Denote by $\mathcal{P} (M)$ the space of smooth plane fields(oriented and transversely oriented) on a given closed and orientable 3-manifold $M$ with the $C^{\infty}$ topology, and by $\mathcal{F}(M)$ the subspace of integrable plane fields.

Edit: oriented means that planes are oriented coherently and transversely oriented means that the normal directions is oriented coherently. An integrable plane field is one that comes from tangent planes of a foliation.

I was wondering what is known about the topology of subspaces of $\mathcal{P} (M)$ , in particular is the inclusion $i: \mathcal{F}(M) \longrightarrow \mathcal{P} (M)$ a (weak) homotopy equivalence? What can be said about other subspaces, for example the one corresponding to taut (res. Reebless) foliations?

Edit: A taut foliation is one that every leaf has a closed transversal and a Reebless foliation is one that has no Reeb component(which is a solid torus foliated in a certain way. Every taut foliation is Reebless)

Existence and classification of overtwisted contact structures in all dimensions, arXiv:1404.6157. $\endgroup$1more comment