Poisson Structures on Smooth 4–Manifolds

Abstract

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 vanishes on the singularities, where we give its local form explicitly.

Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6

“…Then there exists a complete Poisson structure whose symplectic leaves correspond to the fibres of the given fibration structure, and the singularities of both the fibration and the Poisson structures coincide. Moreover, for each singularity, the Poisson bivector and induced symplectic form on the leaves are given by the following equations:Folds: 13) If none of its symplectic leaves are, or contain, 2-spheres, then this Poisson structure is integrable.The existence of a Poisson structure with the stated properties follows from Theorem 2.11, previously shown by the first and third named authors together with García-Naranjo [10]. The proof of this theorem follows from an application of Theorem 2.11 and the definition of completeness.These results allow us to present in section 2.5.1 countably many examples of Poisson structures on the same underlying smooth manifold that are Morita inequivalent.…”

“…The existence of a Poisson structure with the stated properties follows from Theorem 2.11, previously shown by the first and third named authors together with García-Naranjo [10]. The proof of this theorem follows from an application of Theorem 2.11 and the definition of completeness.…”

“…Poisson geometry has newly entered the picture, particularly in dimension 4. A singular Poisson bivector of rank 2 that vanishes on the singularity set of a bLf and whose symplectic foliation matches the fibres of the map can be given [10]. This in turn implies that on any homotopy class of maps from a 4-manifold to S 2 there is such a singular Poisson structure.…”

On Bott-Morse Foliations and their Poisson structures in dimension three

“…We will need to do a smooth interpolation between the singular and the regular Poisson structures. For this we will use the following lemmata (2.8 and 2.9 in [12]). Lemma 6.1.…”

“…In the proof we combine methods to determine the bivectors and symplectic forms of a Poisson structure for which the associated Casimir functions can be described explicitly together with a gluing construction that collects all the possible local Poisson structures into a global one. This involves using the work of the second and fourth named authors [12], as we rely on certain S 1 -invariant Poisson structures of dimension 4 in our arguments. Restricting these to 3-dimensional slices we find the Poisson structures associated to the original Bott-Morse foliation.…”