Poisson structures on smooth 4-manifolds

Abstract

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.

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-Ratiu

References in corpus (5)

• Every 4-Manifold is BLF
• Poisson brackets with prescribed Casimirs
• On genus-1 simplified broken Lefschetz fibrations
• Existence of broken Lefschetz fibrations
• Poisson structures on wrinkled fibrations

Cited by in corpus (7)

• On Bott-Morse Foliations and their Poisson Structures in Dimension 3
• Poisson structures on wrinkled fibrations
• On Computational Poisson Geometry II: Numerical Methods
• On Computational Poisson Geometry I: Symbolic Foundations
• Compatible Poisson Structures on Fibered $5$-Manifolds
• Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6
• Poisson structures of near-symplectic manifolds and their cohomology