Poisson structures on smooth 4-manifolds

TL;DR

This paper proves that every closed oriented smooth 4-manifold can be equipped with a complete singular Poisson structure in each homotopy class of maps to the 2-sphere, with specific singularity characteristics.

Contribution

It demonstrates the existence of such Poisson structures on all closed oriented 4-manifolds, detailing their singularity types and local forms.

Findings

Existence of complete singular Poisson structures on all closed oriented 4-manifolds.

Singularity set consists of finitely many circles and isolated points.

Explicit local form of the Poisson bivector at singularities.

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.