A h-principle for symplectic foliations

TL;DR

This paper extends Gromov's h-principle from symplectic geometry to symplectic foliations, providing a cohomological criterion for homotoping regular bivectors to regular Poisson structures, with practical examples.

Contribution

It introduces an h-principle for symplectic foliations and formulates a cohomological criterion for regular bivectors to become regular Poisson structures.

Findings

Cohomological criterion for homotoping bivectors to Poisson structures

Extension of Gromov's h-principle to symplectic foliations

Example demonstrating the criterion's effectiveness

Abstract

We show that a classical result of Gromov in symplectic geometry extends to the context of symplectic foliations, which we regard as a $h$-principle for (regular) Poisson geometry. Namely, we formulate a sufficient cohomological criterion for a regular bivector to be homotopic to a regular Poisson structure, in the spirit of Haefliger's criterion for homotoping a distribution to a foliation. We give an example to show that this criterion is not too unsharp.