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.

arXiv commentv3: 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

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• Singular Lefschetz pencils
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• Wrinkled fibrations on near-symplectic manifolds
• Lagrangian matching invariants for fibred four-manifolds: I
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• Existence of broken Lefschetz fibrations
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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