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 highdimensional 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.
