Please forgive me if this is too simple for MO; most of my posts on anything contact-structure-related in Math Stack, other sites, have barely received answers (maybe because I'm not an expert in the area and my questions are not clear-enough).

1)What do we mean when we say two contact structures $(M,\xi); (M,\xi')$ are homotopic to each other ( as plane fields), and, 2) what is the precise meaning of Eliashberg's classification of O.T contact structures up to homotopy classes of plane fields?

For the first, naively, I would say that there is a pointwise homotopy of planes $\xi_p \rightarrow \xi'_p$, and the homotopy takes place in the ambient space M. But if this was the case, then there would only be , e.g., one class in $ \mathbb R^3 $ . Do we then consider the orientation of the planes relative to the orientation of the ambient space? This seems too complicated.

For the second, according to results by Eliashberg, the classification of overtwisted structures is equivalent to the classification of homotopy classes of plane fields. Do we mean that homotopy classes of planes coincide with contactomorphic classes of OT structures, i.e., if $(M^3, \xi )$ is overtwisted and H is a homotopy between the contact planes $\xi(p), \xi'(p)$ , then $\xi'$ is contactomorphic with $\xi$?.

Thanks for any help, refs., etc.


I hope I'm not mucking up some definitions below, as I am by no means an expert, but let me attempt to answer your questions. No doubt you have a copy, but just for others, here is a link to the relevant paper of Eliashberg, by the way.

You might also find chapter 4 of Geiges's book "An Introduction to Contact Topology" useful.

(1) Recall that a contact structure on a $(2n+1)$-dimensional manifold is a codimension 1 tangent distribution that can be locally defined by the kernel of a 1-form $\alpha$ satisfying $\alpha\wedge(d\alpha)^n\neq0$. Two rank $k$ distributions (e.g. two contact structures) are said to be homotopic as distributions if there is a smooth family of rank $k$ distributions beginning at the first and ending at the second. Being homotopic as plane fields is just the special case for rank 2 distributions.

If you look at section 1.5 of Eliashberg's paper you'll see that he phrases "homotopic as plane fields" in terms of contact structures lying in the same connected component of the space of 2-plane distributions.

This notion is to be distinguished from being homotopic as contact structures which requires that the family of distributions must satisfy the contact condition throughout. Eliashberg talks about two contact structures lying in the same connected component of the space of contact structures.

As you point out, there is only one homotopy class of plane fields on $\mathbb{R}^3$, however another paper of Eliashberg proves that there are an infinite number of distinct classes of contact structures pairwise non-homotopic as contact structures. Thus the two notions of homotopy are generally quite different.

(2) Eliashberg's theorem on the classification of overtwisted contact structures states that despite the difference in definitions above, when we look at the subset of contact structures that are overtwisted (whose definition I won't give here) the two notions coincide!

Let me try to rephrase your last paragraph. If $(M^3,\xi_0)$ and $(M^3,\xi_1)$ are both overtwisted contact structures and are homotopic as plane fields (i.e. we can find a smooth family of plane fields on $M$ which go from $\xi_0$ to $\xi_1$), then indeed, they are contactomorphic, and we can even find a smooth family of contact structures $(M^3,\xi_t)$ which connect them.

  • $\begingroup$ Is this homotopy a pointwise one, i.e., for fixed p in $M^3$, we have $\xi_0 (p)$ homotopic to $\xi_1 (p) $ , for all p in $M^3$? I would think obviously yes, but I've been wrong way to often when I thought something was "obviously" $\endgroup$ – Kontakt Mar 30 '14 at 23:38
  • $\begingroup$ Sorry for taking so long to get back to you. Yes, that is indeed true; indeed these homotopies depend smoothly on $p$ as well. $\endgroup$ – j.c. Apr 4 '14 at 9:39

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