Question

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.

Answer 1

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.