Symplectic 4–manifolds with a Free Circle Action

Let M be a symplectic 4–manifold admitting a free circle action. In this paper we show that, modulo suitable subgroup separability assumptions, the orbit space N admits a fibration over the circle. The separability assumptions are known to hold in several cases: in particular, this result covers the case where N has vanishing Thurston norm, or is a graph manifold. Furthermore, combining this result with the Lubotzky alternative, we show that if the symplectic structure has trivial canonical bundle then M is a torus bundle over a torus, confirming a folklore conjecture. We also generalize various constructions of symplectic structures on 4–manifold with a free circle action. The combination of our results allows us in particular to completely determine the symplectic cone of a 4–manifold with a free circle action such that the orbit space is a graph manifold.