2
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ @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
  • 1
    $\begingroup$ @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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.