# First Chern Class of Contact Structure which is not Torsion

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?

• 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. – magicker72 Jan 8 at 7:34
• Could you give a reference for "It is known ..."? – abx Jan 8 at 7:44
• 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. – Chris Gerig Jan 8 at 8:12
• @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.) – Chris Gerig Jan 8 at 17:05
• @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?? – Praphulla Koushik Jan 9 at 3:08