I am trying to understand some basic facts about (almost) contact (metric) structure, especially on 3-manifolds 5-manifolds.

(1) I saw statement that "any compact oriented 3-manifold admit contact structure" by J. Martinet, and also statement by Geiges in his book that "the obstruction of almost contact structure is 3rd Stiefel–Whitney class". In my impression, contact structure must be first almost contact, so 3rd Stiefel–Whitney class should also be obstruction for contact structure. Am I wrong about this?

(2) Anyhow, it feels that almost contact structure (ACS) is relatively easy to exist. But how easy? An odd-dimensional manifold always allow nowhere-vanishing vector field. Can I take any of these field and make it the Reeb vector field of certain ACS? The reason why I am asking is I saw paper "ALMOST CONTACT STRUCTURE AND THE CONTACT MAGNETIC FIELD" which proves that one can always find a ACS compatible with any give metric g. So I wonder if the same is possible for any prescribed nonvanishing vector field.

I am a physics student, so, if possible, I hope people could gives some intuitive explanation. I have Geiges's an introduction to contact topology, but it's too big for me to consume.


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    $\begingroup$ You are writing 'Geides', but do you mean 'Geiges'? Regarding (1), certainly any contact manifold must satisfy the conditions to be an almost contact manifold, so you aren't wrong about the 3rd SW-class being a (very weak) obstruction to being contact. Consider $M^5=S^1\times S^4$ and let $X$ be a nonvanishing vector field tangent to the $S^1$-fibers. Then there is no almost contact structure on the orthogonal $4$-plane field on $M$ because $S^4$ does not carry an almost symplectic structure. $\endgroup$ – Robert Bryant Apr 10 '13 at 19:54
  • $\begingroup$ Thanks for you answer and sorry to misspell author's name. I'll edit it. $\endgroup$ – Lelouch Apr 11 '13 at 3:12

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).

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    $\begingroup$ Thank you for your answer! Let me make sure I understand some points you made: (1) So any compact+oriented 3-manifold $\Rightarrow$ Open-book decomposition $\Rightarrow$ contact structure, without any requirement of vanishing the 3rd S-W class? Actually I am mainly interested in $d= 5$ and almost contact structure, as it relates to some physics developing lately. But indeed I can't find much about them through googling. $\endgroup$ – Lelouch Apr 18 '13 at 16:11
  • $\begingroup$ Every oriented 3-manifold has a trivial tangent bundle and so all of it's Steifel-Whitney classes vanish. There's also the fact that an orientation is the same as a homotopy class of almost-complex structure on a 2-plane field: This means that for 3-manifolds, a non-vanishing vecor field (which always exists) determines an oriented 2 plane field $\xi$ -- for example, by taking an orthogonal complement with respect to a metric -- and so $\xi$ is almost contact. As Robert Bryan's example points out, the statement "non-vanishing vector field implies almost contact strcture" breaks down in $\endgroup$ – Russell Apr 19 '13 at 3:16
  • $\begingroup$ higher dimensions. This paper I mentioned earlier arxiv.org/abs/1210.5208 and the references therein (in particular to Geiges papers) will tell you about the characteristic class theory for almost contact structures on 5-manifolds. Go Seawolves! $\endgroup$ – Russell Apr 19 '13 at 3:20

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