I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure.
Can someone provide me with some examples?
[Math] Examples of odd-dimensional manifolds that do not admit contact structure
at.algebraic-topologycontact-geometrydg.differential-geometrydifferential-topology
Related Solutions
I will respond under the impression that you're primarily interested in contact 3-manifolds. Much more is known in this low-dimensional case as techniques such as Dehn surgery etc. can be adapted (with some slightly non-trivial work) to work for contact manifolds in this dimension.
1) One easy way to construct contact manifolds is using the Thurston-Winkelnkemper construction. For dimension-3, this construction asserts that if you have an open book decomposition of your manifold of interest $M=M^{3}$, then that decomposition uniquely determines a contact structure on $M$. This decomposition consists of a link $B\subset M$ and a fibration $\pi:M\setminus B \rightarrow S^{1}$ such that the boundary of every (compactified) fiber $\pi^{-1}(\theta)$ is equal to $B$. I suggest you check out John Etnyre's lectures on open books for some background on this: There you can see that every oriented, compact 3-manifold has an open book, and so a contact structure. This follows from the Lickorish-Wallace theorem, asserting that every compact, oriented 3-manifold has a "nice" surgery presentation. For example, every fibered link in the 3-sphere determines a contact structure on it via this construction. These appear "in nature" when you look at Milnor fibrations.
In higher dimensions, things don't work out so nicely. You can still use open book decompositions to build contact structures, but you need additional hypotheses. Each fiber must be a Liouville domain, and the diffeomorphism of the fiber associated to the open book (which makes sense to discuss as $M\setminus B$ is a mapping torus) must be a symplectomorphism. I guess you can Google the relevant definitions if you're interested :)
If you're interested in less constructive methods of determining whether or not a manifold admits a contact structure, then there are currently not many tools available. There are the characteristic class obstructions to the existence of an almost contact structure on a given $M$ described by Geiges, but little is known as to when an almost contact structure can be deformed to a contact structure. Etnyre (see his most recent paper on the ArXiv) figured out how to do this in dimension 5 -- building upon work of Presas et al -- but it seems nothing is known in higher dimensions.
2) It seems like there are extreme restrictions on what non-vanishing vector fields on a 3-manifold can be the Reeb vector field associated to a contact form on it. A few years ago, Cliff Taubes proved the Weinstein conjecture which says that every Reeb vector field on a closed, oriented 3-manifold has a closed (AKA periodic) orbit. On the other hand, there are lots of examples of vector fields on 3-manifolds without closed orbits. Examples are easy to write down on the 3-torus and there are (highly non-trivial) examples of such vector fields on the 3-sphere (which are counter-examples to the Seifert conjecture). It is expected that this conjecture holds for Reeb vector fields on contact manifolds of all odd dimensions (every Reeb vector field has a closed orbit).
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 closed horizontal curve(in the sense of the contact structure) which bounds a disc tangent to the structure, 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 friend 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.
Best Answer
According to a well known result of Martinet, every compact orientable $3$-dimensional manifold has a contact structure [2], see also [1] for various proofs. On the other hand we have
For $n=2$, $SU(3)/SO(3)$ has no contact structure and for $n>2$, $SU(3)/SO(3)\times\mathbb{S}^{2n-4}$ has no contact structure, see Proposition 2.4 in [3].
[1] H. Geiges, An introduction to contact topology, Cambridge studies in advanced mathematics 109.
[2] J. Martinet, Formes de contact sur les variétés de dimension 3. Proceedings of Liverpool Singularities Symposium, II (1969/1970), pp. 142–163. Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971.
[3] R. E. Stong, Contact manifolds. J. Differential Geometry 9 (1974), 219–238.