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

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

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Cette thèse explore les propriétés de rigidité et de flexibilité des cobordismes lagrangiens immergés entre sous-variétés lagrangiennes de variétés symplectiques. Dans le premier article de cette thèse, intitulé On cobordism groups of Lagrangian immersions, on s’intéresse aux aspects flexibles des cobordismes lagrangiens. On y étudie les groupes de cobordisme d’immersions lagrangiennes \( \Omega^{\operatorname{lag}}(M) \) d’une variété symplectique \( M \). Il s’agit d’un sujet classique dont l’étude a été initiée par Arnold au début des années 80. Étendant un théorème dû à Eliashberg dans le cas des variétés symplectiques exactes, nous démontrons que le calcul de \( \Omega^{\operatorname{lag}}(M) \) se réduit à un problème de théorie de l’homotopie stable. Plus précisément, nous associons à toute variété symplectique \( M \) un spectre de Thom et démontrons que le groupe \( \Omega^{\operatorname{lag}}(M) \) s’exprime en terme des groupes d’homotopie stables de ce spectre. L’ingrédient principal de la preuve est le h-principe de Gromov- Lees, qui a pour conséquence que le problème d’existence des immersions lagrangiennes se réduit à un problème de topologie algébrique. Dans le second article de cette thèse, intitulé Unobstructed Lagrangian cobordism groups of surfaces, on s’intéresse aux aspects rigides des cobordismes lagrangiens dans le cas de surfaces symplectiques \( \Sigma \) de genre \( g \geq 2 \). On y étudie une classe de cobordismes lagrangiens immergés qui satisfont une contrainte sur les disques holomorphes qu’ils bordent, ce qui permet de leur appliquer les techniques de la théorie de Floer. On dit alors de ces cobordismes qu’ils sont non-obstrués. Les principaux résultat de ce second article sont, d’une part, le calcul du groupe de cobordisme non-obstrué \( \Omega_{\operatorname{unob}}(\Sigma) \) et, d’autre part, la construction d’un isomorphisme naturel entre \( \Omega_{\operatorname{unob}}(\Sigma) \) et le groupe de Grothendieck de la catégorie de Fukaya dérivée de \( \Sigma \). Cela résout, dans le cas des surfaces fermées de genre \( g \geq 2\), un problème posé par Biran et Cornea. / This thesis explores the rigidity and flexibility properties of immersed Lagrangian cobordisms between Lagrangian submanifolds of symplectic manifolds. In the first article of this thesis, titled On cobordism groups of Lagrangian immersions, we are interested in the flexible aspects of Lagrangian cobordisms. We study the cobordism group of Lagrangian immersions \( \Omega^{\operatorname{lag}}(M) \) of a symplectic manifold \( M \). This is a classical topic in symplectic topology, whose study was initiated by Arnold in the 80s. Generalizing a theorem due to Eliashberg in the exact case, we show that the computation of \( \Omega^{\operatorname{lag}}(M) \) reduces to a problem in stable homotopy theory. More precisely, we associate to every symplectic manifold \( M\) a Thom spectrum and show that the group \( \Omega^{\operatorname{lag}}(M) \) can be expressed in terms of the stable homotopy groups of this spectrum. The main ingredient of the proof is the celebrated h-principle of Gromov-Lees, which reduces the existence problem for Lagrangian immersions to a purely topological problem. In the second article of this thesis, titled Unobstructed Lagrangian cobordism groups of surfaces, we are interested in the rigid aspects of Lagrangian cobordisms in the case of symplectic surfaces \( \Sigma \) of genus \( g \geq 2 \). We study a class of immersed Lagrangian cobordism satisfying a constraint on the holomorphic disks that they bound, which makes them amenable to Floer-theoretic methods. Such cobordisms are called unobstructed. The main results of the second article are, on one hand, the computation of the unobstructed cobordism group \( \Omega_{\operatorname{unob}}(\Sigma) \) and, on the other hand, the construction of a natural isomorphism between \( \Omega_{\operatorname{unob}}(\Sigma) \) and the Grothendieck group of the derived Fukaya category of \(\Sigma \). This provides an answer, in the case of surfaces of genus \( g \geq 2\), to a question posed by Biran and Cornea.