Construction of symplectic structures on 4-manifolds with a free circle action (Q2882514)

Suppose that \(M\) is a closed, oriented \(4\)-manifold with a free circle action. Denote the orbit space by \(N\) and by \(p:M\rightarrow N\) the quotient map, which gives a principal \(S^1\)-bundle over the \(3\)-manifold \(N.\) Let \(p_*:H^*(M,{\mathbb R}) \rightarrow H^{* -1}(N,{\mathbb R})\) be the push forward map defined by integration along the fibre. The authors prove that for a \(2\)-form \(\psi\in H^2 (M,{\mathbb R}) \) such that \(\psi^2 >0\) and \(p_* (\psi)\) can be represented by a nowhere zero closed \(1\)-form, there exists an \(S^1\)-invariant symplectic form on \(M\) which is cohomologous to \(\psi.\) This generalizes earlier results by Thurston, Bouyakoub and Fernández. The authors use the foliation defined by a nowhere zero closed \(1\)-form on the \(3\)-manifold \(N\) and the Poincaré duality theorem to construct the required symplectic form. There a technical approximation of a cohomology class with real coefficients by a rational and ultimately an integral one is used. Finally, the authors use their results to determine completely the symplectic cone when the principal \(S^1\)-bundle is trivial and a symplectic manifold.