Dans la première partie de cette thèse, on donne, sous certaines hypothèses, une minoration du nombre de points d’intersections d’une sous-variété Lagrangienne monotone L avec son image par une isotopie Hamiltonienne. Dans le cas où L est un espace K(pi, 1), et en particulier à courbure sectionnelle strictement négative, le minorant est 1 + beta1(L), où beta1 est le premier nombre de Betti à coefficients dans Z2. Une autre conséquence est la non-déplaçabilité d’un plongement Lagrangien monotone de RPn × K (où K est une sous-variété à courbure sectionnelle strictement négative telle que H1(K, Z) ≠ 0) dans certaines variétés symplectiques. Dans la seconde partie, on considère une sous-variété Lagrangienne monotone L non déplaçable. En utilisant l’homologie de Floer définie pour les Lagrangiennes qui sont C-1-proches de L, on obtient des informations sur son nombre de Maslov. De plus, si L peut être approchée par une suite de Lagrangiennes déplaçables, alors, sous certaines hypothèses topologiques sur L, l’énergie de déplacement des éléments de cette suite tend vers l’infini. / N the first part of the thesis, we give, under some hypotheses, a lower bound on the intersection number of a closed monotone Lagrangian submanifold L with its image by a generic Hamiltonianisotopy. For monotone Lagrangian submanifolds L which are K(pi, 1) and, in particular with negative sectional curvature, this bound is 1 + beta_1(L), where beta_1 is the first Betti number with coefficients in Z_2. Another consequence, is the non-displaceability of a monotone Lagrangian embedding of RPn x K (where K is a submanifold with negative sectional curvature such that H^1(K, Z) ≠ 0) in some symplectic manifolds. In the second part, given a closed monotone Lagrangian submanifold L, which is not displaceable, we use Floer homology defined on Lagrangians which are C^1 - close to L, to get information about it Maslov number. Besides, if L can be approached by a sequence of displaceable Lagrangians, then, under some topological assumptions on L, the displacement energy of the elements of this sequence converge to infinity.

This thesis is split up into two parts each revolving around Floer homology and quantum cohomology of closed monotone symplectic manifolds. In the first part we consider symplectic manifolds obtained by symplectic reduction. Our main result is that a quantum version of an abelianization formula of Martin holds, which relates the quantum cohomologies of symplectic quotients by a group and by its maximal torus. Also we show a quantum version of the Leray-Hirsch theorem for Floer homology of Lagrangian intersections in the quotient. The second part is devoted to Floer homology of a pair of monotone Lagrangian submanifolds in clean intersection. Under these assumptions the symplectic action functional is degenerated. Nevertheless Frauenfelder defines a version of Floer homology, which is in a certain sense an infinite dimensional analogon of Morse-Bott homology. Via natural filtrations on the chain level we were able to define two spectral sequences which serve as a tool to compute Floer homology. We show how these are used to obtain new intersection results for simply connected Lagrangians in the product of two complex projective spaces. The link between both parts is that in the background the same technical methods are applied; namely the theory of holomorphic strips with boundary on Lagrangians in clean intersection. Since all our constructions rely heavily on these methods we also give a detailed account of this theory although in principle many results are not new or require only straight forward generalizations.

This thesis is split up into two parts each revolving around Floer homology and quantum cohomology of closed monotone symplectic manifolds. In the first part we consider symplectic manifolds obtained by symplectic reduction. Our main result is that a quantum version of an abelianization formula of Martin holds, which relates the quantum cohomologies of symplectic quotients by a group and by its maximal torus. Also we show a quantum version of the Leray-Hirsch theorem for Floer homology of Lagrangian intersections in the quotient. The second part is devoted to Floer homology of a pair of monotone Lagrangian submanifolds in clean intersection. Under these assumptions the symplectic action functional is degenerated. Nevertheless Frauenfelder defines a version of Floer homology, which is in a certain sense an infinite dimensional analogon of Morse-Bott homology. Via natural filtrations on the chain level we were able to define two spectral sequences which serve as a tool to compute Floer homology. We show how these are used to obtain new intersection results for simply connected Lagrangians in the product of two complex projective spaces. The link between both parts is that in the background the same technical methods are applied; namely the theory of holomorphic strips with boundary on Lagrangians in clean intersection. Since all our constructions rely heavily on these methods we also give a detailed account of this theory although in principle many results are not new or require only straight forward generalizations.:1. Introduction 2. Overview of the main results 2.1. Abelianization . 2.2. Quantum Leray-Hirsch theorem 2.3. Floer homology of Lagrangians in clean intersection 3. Background 3.1. Symplectic geometry . 3.2. Hamiltonian action functional 3.3. Morse homology . 3.4. Floer homology 4. Asymptotic analysis 4.1. Main statement . 4.2. Mean-value inequality . 4.3. Isoperimetric inequality 4.4. Linear theory 4.5. Proofs 5. Compactness 5.1. Cauchy-Riemann-Floer equation . 5.2. Local convergence . 5.3. Convergence on the ends 5.4. Minimal energy . 5.5. Action, energy and index estimates 6. Fredholm Theory 6.1. Banach manifold . 6.2. Linear theory 7. Transversality 7.1. Setup 7.2. R-dependent structures 7.3. R-invariant structures . 7.4. Regular points . 7.5. Floer’s ε-norm . 8. Gluing 8.1. Setup and main statement 8.2. Pregluing . 8.3. A uniform bounded right inverse 8.4. Quadratic estimate 8.5. Continuity of the gluing map 8.6. Surjectivity of the gluing map 8.7. Degree of the gluing map 8.8. Morse gluing . 9. Orientations 9.1. Preliminaries and notation 9.2. Spin structures and relative spin structures 9.3. Orientation of caps 9.4. Linear theory . 10.Pearl homology 10.1. Overview . 10.2. Pearl trajectories . 10.3. Invariance . 10.4. Spectral sequences 11.Proofs of the main results 11.1. Abelianization Theorem 11.2. Quantum Leray-Hirsch Theorem . 12.Applications 12.1. Quantum cohomology of the complex Grassmannian 12.2. Lagrangian spheres in symplectic quotients A. Estimates A.1. Derivative of the exponential map A.2. Parallel Transport A.3. Estimates for strips B. Operators on Hilbert spaces B.1. Spectral gap B.2. Flow operator C. Viterbo index D. Quotients of principal bundles by maximal tori D.1. Compact Lie groups D.2. The cohomology of the quotient of principle bundles by maximal tori