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

Abstract

Let $M^4$ be a smooth compact manifold with a free circle action generated by a vector field $X.$ Then for any invariant symplectic form $ω$ on $M$ the contracted form $ι_X ω$ is nondegenerate. Using the map $ω--> ι_X ω$ and the related map to $H^1(M / S^1, R)$ we study the topology of the space $S_{inv}$ of invariant symplectic forms. In particular we give a description of $π_0 S_{inv}(M^4)$ in terms of the unit Thurston ball.