Symplectomorphism groups and isotropic skeletons

The symplectomorphism group of a 2–dimensional surface 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 a decomposition of the symplectic 4–manifold (M,ω) into a disjoint union of an isotropic 2–complex L and a disc bundle over a symplectic surface Σ which is Poincare dual to a multiple of the form ω . We show that then one can recover the homotopy type of the symplectomorphism group of M from the orbit of the pair (L,Σ). This allows us to compute the homotopy type of certain spaces of Lagrangian submanifolds, for example the space of Lagrangian RP ⊂ CP isotopic to the standard one. AMS Classification numbers Primary: 57R17 Secondary: 53D35