Lagrangian Intersections and the Serre Spectral Sequence

For a transversal pair of closed Lagrangian submanifolds L, L of a symplectic manifold M so that π1(L) = π1(L′) = 0 = c1|π2(M) = ω|π2(M) and 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 highdimensional moduli spaces of pseudo-holomorpic strips of finite energy that join L and L. When L and L are hamiltonian isotopic, these pages coincide (up to a horizontal translation) with the terms of the Serre-spectral sequence of the path-loop fibration ΩL → PL → L. Among other applications we prove that, in this case, each point x ∈ L\L belongs to some pseudo-holomorpic strip of symplectic area less than the Hofer distance between L and L.