Lusternik-Schnirelman-theory for Lagrangian intersections

Consider a compact symplectic manifold (M, ω) together with a pair (L, L_1) of isotopic compact Lagrangian submanifolds such that π_2(M, L) = 0 . Using Gromov’s theory of (almost) holomorphic curves the cohomological properties of a family of holomorphic disks are studied. By means of a stretching construction for those disks and a Lusternik–Schnirelman theory in compact topological spaces cuplength estimates for the intersection set L ∩ L_1 are derived. Résumé On examiṅe une variété symplectique compacte (M, ω) munie d’un couple (L_0, L_1) de sous-variétés lagrangiennes isotopes telles que π_2(M, L) = 0 . En utilisant la théorie des courbes presque holomorphes, développée par Gromov, on étudie la cohomologie de certaines familles de disques holomorphes. On en déduit, par la théorie de Lusternik–Schnirelman, des estimations du cuplength de l’intersection L_0 ∩ L_1 .