nLab Haefliger theorem

Idea

On the classification of foliations:

Statement

Theorem

If $X$ is an open manifold, then there is a bijection between

1. codimension q foliations on $X$ up to integral homotopy;

2. homotopy classes of fiberwise surjective vector bundle maps $T X \to N \Gamma_q$,

where $\Gamma_q$ is the topological groupoid of germs of local diffeomorphisms of $\mathbb{R}^q$.

Related concepts

• Haefliger groupoid

References

• John Francis, notes by M. Hoyois, Haefliger’s theorem classifying foliations over open manifold (pdf)