Haefliger structures and symplectic/contact structures

TL;DR

This paper develops a two-step method to realize symplectic and contact structures on manifolds by constructing and regularizing Haefliger structures, extending Gromov's h-principle to broader contexts.

Contribution

It introduces a novel two-step approach to Gromov's h-principle for symplectic and contact structures, utilizing Haefliger structures and primitive jiggling techniques.

Findings

Constructs transversely geometric Haefliger structures from formal data.

Regularizes Haefliger structures to genuine geometric structures on open manifolds.

Extends the applicability of the h-principle to closed and open manifolds for certain geometries.

Abstract

For some geometries including symplectic and contact structures on an n-dimensional manifold, we introduce a two-step approach to Gromov's h-principle. From formal geometric data, the first step builds a transversely geometric Haefliger structure of codimension n. This step works on all manifolds, even closed. The second step, which works only on open manifolds and for all geometries, regularizes the intermediate Haefliger structure and produces a genuine geometric structure. Both steps admit relative parametric versions. The proofs borrow ideas from W. Thurston, like jiggling and inflation. Actually, we are using a more primitive jiggling due to R. Thom.