n.

Denote now by P 0 (τ ) and P 1 (τ ) the polynomials in τ , with coefficients in the ring of functions V → R, defined respectively by the identities

Similarly, denote by P 2 (τ, ) the polynomial in τ and given by P 2 (τ, ) = gr 2 h.

We then have the followings:

Lemma 5.25. For all τ ≥ 0, P 0 (τ ) is non-negative everywhere on V and strictly positive away from M .

Proof (Lemma 5.25). This simply follows from the fact that (W, ω) is a weak filling of (V, η) and that π| V is a branched covering with (upstairs) branching locus M .

Lemma 5.26. There are constants 0 < 0 < and r 0 > 0, such that P 1 (τ ) + P 2 (τ, ) > 0 on M r0 for all 0 ≤ < 0 and all τ ≥ 0.

This lemma will be proven after the end of the proof of Theorem 5.23. Notice that we allow = 0 in the statement of Lemma 5.26. Now, according to Lemmas 5.25 and 5.26, we have that α ∧( ω V + τ d α ) n is a positive volume form on M r0 for all 0 < < 0 and all τ ≥ 0. Remark that here has to be strictly positive.

At this point, we have the following result, whose proof is also postponed:

Lemma 5.27. There is 0 < 1 < 0 such that P 0 (τ ) + [P 1 (τ ) + P 2 (τ, )] > 0 on the complement of Mr 0/2 , for all 0 ≤ < 1 and all τ ≥ 0.

This concludes the proof of Theorem 5.23.

We now give a proof of Lemmas 5.26 and 5.27 above. They are corollaries of the following:

V × [0, ). Notice that we allow = 0 here. Consider the compact set K := M × {0} in S. If (q, 0) ∈ K, then P (q,0) • vol (q,0) = P 1 (τ ) q + P 2 (τ, 0) q • vol (q,0)

which is positive because the restriction of ω to X weakly dominates ξ on M = ∂X. In particular, for (q, 0) ∈ K, P (q,0) has positive leading coefficient and m (q,0) > 0.

Moreover, for each (q, 0) ∈ K,

We are then in the hypothesis of Lemma 5.28, which then tells us, by compactness of K, that there is a neighborhood U of K in S such that m s exists and is positive for all s ∈ U. This is exactly what we wanted because U contains an open set of the form {r < r 0 , < 0 } ⊂ S = V × [0, ).

Proof (Lemma 5.27). We use again Lemma 5.28. Here, S := M c r 0/2 × [0, 0 ), where M c r 0/2 is the complement of Mr 0/2 in V and r 0 , 0 are given by Lemma 5.26. Also, P : S × R ≥0 → R is here defined as P (p, ) (τ ) = P 0 (τ )| p + [P 1 (τ ) + P 2 (τ, )] (p) for (p, ) ∈ S. Notice that once again we allow = 0.

Then, if K := M c r 0/2 × {0}, P q,0 = P 1 (τ ) q for all (q, 0) ∈ K, hence it is positive by Lemma 5.25. In particular, P q,0 has positive leading coefficient and positive minimum m (q,0) for all (q, 0) ∈ K. Moreover, deg τ P (q,0) = n ≥ deg τ (P p, ), for all q, p ∈ M c r 0/2 and ∈ [0, 0 ).

We are then again in the hypothesis of Lemma 5.28, so that, by compactness of K, we can conclude that P (p, ) admits minimum m (p, ) , and that it is positive in a neighborhood of K, which is exactly what we wanted.

FROM CONTACT VECTOR FIELDS TO OPEN BOOKS

The rest of Section 6.1 is devoted to the proof of the above result, which is a more detailed version of the first part of Théorème II.E. To improve the readability, each lemma in this section will be proved right after the conclusion of the part of the proof in which it is used.

We fix for convenience a contact form α for ξ and we denote by f, g : M → R the smooth functions given by L X α = f α and L Y α = gα respectively: these functions exist because X and Y are contact vector fields. Of course, for the proof of point (c) we will need to change this α conveniently.

We also point out the following: Remark 6.2. For all θ ∈ S 1 , X θ , Y θ are contact vector fields and satisfy the identity

Proof (Remark 6.2). This follows from the fact that

α and from the fact that the Lie bracket is anti-symmetric and bilinear.

We may now proceed to prove Proposition 6.1.

Proof of point (a). This proof consists of the following two lemmas.

Lemma 6.3. α (X θ ) is somewhere zero.

The first one means exactly that Σ θ = {α (X θ ) = 0} is non-empty.

The second one tells that α (X θ ) : M → R is transverse to {0} ⊂ R, hence Σ θ is a smooth hypersurface, and that, more precisely, the contact vector field Y θ is transverse to Σ θ , i.e. the latter is ξ-convex.

We now prove Lemmas 6.3 and 6.4.

Proof (Lemma 6.3). Suppose by contradiction this is not the case, i.e. α (X θ ) > 0 without loss of generality. If we define β := 1 α(X θ ) • α, then X θ = R β . By Remark 6.2, we have β ([X θ , Y θ ]) = β ([X, Y ]) < 0. On the other hand, we also have

Here, for the equality (i) we used the fact that β(

) by the formula for the exterior derivative of differential forms, and for the equality (ii), we used that dβ(R β , .) = 0 and β(R β ) = 1. Now, β (Y θ ) is a function defined on a closed manifold, hence it has at least one critical point. This contradicts the fact that β ([X θ , Y θ ]) is everywhere negative.

Proof (Lemma 6.4). Using the formula for the exterior derivative, we compute

(6.1) Also, by Remark 6.2 there are f θ , g θ : M → R such that

CHAPTER 6. OPEN BOOKS AND CONTACT VECTOR FIELDS Now, evaluating these last two equations respectively on Y θ and X θ gives

Substituting inside Equation (6.1), we get dα

), which, using α (X θ ) = 0 (we are interested at points p ∈ Σ θ ), gives -dα

We point out a direct consequence of Lemma 6.4 and another lemma, which we will both need later:

In particular, along Σ θ ∩Σ θ+π/2 (which we will show below to be independent of θ and denote by K), we have both

Lemma 6.6. X θ is tangent to Σ θ . Moreover, it is transverse to ∂F θ = Σ θ ∩ Σ θ+π/2 and points outwards from F θ .

Proof (Lemma 6.6). X θ is tangent to Σ θ because d (α (X θ )) (X θ ) = 0; this last identity comes from the evaluation of the left identity in Equation (6.2) on X θ , at points p ∈ Σ θ . The second part of the statement follows from the fact that α (Y θ ) = 0 along ∂F θ = Σ θ ∩ Σ θ+π/2 (by definition of Σ θ+π/2 ), and that d (α (Y θ )) (X θ ) < 0 along ∂F θ by Corollary 6.5. Indeed, this means that X θ points in the region where α (Y θ ) < 0 along ∂F θ , being always tangent to Σ θ , i.e., by definition of F θ , that it points outwards from F θ along its boundary.

Proof of point (b).

We start by proving that the intersection Σ θ ∩ Σ θ is independent of θ, θ , provided that θ = θ mod π; for this, we will prove that this intersection coincides with ν -1 (0), where ν := (α (X) , α (Y )) :

On the other hand, if p ∈ Σ θ ∩ Σ θ , then, evaluating the same two equations at p, we deduce that ν(p) has to be proportional to (-sin (θ) , cos (θ) ) and to (-sin (θ ) , cos (θ ) ); because θ = θ mod π, this means that ν(p) = 0. In other words, Σ θ ∩ Σ θ ⊂ ν -1 (0) too. We can then denote Σ θ ∩ Σ θ by K in the following.

We now prove that K = Σ θ ∩Σ θ+π/2 is non-empty. Recall that Σ θ+π/2 = {α(Y θ ) = 0} and that Y θ is a contact vector field transverse to Σ θ ; in particular, K = {α(Y θ ) = 0} ∩ Σ θ ⊂ Σ θ is a dividing set for the characteristic foliation Σ θ (ξ). Now, according to [START_REF] Giroux | Convexité en topologie de contact[END_REF], dividing sets are non-empty 2-codimensional contact submanifolds.

Proof of point (c). Consider

Lemma 6.7. φ is transverse to the origin of R 2 and φ -1 (0) = K as subsets of M . Also, ϕ is a submersion and ϕ -1 (θ) = F -θ-π/2 as subsets of M . Moreover, ϕ -1 (θ) is cooriented by the vector Y -θ-π/2 and φ -1 (0), naturally oriented as boundary of ϕ -1 (θ) by definition of ϕ, is also cooriented by the ordered couple of vectors (Y, X).

FROM CONTACT VECTOR FIELDS TO OPEN BOOKS

The proof of this lemma is postponed. We now want to show that the couple (K, ϕ), which is an open book decomposition of M according to Lemma 6.7, is moreover adapted to ξ. Notice that this is enough in order to prove point (c) of Proposition 6.1, because the ϕ in point (c) is just obtained from the ϕ of Lemma 6.7 by post-composing with the rotation S 1 → S 1 of angleπ 2 : they hence have the same set of pages.

Consider on K, F θ the orientations such that φ -1 (0) = K, ϕ -1 (θ) = F -θ-π/2 as oriented manifolds. To show that (K, ϕ) is adapted to ξ, we then need to verify that ξ ∩ T K is a positive contact structure on K and that there is a contact form defining ξ whose differential is a positive symplectic form on each F θ . To prove this, we use the following result, whose proof is postponed: Lemma 6.8 (Giroux). Let (M 2n-1 , ξ) be a contact manifold. Suppose there are an open book decomposition

) and a contact form α defining ξ such that:

(i) ϕ restricted to N \K is the angular coordinate of the projection on the second factor

(ii) ξ induces a positive contact structure on each submanifold

(iii) dα induces a positive symplectic form on each fiber of ϕ |M \N .

Then, the open book decomposition (K, ϕ) is compatible with the contact structure ξ.

Thanks to the above result, the fact that (K, ϕ) supports ξ follows from the following two lemmas: Lemma 6.9. Let Ψ be the map defined by

, where ψ 1 Z denotes the time-1 flow of the vector field Z on M and D 2 δ is the 2-disk of radius δ in R 2 . Then for δ > 0 sufficiently small, we have the followings:

(i) Ψ is a diffeomorphism onto its image;

(ii) if we denote N := Ψ(K × D 2 δ ), then we have the following commutative diagram, where ν is the composition of the projection on D 2 δ \ {0} and the natural angle function

Lemma 6.10. Let N be the neighborhood of K given by Lemma 6.9. Then there is a contact form α defining ξ such that:

(i) α induces a positive contact structure on each submanifold K z of N ; CHAPTER 6. OPEN BOOKS AND CONTACT VECTOR FIELDS Definition 6.11. A smooth function Ψ : M → R 2 is transverse to OB if it is transverse to the binding and to the pages of OB .

Let K := Ψ -1 (0) and ϕ := Ψ/ Ψ :

) is an open book decomposition of M . Then, we say that Ψ defines the open book (K, ϕ).

Moreover, given two functions Ψ, Ψ : M → R 2 transverse to OB , we call them equivalent, and write Ψ ∼ Ψ , if there is a smooth positive function f :

As already stated for instance in [START_REF] Lutz | Sur la géométrie des structures de contact invariantes[END_REF]Page 289], open book decompositions are then equivalent to functions transverse to OB up to equivalence: Proposition 6.12. We have a bijection

Proof. The map in the statement is well defined because two functions M → R 2 transverse to OB in the same equivalence class clearly define the same open book on M . Let's show that the map is surjective. Fix a Riemannian metric on M . Then, given an open book decomposition (K, ϕ), consider the distance function ρ to the submanifold K. Then, (K, ϕ) is the image of the class of Ψ = (ρ cos ϕ, ρ sin ϕ).

Lastly, we show that the map in the statement is injective, i.e. that if Ψ and Ψ define the same open book (K, ϕ), then they are equivalent. Because Ψ/ Ψ = ϕ = Ψ / Ψ , a function f such that Ψ = f Ψ has to be equal to Ψ / Ψ on the complement of K. We then have to check that it smoothly extends at each point p ∈ K. Now, K admits a neighborhood K × D 2 and local coordinates (p, r, θ), with p ∈ K, r ∈ [0, 1), θ ∈ S 1 such that ϕ| K×D 2 (p, r, θ) = θ for r > 0. This implies that Ψ(p, r, θ) = λ(p, r, θ)(cos θ, sin θ) and Ψ (p, r, θ) = λ (p, r, θ)(cos θ, sin θ), where λ, λ : K × D 2 → R are non-negative functions, strictly positive away from K × {0}. Then, f = λ /λ as functions K × D 2 \ {0} → R. Because Ψ, Ψ are transverse to 0 ∈ R 2 , the limits of dλ/dr and dλ /dr for r → 0 exist, are both non-zero for all p ∈ K, θ ∈ S 1 and do not depend on θ. Hence, the function K → R defined by dλ/dr • (dλ /dr) -1 smoothly extends f = λ /λ over K × {0}.

Going back to the contact setting, we have the following converse to Proposition 6.1, which is also a more precise version of the second part of Théorème II.E: Proposition 6.13 (stated in [START_REF] Giroux | The existence problem, contact geometry in high dimensions[END_REF]). Suppose that (B, ϕ) an open book decomposition of M supporting ξ. Denote by α a contact form defining ξ and such that dα is symplectic on the fibers of ϕ. Then, there is a smooth function φ : M → R 2 defining (B, ϕ) and such that the contact vector fields X and Y , associated via α respectively to the contact Hamiltonians φ 1 and -φ 2 , have Lie bracket [X, Y ] negatively transverse to ξ.

Proof. This proof is strongly inspired from the computations in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF], which will be recalled in detail in Section 7.2.

Let φ = (φ 1 , φ 2 ) : M → R 2 be a smooth function defining (B, ϕ). Consider then > 0 such that α ∧ dα n-2 ∧ dφ 1 ∧ dφ 2 is positive on { φ < }. Such an exists because α induces a contact form on B = φ -1 (0).

define, for each v ∈ t, a vector field X v such that

where γ v : (-, ) → T 2 is a curve passing through Id ∈ T 2 and with velocity v ∈ t = T Id T 2 at t = 0. Notice that B := {p ∈ V | ∀v ∈ t, X v ∈ η} is a T 2 -invariant subset of V ; denote then by B its image in M via the bundle projection π : V → M . Now, up to taking an average using a Haar measure on T 2 , we can also write η = ker α with α ∈ Ω 1 (V ) invariant under the action of T 2 . For the rest of the section we will work with this contact form, but at each step we will remark that all the constructions are actually independent of this particular choice of invariant form and dependent only on η.

Denote by t * the dual of t and by St * its spherization, i.e. the quotient of t * \ {0} by the natural R >0 -action. Consider now, for each p ∈ V , the element L p of t * given by v → α p (X v (p)). Notice moreover that, by definition of B, L p is non-zero for all p ∈ V \ B. Denote then by [L p ] its class in St * ; we point out that [L p ] doesn't depend on the choice of T 2 -invariant contact form α for η.

Define then ϕ :

Suppose now that the principal T 2 -bundle π : V → M admits a principal Ehresmann connection F, i.e. a codimension 2 distribution on V which is T 2 -invariant and transverse to each T 2 -orbit. Suppose moreover that F is a codimension 2 foliation on V .

Lastly, notice that F ∩η is T 2 -invariant, because it is the intersection of T 2 -invariant distributions. We point out though that it is a priori only a singular distribution on V , i.e. it has codimension 2 at each point p ∈ V such that F p ⊂ η p and codimension 3 at each point p ∈ V such that η p F p . Denote then by ξ the projection of F ∩ η to M via the differential of π; this projection is well defined because F ∩ η is T 2 -invariant.

Remark 7.1. [Lut79, Page 301] gives an explicit example where F ∩ η fails to be a distribution. Consider on the principal T 2 -bundle π : S 2n-1 × T 2 → S 2n-1 the invariant form

where (θ 1 , θ 2 ) are coordinates on T 2 , i : S 2n-1 → R 2n is the natural inclusion and (x 1 , . . . , x 2n ) are coordinates on R 2n . Then, via the natural flat connection F = T S 2n-1 ⊕ {0}, ker α induces on S 2n-1 the kernel of

This form β is zero on the subset (x 1 , x 2 , 0, . . . , 0) ∈ S 2n ⊂ S 2n , hence doesn't define an hyperplane field on S 2n . Then Lutz' work, reinterpreted using the language of adapted open books introduced by Giroux, gives the following: Theorem 7.2. In the situation described above, we have the following:

c. Let B be B with opposite orientation and ϕ be the composition of ϕ with an arbitrary orientation-reversing diffeomorphism St * → St * . Then, if the conditions in point b. are satisfied, ξ is supported by (B, ϕ).

We recall that a q-codimensional contact foliation on a manifold Y 2n+1+q is a couple (F 2n+1 , ζ 2n ) of a codimension q foliation F on Y and a codimension q + 1 distribution ζ on Y such that, for each leaf L of F, (L, ζ ∩ T L) is a contact manifold. We also point out that the particular choice of ϕ in point c. of Theorem 7.2 is not important, because if ϕ : M \ B → St * is obtained from ϕ by composing with an orientation-preserving diffeomorphism St * → St * , then the set of fibers of ϕ and the set of fibers of ϕ coincide.

The rest of Section 7.1 is devoted to the proof of the above theorem.

Proof (Theorem 7.2). We start by showing point a., i.e. that (B, ϕ) is an open book decomposition of M ; this is the content of [Lut79, Lemme Fondamental (page 286)].

For computational convenience, we denote by X 1 , X 2 the pair of infinitesimal generators of the T 2 -action associated to a choice of coordinates (θ 1 , θ 2 ) on T 2 . In other words, for i = 1, 2,

, with the notation X v for v ∈ t introduced above.

Consider then, for

Lemma 7.3. On all V , we have the following: i. d φ i = -ι Xi dα, for i = 1, 2;

ii. d φ j (X i ) = 0, for {i, j} = {1, 2};

iii. dα (X 1 , X 2 ) = 0.

Proof (Lemma 7.3). By the T 2 -invariance of α, we have L Xi α = 0, for i = 1, 2. In particular, point i. simply follows from the facts that L Xi α = dι Xi α + ι Xi dα and that φ i = ι Xi α. Points ii. and iii. follow directly from point i., from the formula for exterior derivative of differential forms, stating that dα(X, Y ) = X•(α(Y ))-Y •(α(X))-α ([X, Y ]) for all vector fields X, Y on V , and from the fact that [X 1 , X 2 ] = 0, because T 2 is commutative.

Let now ρ : M → R ≥0 be given by ρ := φ . Then, we have

S 1 defined above is just given by ϕ := φ /ρ. Here, the identification St * S 1 is made through the coordinates (θ 1 , θ 2 ) on T 2 .

Let's then prove that φ is transverse to the origin of R 2 and that the linear map

Suppose by contradiction that d p φ :

We then get thatsin θ d p φ 1 + cos θ d p φ 2 = 0 as map T p M → R. Consider now the everywhere non-zero vector field Z := -sin θX 1 + cos θX 2 on V . We have α(Z) |p =

THE BOURGEOIS CONSTRUCTION

We point out that the condition γ ∧ dγ n-2 ∧ dφ 1 ∧ dφ 2 ≥ 0 in point a. of Theorem 7.5 is independent of the choice of form γ defining ξ: indeed, it is equivalent to the fact that ξ induces by restriction a contact structure on φ -1 (z), for each regular value z of φ. Moreover, the contact form α in point b. is clearly invariant under the natural T 2 -action on the principal T 2 -bundle π : M × T 2 → M .

We now recall the proof of Theorem 7.5 in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF], because it will be useful in the proof of Proposition 7.7.

Proof (Theorem 7.5). Let's start by proving point a.. Chose an auxiliary Riemannian metric on M and fix a tubular neighborhood B × D 2 via the exponential map on the normal bundle of B (recall that the binding has trivial normal bundle by definition of open book decomposition). Up to rescaling the metric, we can suppose that each {pt} × D 2 is a geodesic disk of radius 1.

Consider now a map ρ : B ×D 2 → R ≥0 which is smooth away from B ×{0}, equal to r for r ∈ (0, δ /2), strictly increasing with respect to the radial coordinate for r ∈ (0, δ) and equal to 1 on r ∈ [δ, 1]. Here, δ > 0 is so small that all B × {pt} in B × D 2 δ , equipped with the hyperplane field given by the restriction of ξ, are contact submanifolds of (M, ξ). Take then the natural extension ρ : M → R ≥0 of ρ given by the constant function 1 defined outside of B × D 2 .

Considering the fibration ϕ : M \ B → S 1 as a map with values in S 1 ⊂ C, we can now obtain a map φ :

Then, we claim that φ is as in the statement of Theorem 7.5. Indeed, if we denote N := B ×D 2 δ , then φ is 1 on M \N and each B ×{pt} in N is a fiber of φ and a contact submanifold of (M, ξ), so that γ ∧ dγ n-2 ∧ dφ 1 ∧ dφ 2 , with γ defining ξ, is non-negative everywhere on M , as wanted.

We now prove point b. of Theorem 7.5. Let α = β +φ 1 dθ 1 -φ 2 dθ 2 as in the statement. Then, we can compute dα = dβ + dφ 1 ∧ dθ 1 -dφ 2 ∧ dθ 2 , which gives

all the other terms being zero because they contain as a factor dφ 1 ∧ dφ 1 or dφ 2 ∧ dφ 2 .

We then get

Here, we can conclude with the same argument already used in the proof of Proposition 6.13. Indeed, the first summand is everywhere non-negative and strictly positive away from B × T 2 , because dβ is a symplectic form on the pages of the open book decomposition chosen (inducing the same orientation as φ). The second summand is everywhere non-negative by hypothesis, and it is strictly positive on B × T 2 , because B is a contact submanifold of M (with same orientation as that induced by φ). This means that α is a positive contact form on M × T 2 .

Remark 7.6. If φ = (φ 1 , φ 2 ) satisfies point a. of Theorem 7.5, then, for all > 0, the same is true for φ = ( φ 1 , φ 2 ). In particular, the 1-forms α := β + φ 1 dθ 1 -φ 2 dθ 2 always define positive contact structures by point b. of Theorem 7.5, which are moreover

THE BOURGEOIS CONSTRUCTION

depending only on φ , non-increasing in φ , equal to 1/ on the set { φ < } and equal to 1/ φ on the set M \ { φ < 2 }. Now, define β := f β , φ 1 := f φ 1 and φ 2 := f φ 2 . Then, α = f α , i.e. η = ker (α ), and B × {pt} ⊂ N are fibers of φ . Moreover, we have φ = 1 on all M \ N . The computation

analogous to the one recalled in the proof of Theorem 7.5, then tells us that dβ has to be symplectic on the fibers of ϕ| M \N , because α is a contact form and φ is constant on M \ N . Now, Lemma 6.8 tells us that there is g : M → R such that β := gβ , which obviously still defines ξ = ker (β ), is adapted to the open book (B, ϕ). Moreover, from the explicit proof of Lemma 6.8, we see that g can be chosen to be equal to 1 on the complement of N and depending only on φ .

Define then φ 1 := gφ 1 , φ 2 := gφ 2 , α := β + φ 1 dx -φ 2 dy and φ := (φ 1 , φ 2 ). Then, noticing that α = f gα and that each B × {pt} is also a fiber of φ, it is clear that the properties i. to iii. above are satisfied. This concludes the proof of Proposition 7.7.

FIBRATIONS WITH A FLAT CONTACT CONNECTION

For simplicity, in the following we will also denote the data of a contact fiber bundle by (V, B, M, η). Lemma 8.2. [Ler04, Lemma 2.3] Let (V, B, M, η) be a contact fiber bundle and α a 1-form on V defining η. The distribution H defined as the dα| η -orthogonal of ξ b in η is an Ehresmann connection on the bundle (M, V, B) and, at each point p ∈ V , we have η(p) = ξ π(p) (p) ⊕ H(p). Moreover, its holonomy over a path γ : [0, 1] → B is a contactomorphism between ξ γ(0) and ξ γ(1) .

In order to avoid issues about the completeness of the parallel transport, we will always assume M compact here.

We point out that the data of ξ b for all b ∈ B and H also allows to restore the hyperplane field η. Hence, we will call contact fiber bundle the data (V, B, M, η) as well as the data (V, B, M, {ξ b } b∈B , H), without any distinction. Also, if v is a vector of T b B, we denote by v # (p) its horizontal lift to H(p), for all p ∈ π -1 (b).

For the rest of Chapter 8, we actually focus on the case of B an oriented smooth surface Σ, as this will be setting for our future considerations.

Moreover, we remark that in Definition 8.1 we do not require η to be necessarily a contact structure on V . On the other hand, this will be the case we are interested in, hence we need a criterion that establishes when this is the case. In order to give a statement, we point out the following. For a couple (X, Y ) of vector fields on Σ, the vector field

# on V is actually tangent to the fibers. Indeed, it is mapped to zero via the differential dπ because dπ commutes with the Lie bracket and dπ(Z # ) = Z for every vector field Z on Σ. What's more, the restriction of

# on M b depends only on X(b) and Y (b): this can easily be seen with the tensoriality criterion (which is stated precisely in [KMS93, Lemma 7.3] for instance). Hence, for every oriented basis (u, v) of T b Σ, we have a well-defined vector field

# on M p . Then, we have the following:

Proposition 8.3 ([Ler04, Proposition 3.1]). Let (V, Σ, M, η) a contact fiber bundle over an oriented surface Σ. The hyperplane field η is a positive contact structure on V if and only if for every p ∈ Σ and every oriented basis (u, v) of T p Σ, the vector field

# on M p is a negative contact vector field for ξ p .

Recall that a contact vector field is called negative if it is negatively transverse everywhere to the contact structure. We also remark that [Ler04, Proposition 3.1] is actually more general, because the base space B is not assumed to be a surface but can have dimension 2n ≥ 2.

Fibrations with a flat contact connection

In this section we give an adaptation of [Ler04, Remark 3.2 and Theorem 3.6] to the case of flat contact connections. For the reader interested in the details, [Ler04, Section 3.2] deals with the case of principal G-bundles using the notion of contact moment map, under the implicit assumption of the Lie group G having dimension at least 1. The case of flat contact connection that follows corresponds then to the case of G = π 1 (Σ) of dimension 0, where Σ is the surface which is base of the contact fiber bundle. For this reason, we try to keep our description as self contained as possible.

In the following, we call fiber bundle with contact fibers, and denote it (V, Σ, M ξ ), the data of a fiber bundle (V, Σ, M ) and a collection of a contact structure ξ p on each fiber CHAPTER 8. BOURGEOIS' CONSTRUCTION REVISITED M p , depending smoothly on p ∈ Σ. In particular, on a fiber bundle with contact fibers (V, Σ, M ξ ), the structure of a contact fiber bundle, with the prescribed contact structures on the fibers, is equivalent to the data of a contact connection.

We focus in this section on those fiber bundles with contact fibers (V, Σ, M ξ ) admitting a structure of contact fiber bundle (V, Σ, M, η 0 ) with flat connection, i.e. such that the associated connection H 0 has zero curvature. We recall that the curvature of an Ehresmann connection H on a fiber bundle (V, Σ, M ) is defined as follows. Via H, each vector field Z on V can be uniquely decomposed as Z = Z h + Z v , where Z h is horizontal, i.e. everywhere tangent to H, and Z v is vertical, i.e. tangent everywhere to the fibers of the fibration. The curvature R of H is then the 2-form on V with values in T V , i.e. R ∈ Ω 2 (V, T V ), defined by R (Z, W ) = [Z h , W h ] v for all vector fields Z, W on V . Frobenius' theorem gives the following equivalence: R is everywhere zero if and only if the connection H is a foliation on V .

The flatness of H 0 allows us to give a nice presentation of (V, Σ, M, η 0 ). Indeed, once fixed a certain fiber (M, ξ) of (V, Σ, M, η 0 ), we can define a representation ρ : π 1 (Σ) → Diff(M, ξ), where Diff(M, ξ) is the space of contactomorphisms of (M, ξ): for each class c ∈ π 1 (Σ), we consider the monodromy of the connection H 0 over a (smooth immersed) representative of c. This gives a well defined ρ: indeed, the monodromy doesn't depend on the representative chosen, because H 0 is a foliation, and it is a contactomorphism of the fibers, by Lemma 8.2.

If we denote by π Σ : Σ → Σ the universal cover of Σ, we get a well defined homomorphism F : Σ × M → V of fibrations over Σ given simply by F (c, q) := ρ c (q). Moreover, the differential of F sends the connection T Σ ⊕ {0} of Σ × M → Σ to the connection H 0 of π : V → Σ, and the contact structure {0} ⊕ ξ on the fiber of Σ × M over p ∈ Σ to the contact structure ξ p of the fiber M p of V over p = π Σ ( p).

Also, if we denote by ρ the diagonal action of π 1 (Σ) on Σ × M induced by the natural action on the first factor and by the action ρ on the second factor, F induces an isomorphism f : Σ × ρ M → V of fiber bundles over Σ, where Σ × ρ M is the quotient of Σ × M by ρ. Moreover, f sends the tangent space of Σ to H 0 and the contact structure {0} ⊕ ξ on the fiber of Σ× ρ M over p exactly to the contact structure ξ p of the fiber M p of V over p.

On this fiber bundle with contact fibers (Σ × ρ M, Σ, M ξ ), we can moreover explicitly describe all the contact fiber bundles giving the prescribed contact structure on the fibers. For this, we introduce the notion of potential of a connection. With a little abuse of notation and thanks to the isomorphism of contact vector bundles found above, we will write V = Σ × ρ M in the following.

Let's consider a contact connection H on (V = Σ × ρ M, Σ, M ξ ) and take the 1-form A, with values in the space X f ib (V ) of vector fields on a fiber, which is defined on Σ as follows: take a point p ∈ Σ, a vector v ∈ T p Σ and define A v := v # -v, where v # is the lift of v to H and v is the lift of v to H 0 . This differential form A is called potential of H relative to H 0 . We point out that it clearly allows to recover H from H 0 . Moreover, A has actually values in the space X cont f ib (V ) of contact vector fields on a fiber: more precisely, for each v ∈ T p Σ the vector field A v on the fiber M p is a contact vector field for the contact manifold (M p , ξ p ).

Vice versa, each differential form A ∈ Ω 1 (Σ; X cont f ib (V )), such that for each v ∈ T p Σ the vector field A v is contact for (M p , ξ p ), actually defines, together with the flat contact connection H 0 , a contact connection H, hence a contact bundle η on the fibration with contact fibers (V, Σ, M ξ ). Now, between all these contact fiber bundles, we are interested in particular in those 8.4. BOURGEOIS' CONTACT STRUCTURES the condition d 0 A = 0 is equivalent to the fact that 1 R → 0 for → 0, which was the condition used to introduce Bourgeois contact structures in Chapter 1.

We now point out that Definition 8.6 is a non-trivial generalization of Bourgeois' construction, i.e. the class of Bourgeois contact structures is not exhausted by the examples on M × T 2 from [Bou02]: Proposition 8.7. There is a flat contact fiber bundle (V, T 2 , M, η 0 ) that admits a Bourgeois contact structure and is non trivial, i.e. not isomorphic (as flat contact fiber bundle) to (M × T 2 , T 2 , M, ξ M ⊕ T T 2 ).

We deduce Proposition 8.7 from the following: Lemma 8.8. Let (M, ξ) be a contact manifold, G a subgroup of the group of contactomorphisms of (M, ξ), and ρ :

Let's also denote by β a G-invariant contact form for ξ on M such that dβ is symplectic on the fibers of ϕ, and by η 0 the flat contact bundle induced on M × ρ T 2 → Σ by the flat contact bundle ξ ⊕ T R 2 on M × R 2 → Σ. Here, ρ is the action of π 1 (T 2 ) on M × R 2 given by ρ on the first factor and by the natural action on the universal cover R 2 → T 2 on the second factor. Then, the hyperplane field η on M × ρ T 2 , induced by ker(β

Once noticed that the form β + φ 1 dθ 1 -φ 2 dθ 2 on M × R 2 is invariant under the action ρ, the above lemma can be shown exactly as Theorem 7.5; the proof is hence omitted.

Proof (Proposition 8.7). We recall that van Koert and Niederkrüger exhibited in [START_REF] Van Koert | Open book decompositions for contact structures on Brieskorn manifolds[END_REF] a particular open book decomposition for each Brieskorn manifold W 2n-1 k ⊂ C n+1 , with supporting form α k . In particular, the adapted open book decomposition is defined by a map φ : W 2n-1 k → R 2 which is invariant under the action of a subgroup SO(n) of strict contactomorphisms for the strict contact manifold (W 2n-1 k , α k ). More precisely, if (z 0 , . . . , z n ) are the coordinates of C n+1 , SO(n) is just the subgroup of linear transformations of C n+1 fixing the first coordinate z 0 and acting on the sub-vector (z 1 , . . . , z n ) made of the other n coordinates by matrix multiplication; see [START_REF] Van Koert | Open book decompositions for contact structures on Brieskorn manifolds[END_REF] for the details. For simplicity, we denote the couple

where f is any element of SO(n) of order 2. Then, Lemma 8.8 tells us that the η on M × ρ T 2 , induced by ker(β

A direct computation shows also that the potential A associated to the kernel of α := β + φ 1 dx -φ 2 dy on M × R 2 with respect to the trivial flat contact bundle η 0 := ξ ⊕ T R 2 is given by A = -X ⊗ dx -Y ⊗ dy, where X, Y are respectively the contact vector fields on (M, ξ) such that β(X) = φ 1 and β(Y ) = -φ 2 . In particular, it satisfies d 0 A = 0. Hence, the potential A of η = ker(β) on V (with respect to the flat connection η 0 on V induced by η 0 on M × R 2 ) will also satisfy d 0 A = 0. In other words, this is an example of Bourgeois contact structure on the flat contact fiber bundle (V, T 2 , M, η 0 ), as wanted.

The only thing left to show is that (V, T 2 , M, η 0 ) is not isomorphic to the trivial flat contact bundle (M × T 2 , T 2 , M, ξ ⊕ T T 2 ). Recall that flat connections define foliations, according to Frobenius' theorem. In particular, the connection H 0 associated to η 0 defines a foliation F 0 by tori T 2 on the total CHAPTER 8. BOURGEOIS' CONSTRUCTION REVISITED Proof (Corollary 8.10). Given b ∈ Σ, consider an ordered basis (u, v) of T b Σ and define ψ η (b) as the class of Ψ η (u, v) under the relation ∼. We then need to show that this is well defined. Suppose (u , v ) is another ordered basis of T b Σ; we want to show that Ψ η (u, v) ∼ Ψ η (u , v ). Choose a curve γ : [0, 1] → F Σ with image contained in the fiber pr -1 (b) of pr : F Σ → Σ and such that γ(0) = (u, v) and γ(1) = (u , v ). Then, according to point ii. of Proposition 8.9, Ψ η • γ gives a smooth

The last statement about the construction by Bourgeois follows directly from the definition of Ψ η and from point (c) of Proposition 6.1. Indeed, let η = ker(β + φ 1 dθ 2φ 2 dθ 2 ) be the Bourgeois contact structure on the flat contact bundle (M × T 2 , T 2 , M, ξ ⊕ T T 2 ) given by Theorem 7.5 starting from open book (B, ϕ) of M adapted to ξ. As already observed in the beginning of Section 8.4, we can compute that A ∂ θ 1 and A ∂ θ 2 are respectively the contact vector fields on (M, ξ) of contact hamiltonians -φ 1 and φ 2 (via β), with φ = (φ 1 , φ 2 ) defining (B, ϕ). Then, we can see that point (c) of Proposition 6.1 with X := A ∂ θ 1 and Y = A ∂ θ 2 gives exactly the open book (B, ϕ ), where ϕ is obtained from ϕ by composition with the antipodal map S 1 → S 1 . In other words, for all b ∈ T 2 , if (∂ θ1 , ∂ θ2 ) is the oriented base of T b T 2 coming from the choice of coordinates (θ 1 , θ 2 ) ∈ T 2 as in the statement of Theorem 7.5, then Ψ η (∂ θ1 , ∂ θ2 ) = (B, ϕ ). In particular, ψ η (b) is the isotopy class of (B, ϕ ), which coincides with that of (B, ϕ).

We now derive Proposition 8.9 as a consequence of Proposition 6.1: Proof (Proposition 8.9). Let's start by defining Ψ η . Let A be the potential for η relative to the flat contact connection H 0 of η 0 . Then, for each b ∈ Σ and each positive basis (u, v) of T b Σ, A u and A v are two vector fields on M b which are contact for ξ b . Moreover, according to Proposition 8.4 (and by definition of Bourgeois contact structure), [A u , A v ] is negatively transverse to ξ b ; then, Proposition 6.1 gives an open book decomposition OBD (u,v) for M b supporting ξ b . Because OBD (u,v) is also a fiber adapted open book, we can define Ψ η (u, v) := OBD (u,v) . In particular, it is clear that point i. of Proposition 8.9 is satisfied.

Let's now prove point ii.. Consider b ∈ Σ and a basis (u, v) of T b Σ. Let α be a 1-form defining η on V and denote α b its restriction to the fiber M b of V → Σ. From the explicit proof of point (c) of Proposition 6.1, we can see that

By definition of pullback smooth bundle and pullback contact bundle, Ψ η • γ describes then the smooth [0, 1]-family of open books in γ * pr * V adjusted to γ * pr * η which is given by the conformal class of

where, for each t ∈ [0, 1], γ 1 (t) and γ 2 (t) are the two vectors of the (ordered) basis γ(t) ∈ F Σ and where µ : γ * pr * V → V is just the restriction of the projection pr V : [0, 1]×V → V to γ * pr * V . Notice that A γ1(t) (p) and A γ2(t) (p) are well defined because (t, p) ∈ γ * pr * V . This concludes the proof of point ii. of Proposition 8.9.

Chapter 9

Applications

Virtually overtwisted contact structures in high dimensions

In Section 9.1.1, we prove Proposition II.C from Chapter 1, stating that a Bourgeois contact structure on a fiber bundle with total space M × T 2 is weakly fillable provided that the same is true for the fiber (M, ξ). Then, Section 9.1.2 contains the proof of Théorème II.F, also stated in Chapter 1, about the existence of virtually overtwisted manifolds in all odd dimensions.

Bourgeois contact structures and weak fillability

Consider (M 2n-1 , ξ) a contact manifold and take the flat contact bundle (M ×T 2 , T 2 , M, η 0 ), where η 0 = ξ ⊕ T T 2 and the fibration map M × T 2 → T 2 is just the projection on the second factor.

Proposition 9.1. Let η be a Bourgeois contact structure on (M × T 2 , T 2 , M, η 0 ). If (M, 2n-1 ξ) is weakly filled by (X 2n , ω), then (M × T 2 , η) is weakly filled by (X × T 2 , ω + ω T 2 ), where ω T 2 is an area form on T 2 .

We point out that the result is already known in the case of η obtained by Bourgeois' construction [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF]: the statement and the idea of the proof in that case already appeared in [MNW13, Example 1.1]; see also [LMN18, Theorem A.a] for an explicit proof.

Proof (Proposition 9.1). We start by choosing a convenient contact form for η. If β is a form on M defining ξ, we claim that we can write η = ker (α), where α :

. Now, because η induces ξ on each fiber, there is a positive function h : M × T 2 → R >0 such that γ = hβ. Then, the division by h gives a contact form for η of the announced form. Moreover, we recall from Section 8.4 that if A denotes the potential of η with respect to η 0 then, for each > 0, the family of potentials A := A define a family η of Bourgeois contact structure that are all isotopic between Bourgeois contact structures (hence in particular between contact structures). We then claim that η is just the kernel of α = β + φ 1 dθ 1 -φ 2 dθ 2 . Indeed, φ 1 and

VIRTUALLY OVERTWISTED CONTACT STRUCTURES IN HIGH DIMENSIONS

We also point out another result in a similar vein from [START_REF] Lisi | Some properties of the Bourgeois contact structures[END_REF], that deals more precisely with the specific construction from [Bou02]: Let's now come back to the results we need in order to exhibit examples of virtually overtwisted manifolds in all odd dimensions. Theorem 5.23 and Proposition 9.1 have then the following immediate corollary: Proposition 9.7. Consider a branched covering Σ g → T 2 , where Σ g is the closed genus g ≥ 2 surface, and the naturally induced branched covering M × Σ g → M × T 2 . Let η g on M × Σ g be a contact branched covering of a Bourgeois contact structure η on the the contact bundle (M × T 2 , T 2 , M, ξ ⊕ T T 2 ), where ξ is a contact structure on the fiber M . Then, if (M, ξ) admits a weak filling (X, ω), there is a symplectic form Ω on X × Σ g weakly dominating η g on M × Σ g = ∂X × Σ g .

We point out that the explicit proof of Theorem 5.23 actually shows that, up to isotopy, Ω can be chosen to be of the form ω + ω g , for a certain area form ω g on Σ g .

High dimensional virtually overtwisted manifolds

Let π : Σ g → T 2 be a covering map and suppose here that it is branched along two points. Let also (Id, π) : M × Σ g → M × T 2 be the induced branched covering. We remark that g equals here the branching index along each of the two connected components of the upstairs branching locus of (Id, π).

Proposition 9.8. Let η be a Bourgeois contact structure on the flat contact bundle (M ×T 2 , T 2 , M, η 0 ) and consider a contact branched covering η g of η on M ×Σ g . If (M, ξ) is weakly fillable and virtually overtwisted, then, for g ≥ 2 big enough, (M × Σ g , η g ) is weakly fillable and virtually overtwisted. Then, starting for example from the base case of a holomorphically fillable virtually overtwisted contact structure on lens spaces, that exist by [Gom98, Proposition 5.1] (see also [Gir00, Theorem 1.1]), and using the construction in [START_REF] Bourgeois | Odd dimensional tori are contact manifolds[END_REF], a proof by induction on the dimension 2n -1 of M easily shows that Proposition 9.8 implies the following result, already stated in Section 1.2: Théorème II.F. Les structures virtuellement vrillée existent en toutes dimensions ≥ 3.

Proof (Proposition 9.8). Proposition 9.7 tells us that (M × Σ g , η g ) is weakly fillable for all g ≥ 2. We then have to show that, for g sufficiently big, this contact manifold admits a finite cover which is overtwisted.

By hypothesis, there is a finite cover p : M → M such that (M , ξ := p * ξ) is overtwisted. Consider then the following commutative diagram of smooth maps:

BOURGEOIS CONSTRUCTION AND REEB DYNAMICS

i. on M \ B, it is tangent to the fibers of ϕ,

ii. on the binding B, it is equal, up to a non-zero constant factor, to the Reeb vector field R B of the restriction of β to B;

b. f, g : M → R are smooth functions such that f = g = 0 on B and such that (f, g) : M → R 2 is positively proportional to (cos ϕ, sin ϕ) on M \ B.

Proof (Proposition 9.11). We start by finding the adapted form β in the statement as well as a particular normal neighborhood N of the binding and a particular smooth map φ : M → R 2 defining (B, ϕ).

For this, we use the following result, whose proof can be found, for example, in [DGZ14, Section 3]: Lemma 9.12 (Giroux). Let D 2 ⊂ R 2 be the disk centered at the origin with radius 1 and β be a contact form on B × D 2 with the following properties:

2. For each ϕ ∈ S 1 , dβ| T Σϕ is a symplectic form on Σ ϕ \ B, where

3. With the orientations of B and Σ ϕ induced, respectively, by β B and dβ| T Σϕ , B is oriented as the boundary of Σ ϕ .

Then, for a sufficiently small δ > 0, if we denote by D 2 δ ⊂ R 2 the disk centered at the origin and of radius δ > 0, there is an embedding B × D 2 δ → B × D 2 which preserves the angular coordinate ϕ on the second factor, which is the identity on B × {0} and which pulls back a convenient isotopic modification β of β (with an isotopy between contact forms that satisfy Hypothesis 1, 2 and 3 above) to a 1-form h 1 (r) • β B + h 2 (r) • dϕ, such that: i. h 1 (0) > 0 and h 1 (r) = h 1 (0) + O(r 2 ) for r → 0, ii. h 2 (r) ∼ r 2 for r → 0,

r > 0 ∀r ≥ 0 (contact condition); iv. h 1 (r) < 0 for r > 0, (symplectic condition on Σ ϕ ).

Then, we choose β, N and φ as follows. Take a normal neighborhood B × D 2 of the binding B in M such that ϕ : B × D 2 \ {0} → S 1 becomes the angular coordinate of D 2 \ {0}: such neighborhood exists by definition of open book decomposition. Also, take a contact form β 0 which defines ξ and is adapted to the open book (B, ϕ) on M . Then, Lemma 9.12 gives an isotopic modification β of β 0 , still adapted to the same open book, and of the form

δ , equal to 1 on the complement in M of the open set {r < 2δ /3} ⊂ N , and depending only on r and strictly increasing in it on the set δ /3 < r < 2δ /3.

Then, we define φ := ρ • (cos ϕ, sin ϕ). Remark that such a φ is indeed well defined and smooth on all M , and defines the open book (B, ϕ).

CHAPTER 9. APPLICATIONS

We now define two functions λ, µ : M → R as follows:

We point out that they are well defined smooth functions on all M × T 2 . Indeed, ρ smoothly extends as 1 at r = 0, h 1 = O(r) near r = 0 (by point i. of Lemma 9.12) and the denominator ρ h 1 -ρh 1 is positive for r > 0 and smoothly extends as h 1 (0) at r = 0.

Consider then Z := λR B and (f, g) := µ(cos ϕ, sin ϕ). Here, R B is seen as as a vector field on N = B × D 2 δ tangent to the first factor and λ has support contained inside N , hence λR B is well defined on all M . Similarly, f, g are well defined because µ is zero on B.

It is now easy to check that such Z, f, g satisfy points a. and b. of Proposition 9.11.

Lastly, we have to choose a contact form α defining a Bourgeois contact structure η on the flat contact bundle (M × T 2 , T 2 , M, ξ ⊕ T T 2 ), as in the statement of Proposition 9.11. Let α := β + φ 1 dx -φ 2 dy, i.e. the one obtained from Theorem 7.5 with the choices of φ and N made above. We already know that the contact structures obtained from Theorem 7.5 are particular cases of Bourgeois structures. The only thing left to check is that the Reeb vector field of this contact form α can be written as in the statement of Proposition 9.11.

Claim 9.13. The Reeb vector field for α on

Claim 9.14. The Reeb vector field for α on N × T 2 is

By the definition of Z, f and g, this concludes the proof of Proposition 9.11.

We now prove the identities used above:

Proof (Claim 9.13). In this region we simply have α = β + cos ϕ dx -sin ϕ dy and dα = dβ-sin ϕ dϕ∧dx-cos ϕ dϕ∧dy. This easily gives α(R α ) = 1 and ι Rα dα = cos ϕ sin ϕ dϕsin ϕ cos ϕ dϕ = 0.

Proof (Claim 9.14). Here, we have α = h 1 β B + h 2 dϕ + ρ cos ϕ dx -ρ sin ϕ dy, so that

We then have the following result on the Reeb dynamics:

BOURGEOIS CONSTRUCTION AND REEB DYNAMICS

Corollary 9.15. Let α on M × T 2 be the contact structure given by Proposition 9.11. Then, the closed contractible orbits of R α in M × T 2 are of the form O B q0 × {(x 0 , y 0 )}, where (x 0 , y 0 ) ∈ T 2 and O B q0 is a closed orbit of R B in B which is contractible in M .

We remark that even if the orbits are contained in B, we are interested here in their homotopy class as loops in M .

Proof (Corollary 9.15). We start by remarking that if p 0 ∈ B × T 2 ⊂ M × T 2 is of the form (q 0 , x 0 , y 0 ), with q 0 ∈ B and (x 0 , y 0 ) ∈ T 2 then O p0 = O B q0 × {(x 0 , y 0 )}, where O B q0 is the orbit of q 0 under the flow of R B on B. Indeed, along B × T 2 , R α is just proportional (via a non zero constant) to R B on the B factor of B × T 2 , so that the orbit of p 0 is tangent to B × {(x 0 , y 0 )} and actually coincides with O B q0 × {(x 0 , y 0 )}. It's then enough to show that, for each

Moreover, the function ϕ is invariant under the restriction of the flow φ t of R α to (M \ B) × T 2 , because Z is tangent to the fibers of ϕ on M \ B, according to Proposition 9.11. In particular, if we write p 0 = (q 0 , x 0 , y 0 ), where q 0 ∈ M \ B, with let's say ϕ(q 0 ) = ϕ 0 , and (x 0 , y 0 ) ∈ T 2 , then ϕ(O p0 ) = ϕ 0 . Finally, the component of R α tangent to the T 2 -factor is f ∂ x -g∂ y , which is non-zero along all M \ B, because (f, g) is positively proportional to (cos ϕ, sin ϕ) there. In particular, the projection (x t , y t ) of φ t (p 0 ) via π : M × T 2 → T 2 has velocity ( ẋt , ẏt ) positively proportional to (cos ϕ 0 , -sin ϕ 0 ). Hence, in order for O p0 to be closed, the angle ϕ 0 ∈ [0, 2π[ has to be a rational multiple of π and {(x t , y t ) ∈ T 2 |t ∈ R} has to be the circle on T 2 = R 2 /Z 2 passing through (x 0 , y 0 ) and with slope (cos ϕ 0 , -sin ϕ 0 ). But this means that O p0 can't be contractible, because it has a homotopically non trivial projection on T 2 .

Contact embeddings of 3-manifolds

We start with a proposition on (topological) open book decomposition of 3-manifolds: Proposition 9.16. Let M be a 3-manifold with non-trivial π 1 (M ). Then, every open book decomposition (K, ϕ) of M can be transformed, by a sequence of positive stabilizations, to an open book decomposition (K , ϕ ) with binding K having at most two connected components, each of which is not contractible in M .

According to Perelman's proof of Poincaré's conjecture (see for instance [START_REF] Morgan | Recent progress on the Poincaré conjecture and the classification of 3-manifolds[END_REF] for a summary), M has trivial fundamental group if and only if it is diffeomorphic to S 3 . Proof. We start by applying a sequence of stabilizations to reduce the number of connected components of the boundary of the pages to one. We can thus suppose that the open book decomposition (K, ϕ) has connected binding K. Denote by Σ the page of (K, ϕ). Let p ∈ K. Notice that if c := [K] ∈ π 1 (M, p) is non-trivial, then we have nothing to prove. We can hence suppose that it is trivial.

We now need a presentation of π 1 (M, p) in terms of the fundamental group of the page and the monodromy of (K, ϕ). Such a presentation can be found using Van-Kampen theorem as described, for example, in [START_REF] Etnyre | Invariants of contact structures from open books[END_REF].

BOURGEOIS CONSTRUCTION AND REEB DYNAMICS

2. the core curve of A bounds a disk in M \ Σ and the linking number of the boundary components is 1.

In particular, we know that the core of the annulus A bounds a disk in M \ Σ. Now, the null-homotopy defined by this disk gives a homotopy between one of the two boundary components of Σ , which we can call K 1 , and β g (seen as a curve on Σ via the natural inclusion Σ ⊂ Σ as surfaces embedded in M ). Hence, K 1 is homotopically non-trivial, because b g = [β g ] is non-zero.

Similarly, the fact that the core of A bounds a disk implies that the other connected component of ∂Σ , which we can call K 2 , is homotopic to K * β -1 g . Here, * is the concatenation of paths and . -1 is the inverse of a path. In particular, K 2 is homotopically non-trivial, because K is trivial and β g is not.

Consider now a 3-dimensional contact manifold (M, ξ) and an open book decomposition (Σ, ϕ) supporting ξ. Corollary 9.15 and Proposition 9.16 have the following direct consequence: Proposition 9.17. If π 1 (M ) = {1}, there exists an open book (Σ , ϕ ) supporting ξ, obtained from (Σ, ϕ) by a sequence of positive stabilizations, such that Proposition 9.11 gives a hypertight Bourgeois contact structure on M × T 2 .

Let's denote by D 2 R the ball of radius R > 0 centered at the origin in R 2 and by (r, ϕ) the polar coordinates on it. At this point, we can deduce from Proposition 9.17 the following result, which consists of Théorème II.G and Corollaire II.H stated in Section 1.2: Theorem 9.18. Every closed 3-dimensional contact manifold (M, ξ), with π 1 (M ) nontrivial, can be embedded, with trivial conformal symplectic normal bundle, in a hypertight closed 5-dimensional contact manifold (N, η). In particular, for each contact form α defining ξ on M , there is an > 0 such that M × D 2 , ker α + r 2 dϕ is tight.

As already remarked in the introduction, we point out that the recent paper [START_REF] Hernández-Corbato | Tight neighborhoods of contact submanifolds[END_REF] deals with the higher dimensional case; more precisely, it contains a generalization of the second part of this result, as well as an analogue (with no control on the codimension) of the first part of it.

Proof. Consider an arbitrary contact 3-manifold (M, ξ) and take one of the hypertight contact manifolds (M × T 2 , η) given by Proposition 9.17. Each (M × {pt}, η ∩ T (M × {pt}))) is then exactly (M, ξ) and it has topologically trivial normal bundle, hence trivial conformal symplectic normal bundle. Indeed, a symplectic vector bundle of rank 2 is symplectically trivial if and only if it is topologically trivial.

As far as the second part of the statement is concerned, according to the standard neighborhood theorem for contact submanifolds [Gei08, Theorem 2.5.15], the contact submanifold (M, ξ = ker(α)) = (M × {pt}, η ∩ T (M × {pt}))) of (M × T 2 , η) has a contact neighborhood of the form M × D 2 , ker α + r 2 dϕ , for a certain real > 0. Moreover, each hypertight high dimensional contact manifold is in particular tight, according to [START_REF] Albers | On the Weinstein conjecture in higher dimensions[END_REF][START_REF] Casals | Geometric criteria for overtwistedness[END_REF]; in particular, (M × T 2 , η) is tight. Then, M × D 2 , ker α + r 2 dϕ is tight too, because it embeds (in codimension 0) in a tight contact manifold. Titre : Sur quelques constructions de vari ét és de contact Mots cl és : vari ét é de contact, structure de contact, construction R ésum é : Cette th èse est subdivis ée en deux parties. La premi ère partie porte sur l' étude de la topologie de l'espace des contactomorphismes pour quelques exemples explicites de vari ét és de contact en grandes dimensions. Plus pr écis ément, en utilisant des constructions et r ésultats dus à Massot, Niederkr üger et Wendl, on construit, en chaque dimension impaire, une infinit é d'exemples de contactomorphismes de vari ét és de contact vrill ées ferm ées qui sont lissement isotopes mais pas contact-isotopes à l'identit é. On donne aussi, en toutes dimensions impaires, des exemples de vari ét és de contact tendues ferm ées qui admettent un contactomorphisme tel que tous ses it ér ées sont lissement isotopes mais pas contacto-isotopes à l'identit é; ceci generalise un r ésultat en dimension 3 d û à Ding et Geiges. Dans la deuxi ème partie, on construit des exemples de vari ét és de contact ferm ées en grandes dimensions avec des propri ét és particuli ères. Ceci nous am ène à l'existence de structures tendues virtuellement vrill ées en toutes dimensions impaires, et au fait que chaque vari ét é de contact ferm ée de dimension 3 se plonge dans une vari ét é de contact tendue ferm ée de dimension 5 avec fibr é normal trivial. Pour cela, on utilise des constructions dues à Bourgeois (sur des produits avec des tores) et à Geiges (sur des rev êtements ramifi és). On passe de ces constructions à des d éfinitions; ceci permet de prouver un r ésultat d'unicit é dans le cas des rev êtements ramifi és de contact, et d' étudier leurs propri ét és globales, en montrant qu'elles ne d épendent d'aucun choix auxiliaire fait dans les proc édures. Un deuxi ème but permis par ces d éfinitions est l' étude des relations entre ces constructions et les notions de livre ouvert porteur, due à Giroux, et de fibr é de contact, due à Lerman. Par exemple, on donne une d éfinition de structure de contact de Bourgeois qui est locale, inclue (strictement) les r ésultats de la construction de Bourgeois et permet de r écup érer une classe d'isotopie de livres ouverts porteurs sur les fibres; ceci suit d'une r éinterpr étation, inspir ée par une id ée de Giroux, des livres ouverts porteurs en termes de paires de champs de vecteurs de contact. Title : On some constructions of contact manifolds Keywords : contact manifold, contact structure, construction Abstract : This thesis is divided in two parts. The first part focuses on the study of the topology of the contactomorphism group of some explicit high dimensional contact manifolds. More precisely, using constructions and results by Massot, Niederkr üger and Wendl, we construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity. We also give examples of tight high dimensional contact manifolds admitting a contactomorphism whose powers are all smoothly isotopic but not contact-isotopic to the identity; this is a generalization of a result in dimension 3 by Ding and Geiges. In the second part, we construct examples of higher dimensional contact manifolds with specific properties. This leads us to the existence of tight virtually overtwisted closed contact manifolds in all dimensions and to the fact that every closed contact 3-manifold embeds with trivial normal bundle inside a tight clo-sed contact 5-manifold. This uses known construction procedures by Bourgeois (on products with tori) and Geiges (on branched covering spaces). We pass from these procedures to definitions; this allows to prove a uniqueness statement in the case of contact branched coverings, and to study the global properties (such as tightness and fillability) of the results of both constructions without relying on any auxiliary choice in the procedures. A second goal allowed by these definitions is to study relations between these constructions and the notions of supporting open book, due to Giroux, and of contact fiber bundle, due to Lerman.