# Tight vs. overtwisted contact structure

I know the familiar differences between tight and overtwisted contact structures. For example, each homotopy class of plane-bundles on a three-manifold has an overtwisted representative but tight contact structures are less common. Also a tight contact structure gives a genus bound on an embedded closed surface and therefore can tell things about the topology of the ambient manifold. My question is what were the motivations behind distinguishing between the two originally? Is there possibly an intuition which makes the two fundamentally different and then proving the above differences comes as a consequence or it was these differences which led to distinguishing between the two? In other words I am trying to see the difference between the two without appealing to the statements like the above.

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Your question will be answered in Geiges' An Introduction to Contact Topology book, section 4.5 "Tight and Overtwisted". –  Chris Gerig Jun 1 '13 at 21:07
I believe taut foliations en.wikipedia.org/wiki/Taut_foliation were studied much earlier, and that this even motivated the name. –  Douglas Zare Jun 2 '13 at 10:26

As mentioned in the previous answer, the first use of the overtwisted/tight dichotomy is most certainly Bennequin's Theorem that there are non-isomorphic contact structures on $\mathbb{R}^3$ and $\mathbb{S}^3$, a landmark result.

However, the relevance of this dichotomy goes now far beyond this. As you probably know, contact topology has a Darboux theorem: locally, all contact structures are the same, isomorphic to the standard contact structure on $\mathbb{R}^3$. So, all of them are "locally tight", and having an overtwisted disc must be a global condition. A global way of distinguishing objects which are locally the same is tremendously important, here it can be thought of as some analogue of Gromov's non-squeezing theorem in symplectic geometry.

Moreover, while the classification of overtwisted structure has been achieved quite early by Eliashberg (Inventiones 1989), the tight contact structures happened to be very rich (see e.g. Giroux, Inventiones 1999 and its introduction -- in French) : certain manifold have only one of them, but infinitely many are shown to bear infinitely many non-isomorphic tight contact structure in the paper cited. The relation with symplectic fillings is one more indication of the relevance of this dichotomy. One can now consider that the tight contact structure are the one to study, as overtwisted ones are pretty well understood.

Note that the relevance of this dichotomy has recently been more firmly established in higher dimensions too by Niederkrüger, Massot and Wendl (Inventiones 2013).

About Bennequin's theorem, I can not explain its proof, although I learned one in my graduate years. I cannot give a complete account of it any more, but I can give some flavor. Consider the usual cylindrically symmetric version of the standard contact structure, and decompose $\mathbb{R}^3$ as a trivial open book (the vertical line through the origin is the binding, and each half plane it bounds is a page). Assume there is a horizontal (in the sense of the contact structure) which bounds a disc, which you assume in general position, and look at the way this disc intersects the pages. In each page the non-transversal points are vertices of a graph, so you get combinatorial objects to work with. That's pretty much what I remember, but there are several proofs in the literature (at least one by Giroux, but likely to be written in French; it is probably not a waste of time to learn enough French to read mathematics if you are planning to work in contact topology). In fact, a friedn of mine working in this area once told me that it was almost a duty for anyone working in this area to find its own proof of Bennequin's theorem. You should be ale to locate several proofs through the literature, but I am too far from this field to help you there.

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Perhaps I'm wrong but I thought that historically the notions were considered first in the work of Bennequin who proved the existence of contact structures not equivalent to the standard contact structure on R^3 just by using that they have overtwisted disks and by showing (I think this was his main result) that the standard contact structure does not have overtwisted disks, i.e., is tight. see Seminaire Bourbaki

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I cannot read french. Can you explain a bit how he proved there is no overtwisted disk ? I think symplectic filling implies tightness was discovered later... –  nikita Jun 2 '13 at 3:01
The boundary of an overtwisted disc violates Bennequin's inequality. Bennequin didn't have the definition of OT disc or tightness: he just proved the inequality $tb-r\le 2g-1$ for $\xi_{st}$, which in retrospective shows that the standard $S^3$ is tight. I guess that a possible answer to your question is that the OT contact structures on $S^3$ were considered to be exotic, and their common feature was the violation of Bennequin's inequality for the unknot (i.e. the existence of an overtwisted disc), and this feature was shown to be the "right" one by Eliashberg (keyword: flexibility). –  Marco Golla Jun 2 '13 at 5:53