A spectral sequence for Lagrangian Floer homology
We prove the existence of a spectral sequence for Lagrangian Floer homology which converges to the Floer homology of the image of a Lagrangian submanifold under multiple fibred Dehn twists. The $E_1$ term of the sequence is given by the hypercube of "resolutions" of the Dehn twists involved. The proof relies on the exact triangle for fibered Dehn twists due to Wehrheim and Woodward. As applications we obtain a spectral sequences from Khovanov homology to symplectic Khovanov homology. Also when a 3-manifold $M$ is given by gluing two handlebodies by a surface diffeomorphism $ϕ$, we obtain a spectral sequence converging to the Heegaard-Floer homology of $M$, whose $E_1$ term is a hypercube obtained from different ways of resolving the Dehn twists in $ϕ$. This latter sequence generalizes the spectral sequence of branched double covers to general closed 3-manifolds (i.e. those which are not branched double covers of links) however its $E_2$ term is not a 3-manifold invariant. This gives upper bounds on the rank of HF-hat of closed 3-manifolds.