  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'.
