Lagrangian intersections and the Serre spectral sequence

Abstract

For a transversal pair of closed Lagrangian submanifolds L, L′ of a symplectic manifold M such that π1(L) = π1(L′) = 0 = c1|π2(M) = ω|π2(M) and for a generic almost complex structure J, we construct an invariant with a high homotopical content which consists in the pages of order ≥ 2 of a spectral sequence whose differentials provide an algebraic measure of the high-dimensional moduli spaces of pseudo-holomorpic strips of finite energy that join L and L′. When L and L′ are Hamiltonian isotopic, we show that the pages of the spectral sequence coincide (up to a horizontal translation) with the terms of the Serre spectral sequence of the path-loop fibration ΩL → PL → L and we deduce some applications.