Poisson Structures on Smooth 4–Manifolds

We show that every closed oriented smooth 4-manifold admits a complete singular Poisson structure in each homotopy class of maps to the 2-sphere. The rank of this structure is 2 outside a small singularity set, which consists of finitely many circles and isolated points. The Poisson bivector has rank 0 on the singularities, where we give its local form explicitly.Comment: v3: 17pgs. We shortened, and streamlined, both the proof and the exposition. The main result now follows from a formula used by Damianou-Petalidou, attributed to Flaschka-Rati