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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.

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up vote 15 down vote accepted

Let M5 be a closed and oriented. A contact form α gives a 4-plane 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 Stiefel-Whitney class is zero (Gray, "Some global properties of contact structures", MR0112161). Equivalently, M is almost contact iff w2(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 1-connected 5-manifolds", MR1147828)
  • $\pi_1(M)=\mathbb{Z}/2$ and M spin (Geiges-Thomas, "Contact topology and the structure of 5-manifolds with $\pi_1=\mathbb{Z}_2$", MR1656012)
  • $\pi_1(M)$ finite, of odd order with some other restrictions (Geiges-Thomas, "Contact structures, equivariant spin bordism, and periodic fundamental groups", MR1857135)
  • M a product of lower-dimensional manifolds (Geiges-Stipsicz, "Contact structures on product five-manifolds and fibre sums along circles", arXiv:0906.5242)
  • M a circle bundle over a symplectic (X4, ω) with Euler class [ω] (Boothby-Wang, "On contact manifolds", MR0112160)

I'm not an expert, but I don't think there are any almost contact 5-manifolds which are known to not be contact.

Update (7/25/15): The problem has been completely solved since my original answer in 2011. For 5-manifolds, 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.

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I seem to remember Geiges saying, during a lecture, that he didn't have any example of almost contact 5-manifolds that were not contact, at least not last year. – Marco Golla Apr 3 '11 at 15:35
Steven, can you update your answer with pointers to the latest developments? – Marco Golla Jul 25 '15 at 7:51
Done, unless you had any other developments in mind. – Steven Sivek Jul 25 '15 at 17:08

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