'Contactization' and Symplectization - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:14:18Z http://mathoverflow.net/feeds/question/107302 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107302/contactization-and-symplectization 'Contactization' and Symplectization Chris Gerig 2012-09-16T05:50:42Z 2012-09-19T19:46:32Z <p>Given a contact manifold $(M,\lambda)$ we can pass to the <em>symplectization</em> $(\mathbb{R}\times M,\omega=d(e^s\lambda))$ and this is great to bring the machinery of symplectic geometry into the contact world. <strong></strong>: <em>As pointed out in comments, there is another notion of symplectization (although less used)</em>. I am then naturally curious about the other direction.<br> In particular, for Seiberg-Witten theory and Embedded Contact Homology you can talk about symplectic cobordisms (4-manifolds) with contact boundary. But this assumes that I already have some contact geometry at the ends of my symplectic world.</p> <p>The immediate question in my mind was: <strong>Is there a</strong> <em>contactization</em> <strong>to pass from a given symplectic manifold to a contact one?</strong> Or something in the spirit of it?</p> <p>And then I come across a paper of Eliashberg-Hofer-Salamon (<a href="http://www.math.ethz.ch/~salamon/PREPRINTS/ehs95.pdf" rel="nofollow">Lagrangian Intersections in Contact Geometry</a>), and in certain scenarios we do indeed have one. If our symplectic manifold $M$ is $\ast$exact$\ast$, i.e. $\omega=d\alpha$, then $(M\times S^1,dz-\alpha)$ is a contact manifold. Now if we don't have exactness, there is at least a way to contactize $M$ when some positive multiple of $\omega$ represents an $\ast$integral$\ast$ cohomology class in $H^2(M)$, and this is some principal $S^1$-bundle called ''pre-quantization''.</p> <p>So my ultimate two questions:<br> 1) <strong>Is ''pre-quantization'' the only way to contactize here?</strong><br> 2) <strong>Is there some useful notion of contactization in other scenarios?</strong><br> I should clarify that I am not interested in getting contact manifolds in general; I'm not looking for an analog of $T^*M$ (which is a symplectic manifold for <em>any</em> $M$), or more explicitly $\mathbb{P}T^*M$ (which is a contact manifold for <em>any</em> $M$). So while symplectization requires the contact form, contactization <em>should make use of</em> the symplectic form.</p> http://mathoverflow.net/questions/107302/contactization-and-symplectization/107388#107388 Answer by Eugene Lerman for 'Contactization' and Symplectization Eugene Lerman 2012-09-17T14:43:09Z 2012-09-17T14:43:09Z <p>The "pre-quantization" construction of a contact manifold out of symplectic manifold predates prequantization by a couple of decades: see Boothby, W. M.; Wang, H. C. On contact manifolds. Ann. of Math. (2) 68 1958 721–734. The analogue of the theorem for symplectic orbifolds is due to Thomas: Thomas, C. B. Almost regular contact manifolds. J. Differential Geometry 11 (1976), no. 4, 521–533.</p> <p>You may think of the Boothby-Wang construction as constructing a contact fiber bundle over a symplectic manifold with fiber $S^1$. If we look at the construction this way, it can be generalized. See my paper Contact fiber bundles. J. Geom. Phys. 49 (2004), no. 1, 52–66. </p>