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$
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Hassan JolanyJun 22 '13 at 21:18

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it was my first experience to working on this website.
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ArashJun 23 '13 at 11:36

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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.
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Douglas ZareJun 23 '13 at 23:04

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@Arash: welcome here then; you can find relevant information on "how to ask" by following the corresponding link on top of the pages.
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Benoît KloecknerJun 24 '13 at 11:09

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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.
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RussellJul 3 '13 at 22:17