On the classification of foliations:
If $X$ is an open manifold, then there is a bijection between
codimension q foliations on $X$ up to integral homotopy;
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 $\mathbb{R}^q$.
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