On some constructions of contact manifolds

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, Niederkrüger and Wendl, we construct (infinitely many) examples in all dimensions of contactomor-phisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopicto 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 nor-mal bundle inside a tight closed contact 5-manifold. This uses known construction procedures byBourgeois (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. For instance,we give a definition of Bourgeois contact structures on flat contact fiber bundles which is local,(strictly) includes the results of Bourgeois’ construction, and allows to recover an isotopy class of supporting open books on the fibers. This last point relies on a reinterpretation, inspired by anidea by Giroux, of supporting open books in terms of pairs of contact vector fields.
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Fabio Gironella. On some constructions of contact manifolds. Differential Geometry [math.DG]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLX045⟩. ⟨tel-01852415⟩

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