Thurston's h-principle and Flexibility of Poisson Structures

TL;DR

This paper extends Thurston's h-principle to higher-dimensional manifolds with fiber-wise non-degenerate 2-forms, revealing new flexibility properties of Poisson structures in both open and closed 4-manifolds.

Contribution

It establishes an analogue of Thurston's h-principle for 2-dimensional foliations with non-degenerate forms, advancing understanding of Poisson structure flexibility.

Findings

Flexibility of rank 2 regular Poisson structures on open manifolds

Flexibility of Poisson structures on closed 4-manifolds

Extension of Thurston's h-principle to higher dimensions

Abstract

We prove an analogue of Thurston's h-principle for $2$-dimensional foliations on manifolds of dimension bigger or equal to $4$, in the presence of a fiber-wise non-degenerate $2$-form. This helps us understand the flexibility of rank $2$ regular Poisson structures on open manifolds with dimension bigger or equal to $4$ and it also helps us understand the flexibility of Poisson structures (not regular) on closed $4$-manifolds.