We establish a long exact sequence for Legendrian submanifolds $L\subset P\times \mathbb{R}$, where $P$ is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of $L$ to $P$ off of itself. In this sequence, the singular homology $H_{*}$ maps to linearized contact cohomology ${\mathrm{CH}}^{*}$, which maps to linearized contact homology ${\mathrm{CH}}_{*}$, which maps to singular homology. In particular, the sequence implies a duality between $\mathrm{Ker}({\mathrm{CH}}_{*}\to H_{*})$ and ${\mathrm{CH}}^{*}/\mathrm{Im}(H_{*})$. Furthermore, this duality is compatible with Poincaré duality in $L$ in the following sense: the Poincaré dual of a singular class which is the image of $a\in {\mathrm{CH}}_{*}$ maps to a class $\alpha \in {\mathrm{CH}}^{*}$ such that $\alpha (a)=1$.

The exact sequence generalizes the duality for Legendrian knots in ${\mathbb{R}}^3$ (see [26]) 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 [7]

We show that the zero locus of a normal function on a smooth complex algebraic variety $S$ 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 [22], generalizes a previous result proved by the authors for admissible normal functions on curves [4]

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 $1/4$. Le premier résultat de l'article est un énoncé du type “trou spectral”, son titre, dans lequel ${\lambda }_j$ est la $j$-ième des valeurs propres ${\lambda }_0=0<{\lambda }_1\le \cdot \cdot \cdot \le {\lambda }_j\le \cdot \cdot \cdot$ du Laplacien et $g$ 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 $1/4$. The first result of the article is a “spectral gap” statement, namely, its title where ${\lambda }_j$ is the $j$th of the eigenvalues ${\lambda }_0=0<{\lambda }_1\le \cdot \cdot \cdot \le {\lambda }_j\le \cdot \cdot \cdot$ of the Laplace operator and where $g$ 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 $\Lambda (M)$ of a compact Riemannian manifold $M$ are the closed geodesics on $M$. The length function gives a filtration of the homology of $\Lambda (M)$, and we show that the Chas-Sullivan product ${\displaystyle H_i(\Lambda )\times H_j(\Lambda )\stackrel{*}{\to }{H}_{i+j-n}(\Lambda )}$ is compatible with this filtration. We obtain a very simple expression for the associated graded homology ring $\mathrm{Gr}{H}_{*}(\Lambda (M))$ when all geodesics are closed, or when all geodesics are nondegenerate. We also interpret Sullivan's coproduct $\vee$ (see [Su1], [Su2])) on $C_{*}(\Lambda )$ as a product in cohomology ${\displaystyle H^i(\Lambda ,{\Lambda }_0)\times H^j(\Lambda ,{\Lambda }_0)\stackrel{⊛}{\to }{H}^{i+j+n-1}(\Lambda ,{\Lambda }_0)}$ (where ${\Lambda }_0=M$ is the constant loop). We show that $⊛$ is also compatible with the length filtration, and we obtain a similar expression for the ring $\mathrm{Gr}{H}^{*}(\Lambda ,{\Lambda }_0).$ The nonvanishing of products ${\sigma }^{*n}$ and ${\tau }^{⊛n}$ 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 $(H^{*}(\Lambda ,{\Lambda }_0),⊛)$ for spheres $M=S^n$, $n\ge 3$