Let $(M,\xi)$ be a closed connected $3-$dimensional contact manifold with contact structure $\xi$. It is known that the first Chern class $c_{1}(\xi)$ defines an element in $H^{2}(M;\mathbb{Z})$ and is even.

Question: Do there exists examples of contact manifolds $(M,\xi)$ where $c_{1}(\xi)$ is not torsion?

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    $\begingroup$ You might want the Lutz–Martinet theorem on the existence of contact structures in every homotopy class of cooriented plane field on closed oriented 3–manifolds. $\endgroup$ – magicker72 Jan 8 at 7:34
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    $\begingroup$ Could you give a reference for "It is known ..."? $\endgroup$ – abx Jan 8 at 7:44
  • $\begingroup$ Yes, $S^1\times S^2$ has an overtwisted contact structure with Chern class $\pm2$ (after orienting). It is the one which appears in near-symplectic geometry, as the boundary of a tubular neighborhood of the circles of degeneracy (of the near-symplectic 2-form). I think it is $(1-3\cos^2\theta)dt + \sqrt6\cos\theta\sin^2\theta d\phi$, but might be off a bit. $\endgroup$ – Chris Gerig Jan 8 at 8:12
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    $\begingroup$ @abx It's a small exercise in algebraic topology: $c_1\equiv w_2$ mod 2, while $w_2(TM)=w_1(TM)=0$ always for oriented 3-manifolds, and $w_2(TM)=w_2(\xi)$. (And Koushik, yes.) $\endgroup$ – Chris Gerig Jan 8 at 17:05
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    $\begingroup$ @ChrisGerig what does it mean to say it is even?. Does that means image of $c_1(\xi)$ under the obvious map $H^2(M,\mathbb{Z})\rightarrow H^2(M,\mathbb{Z}_2)$ is zero?? $\endgroup$ – Praphulla Koushik Jan 9 at 3:08

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