Symplectic forms invariant under free circle actions on 4-manifolds

Let $M$ be a smooth closed 4-manifold with a free circle action generated by a vector field $X.$ Then for any invariant symplectic form $\omega$ on $M$ the contracted form $\iota_X\omega$ is non-vanishing. Using the map $\omega \mapsto \iota_X\omega$ and the related map to $H^1(M \slash S^1,\mathbb{R})$ we study the topology of the space $S_{inv}(M)$ of invariant symplectic forms on $M.$ For example, the first map is proved to be a homotopy equivalence. This reduces examination of homotopy properties of $S_{inv}$ to that of the space $\mathcal{N}_L$ of non-vanishing closed 1-forms satisfying certain cohomology conditions. In particular we give a description of $\pi_0S_{inv}(M)$ in terms of the unit ball of Thurston's norm and calculate higher homotopy groups in some cases. Our calculations show that the homotopy type of the space of non-vanishing 1-forms representing a fixed cohomology class can be non-trivial for some torus bundles over the circle. This provides a counterexample to an open problem related to the Blank-Laudenbach theorem (which says that such spaces are connected for any closed 3-manifold). Finally, we prove some theorems on lifting almost complex structures to symplectic forms in the invariant case.