Hello, Which manifolds in dimension five admit contact structures? I am not too familiar with the contact realm so any references to look at would be much appreciated.
1 Answer
Let M^{5} be a closed and oriented. A contact form α gives a 4plane distribution with symplectic form dα, reducing the structure group of TM to U(2)×1; such a reduction is called an almost contact structure, and it exists iff the integral third StiefelWhitney class is zero (Gray, "Some global properties of contact structures", MR0112161). Equivalently, M is almost contact iff w_{2}(M) lifts to an integral cohomology class.
The existence problem for actual contact structures is much harder, though. Contact structures are known to exist in at least the following cases:
 $\pi_1(M)=0$ (Geiges, "Contact structures on 1connected 5manifolds", MR1147828)
 $\pi_1(M)=\mathbb{Z}/2$ and M spin (GeigesThomas, "Contact topology and the structure of 5manifolds with $\pi_1=\mathbb{Z}_2$", MR1656012)
 $\pi_1(M)$ finite, of odd order with some other restrictions (GeigesThomas, "Contact structures, equivariant spin bordism, and periodic fundamental groups", MR1857135)
 M a product of lowerdimensional manifolds (GeigesStipsicz, "Contact structures on product fivemanifolds and fibre sums along circles", arXiv:0906.5242)
 M a circle bundle over a symplectic (X^{4}, ω) with Euler class [ω] (BoothbyWang, "On contact manifolds", MR0112160)
I'm not an expert, but I don't think there are any almost contact 5manifolds which are known to not be contact.
Update (7/25/15): The problem has been completely solved since my original answer in 2011. For 5manifolds, Casals, Pancholi, and Presas and Etnyre proved that every homotopy class of almost contact structure contains a contact structure. This was then generalized to all odd dimensions by Borman, Eliashberg, and Murphy, who gave a definition of overtwistedness in higher dimensions and showed that every homotopy class of almost complex structure contains a unique overtwisted contact structure up to isotopy.

$\begingroup$ I seem to remember Geiges saying, during a lecture, that he didn't have any example of almost contact 5manifolds that were not contact, at least not last year. $\endgroup$ Commented Apr 3, 2011 at 15:35

$\begingroup$ Steven, can you update your answer with pointers to the latest developments? $\endgroup$ Commented Jul 25, 2015 at 7:51

1$\begingroup$ Done, unless you had any other developments in mind. $\endgroup$ Commented Jul 25, 2015 at 17:08