À toute variété de contact, on peut associer canoniquement une variété symplectique appelée sa symplectisation de sorte que la géométrie de contact peut se reformuler en termes de géométrie symplectique équivariante. Au sujet de cette construction fondamentale, une question basique restait ouverte : si deux variété de contact ont des symplectisations isomorphes sont-elles isomorphes ? On construit dans cette thèse des contre-exemples à cette question. Il existe en effet, en toute dimension impaire supérieure ou égale à 5, des variétés de contact non difféomorphes admettant pourtant des symplectisations isomorphes. On construit également, sur une même variété deux structures de contact non conjuguées par un difféomorphisme mais admettant des symplectisations isomorphes. Les démonstrations sont basées sur un phénomène bien connu en topologie différentielle (l'existence de h-cobordismes non triviaux, détectée par la torsion de Whitehead) ainsi que sur des résultats de flexibilité en géométrie symplectique dus à Cieliebak et Eliashberg. Un autre résultat de cette th?e affirme que ces variété de contact, bien que non isomorphes, le deviennent toutefois après un nombre suffisant de sommes connexes avec un produit de sphères. / To any contact manifold one can associate a symplectic manifold called its symplectisation in such a way that contact geometry can be reformulated in terms of equivariant symplectic geometry. Concerning this fundamental construction, a basic question remained open : if two contact manifolds have isomorphic symplectizations, are they isomorphic ? In this thesis, we construct counter-examples to this question. Indeed, in any odd dimension greater than or equal to 5, there exist non-diffeomorphic contact manifolds with isomorphic symplectisations. In addition, we construct two contact structures on a closed manifold that are not conjugate by a diffeomorphism though their symplectizations are isomorphic. The proofs are based on a well-known phenomenon in differential topology (the existence of non-trivial h-cobordisms, detected by Whitehead torsion) as well as flexibility results in symplectic geometry due to Cieliebak and Eliashberg. Another result from this thesis asserts that though these contact manifolds are not isomorphic, they become so after sufficiently many connect sum with a product of spheres.

No description available.

[pt] Enquanto a topologia do espaço de todas as curvas suaves imersas em 2-esfera começando e terminando em pontos dados e direções dadas é bem conhecido, é uma questão aberta entender o tipo de homotopia e dos seus subespaços consistindo as curvas com a curvatura restrita a um intervalo próprio aberto prescrito. Neste tese provamos que, sob certas circunstancias para os pontos e as direções inicial e final, estes subespaços não são homotopicamente equivalente ao espaço todo. Adicionalmente, fornecemos uma construção explicita dos geradores exóticos para algum grupo de homotopia e cohomologia. As dimensões desses geradores dependem das posições e das direções nas extremidades. Uma versão do princípio h foi usada na prova desses resultados. / [en] While the topology of the space of all smooth immersed curves in 2-sphere that start and end at given points in given direction is well known, it is an open problem to understand the homotopy type of its subspaces consisting of the curves whose geodesic curvatures are constrained to a prescribed proper open interval. In this article we prove that, under certain circumstances for endpoints and end directions, these subspaces are not homotopically equivalent to the whole space. Moreover, we give an explicit construction of exotic generators for some homotopy and cohomology groups. It turns out that the dimensions of these generators depend on endpoints and end directions. A version of the h-principle is used to prove these results.