Symplectomorphism groups and isotropic skeletons

Abstract

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 $\left(M,\omega \right)$ into a disjoint union of an isotropic 2–complex $L$ and a disc bundle over a symplectic surface $\Sigma$ which is Poincare dual to a multiple of the form $\omega$. We show that then one can recover the homotopy type of the symplectomorphism group of $M$ from the orbit of the pair $\left(L,\Sigma \right)$. This allows us to compute the homotopy type of certain spaces of Lagrangian submanifolds, for example the space of Lagrangian ${ℝ\mathbb{P}}^2\subset {ℂ\mathbb{P}}^2$ isotopic to the standard one.