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We establish a long exact sequence for Legendrian submanifolds , where is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of to off of itself. In this sequence, the singular homology maps to linearized contact cohomology , which maps to linearized contact homology , which maps to singular homology. In particular, the sequence implies a duality between and . Furthermore, this duality is compatible with Poincaré duality in in the following sense: the Poincaré dual of a singular class which is the image of maps to a class such that .
The exact sequence generalizes the duality for Legendrian knots in (see ) and leads to a refinement of the Arnold conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in 
We show that the zero locus of a normal function on a smooth complex algebraic variety is algebraic provided that the normal function extends to an admissible normal function on a smooth compactification such that the divisor at infinity is also smooth. This result, which has also been obtained recently by M. Saito using a different method , generalizes a previous result proved by the authors for admissible normal functions on curves 
Nous étudions l'influence de la topologie d'une surface hyperbolique sur le nombre des valeurs propres de son Laplacien qui sont inférieures ou égales à . Le premier résultat de l'article est un énoncé du type “trou spectral”, son titre, dans lequel est la -ième des valeurs propres du Laplacien et est le genre de la surface. Une construction classique dûe à Buser montre que ce résultat est optimal. Nous donnons aussi un énoncé du même type pour les surfaces de volume fini. Les méthodes prolongent celles de [O], qui utilisaient de manière essentielle l'approche topologique par Sévennec de la question de la majoration de la multiplicité de la deuxième valeur propre des opérateurs de Schrödinger [Se].
We study the influence of the topology of a hyperbolic surface on the number of its Laplace eigenvalues that are at most . The first result of the article is a “spectral gap” statement, namely, its title where is the th of the eigenvalues of the Laplace operator and where is the genus of the surface. A classical construction due to Buser shows that this result is sharp. We give a similar statement for finite volume surfaces. The methods develop those found in [O], which used in an essential way the topological approach of Sévennec to the question of bounding the multiplicity of the second eigenvalue for Schrödinger operators [Se]
The critical points of the length function on the free loop space of a compact Riemannian manifold are the closed geodesics on . The length function gives a filtration of the homology of , and we show that the Chas-Sullivan product is compatible with this filtration. We obtain a very simple expression for the associated graded homology ring when all geodesics are closed, or when all geodesics are nondegenerate. We also interpret Sullivan's coproduct (see [Su1], [Su2])) on as a product in cohomology (where is the constant loop). We show that is also compatible with the length filtration, and we obtain a similar expression for the ring The nonvanishing of products and is shown to be determined by the rate at which the Morse index grows when a geodesic is iterated. We determine the full ring structure for spheres ,
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