Skip to main content Accessibility help
×
Home
  • Print publication year: 2008
  • Online publication date: November 2009

6 - Contact surgery

Summary

‘The afternoon he came to say goodbye there was a positively surgical atmosphere in the flat.’

Christopher Isherwood, Goodbye to Berlin

The proof of the Lutz—Martinet theorem in Chapter 4 was based on Dehn surgery along transverse knots in a given contact 3—manifold. This construction does not admit any direct extension to higher dimensions. In 1982, Meckert [178] developed a connected sum construction for contact manifolds. Now, forming the connected sum of two manifolds is the same as performing a surgery along a 0—sphere (i.e. two points, one in each of the two manifolds we want to connect). Since a point in a contact manifold is the simplest example of an isotropic submanifold, this intimated that there might be a more general form of ‘contact surgery’ along isotropic submanifolds. On the other hand, Meckert's construction is so complex that such a generalisation did not immediately suggest itself.

Then, in 1990, Eliashberg [65] did indeed find such a general form of contact surgery. In fact, he solved a much more intricate problem about the topology of Stein manifolds, involving the construction of complex structures on certain handlebodies such that their boundaries are strictly pseudoconvex (and hence inherit a contact structure), see Example 2.1.7 and Section 5.3.