### Abstract

For a transversal pair of closed Lagrangian submanifolds $L, L’$ of a symplectic manifold $M$ such that $\pi_{1}(L)=\pi_{1}(L’)=0=c_{1}|_{\pi_ {2}(M)}=\omega|_{\pi_{2}(M)}$ and for a generic almost complex structure $J$, we construct an invariant with a high homotopical content which consists in the pages of order $\geq 2$ 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, 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 $\Omega L\to PL\to L$ and we deduce some applications.