Symplectic forms invariant under free circle actions on 4-manifolds

Abstract

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 ω on M the contracted form Xω is non-vanishing. Using the map ω? iXω and the related map to H 1 (M/S 1 , 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 M L of non-vanishing closed 1-forms satisfying certain cohomology conditions. In particular we give a description of π 0 S 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.