Symplectomorphism Groups and Isotropic Skeletons

(Accepted by Geometry and Topology) The symplectomorphism group of a 2-dimensional surface S is homotopy equivalent to the orbit of a filling system of curves. We give a generalization of this statement to dimension 4. The filling system of curves is replaced by the Biran decomposition of the symplectic 4 manifold M into a disjoint union of an isotropic 2-complex L and a disc bundle over a symplectic surface \Sigma, Poincare dual to a multiple of the form. We show that one can recover the homotopy type of the symplectomorphism group of M from the orbit of the pair (L,\Sigma). This allows us to compute the homotopy type of certain spaces of Lagrangian submanifolds, for example the space of Lagrangian RP^2 in CP^2 which are isotopic to the standard one.math.SG/0404496: