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