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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)

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  • $\begingroup$ I have deleted my wrong answer. Also I guess you are aware of Thurston's theorem on foliations. mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0547.0550.ocr.pdf $\endgroup$ – Lev Soukhanov Nov 19 '14 at 23:13
  • $\begingroup$ Thanks for the reference, yes, I've seen that paper. $\endgroup$ – Mehdi Yazdi Nov 19 '14 at 23:15
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    $\begingroup$ In the following paper it is proved that there are only finite number of homotopy classes of plane fields that contain a reebless foliation: D. Gabai, Essential laminations and Kneser normal form. J. Diff. Geom. 53 (1999), 517–574. So your map is not a homotopy equivalence when restricted to reebless (or even to taut) foliations. Eliashberg proved that inclusion of overtwisted contact structures to the space of plane fields is weak homotopy equivalence. I can not give a precise link though. $\endgroup$ – Maxim Prasolov Nov 21 '14 at 6:29
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    $\begingroup$ Eliashberg's result on overtwisted contact structures mentioned by Maxim above has now been generalized to higher dimensions. This is contained in the paper by Borman-Eliashberg-Murphy: Existence and classification of overtwisted contact structures in all dimensions, arXiv:1404.6157. $\endgroup$ – Oldřich Spáčil Nov 21 '14 at 12:33
  • $\begingroup$ Thanks @MaximPrasolov. Actually Eliashberg's result was the motivation for my question. $\endgroup$ – Mehdi Yazdi Nov 21 '14 at 16:21

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