Lagrangian Intersections and the Serre Spectral Sequence

For a transversal pair of closed Lagrangian submanifolds L, L' of a symplectic manifold M so that \(\pi_{1}(L)=\pi_{1}(L')=0=c_{1}|_{\pi_{2}(M)}=\omega|_{\pi_{2}(M)}\) and a generic almost complex structure J we construct an invariant with a high homotopical content which consists in the pages of order \(\geq 2\) 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, these pages coincide (up to a horizontal translation) with the terms of the Serre-spectral sequence of the path-loop fibration \(\Omega L\to PL\to L\). Among other applications we prove that, in this case, each point \(x\in L\backslash L'\) belongs to some pseudo-holomorpic strip of symplectic area less than the Hofer distance between L and L'.