# contact structure on 3 manifolds

every orientable closed 3-manifold admits a contact structure. how we can construct this contact structure? if the manifold is prime what will happen?

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In fact , recently, all contact 3-manifolds have been classified by 1)Lens spaces 2)Torus bundles 3)Circle bundles 4)$S^3$,$S^2 \times S^1$ – Hassan Jolany 海桑乔朗丽 Jun 22 '13 at 21:18
it was my first experience to working on this website. – Arash Jun 23 '13 at 11:36
If you just want to construct one contact structure, take a Dehn-surgery description of the 3-manifold as surgery on a link in $S^3$ (for each connected component), make the link transverse to the standard contact structure on $S^3$, and add an "overtwisted torus" along each link component. Then take out part of this overtwisted torus as you perform the surgery. This will generally produce an overtwisted contact structure on the manifold. – Douglas Zare Jun 23 '13 at 23:04
@Arash: welcome here then; you can find relevant information on "how to ask" by following the corresponding link on top of the pages. – Benoît Kloeckner Jun 24 '13 at 11:09
Lutz and Martinet also proved the existence of contact structures on 3-manifolds using the existence of triangulations -- rather then then the existence of Dehn surgery presentations -- and some ideas from foliation theory. The ones described in the proof are all overtwisted although that terminology didn't exist at the time. For a sketch see Honda's notes www-bcf.usc.edu/~khonda/math599/notes.pdf section 5.3. – Russell Jul 3 '13 at 22:17