CR structures on open manifolds

TL;DR

This paper demonstrates that on open manifolds, the vanishing of higher homology groups guarantees that almost CR structures can be homotoped to genuine CR structures, extending foliation methods to CR geometry.

Contribution

It adapts Haefliger's foliation technique to show the homotopy equivalence of almost CR and CR structures under homological conditions on open manifolds.

Findings

Higher homology vanishing implies CR structures from almost CR structures

Method adapts foliation techniques to CR geometry

Results apply specifically to open manifolds

Abstract

We show that the vanishing of the higher dimensional homology groups of a manifold ensures that every almost CR structure of codimension $k$ may be homotoped to a CR structure. This result is proved by adapting a method due to Haefliger used to study foliations (and previously applied to study the relation between almost complex and complex structures on manifolds) to the case of (almost) CR structures on open manifolds.