Symplectic forms invariant under free circle actions on 4--manifolds

Let $M^4$ be a smooth compact manifold with a free circle action generated by a vector field $X\mathrm{.}$ Then for any invariant symplectic form $\omega$ on $M$ the contracted form ${\iota }_X\omega$ is nondegenerate. Using the map $\omega --> \iota_X \omega$andtherelatedmapto$H^1(M / S^1, R)$ we study the topology of the space $S_{inv}$ of invariant symplectic forms. In particular we give a description of ${\pi }_0{S}_{inv}(M^4)$ in terms of the unit Thurston ball.