Contact topology and CR geometry in three dimensions

Details Contact topology and CR geometry in three dimensions Contact topology and CR geometry in three dimensions

2001-06-11

arxiv.org/abs/math/0106071v1

Proceedings of the 3rd Asian Mathematical Conference 2000 (Quezon City, Philippines) ed. T. Sunada, P. W. Sy, Yang Lo (2002),

Mathematics

Symplectic Geometry

11 pages. Based on the author's notes for his talks at the third Asian Mathematical Conference (Quezon City, Oct. 2000) and th

Scientific paper

We study low-dimensional problems in topology and geometry via a study of contact and Cauchy-Riemann ($CR$) structures. A contact structure is called spherical if it admits a compatible spherical $CR$ structure. We will talk about spherical contact structures and our analytic tool, an evolution equation of $CR$ structures. We argue that solving such an equation for the standard contact 3-sphere is related to the Smale conjecture in 3-topology. Furthermore, we propose a contact analogue of Ray-Singer's analytic torsion. This ''contact torsion'' is expected to be able to distinguish among ''spherical space forms'' $\{\Gamma\backslash S^{3}\}$ as contact manifolds. We also propose the study of a certain kind of monopole equation associated with a contact structure. In view of the recently developed theory of contact homology algebras, we will discuss its overall impact on our study.

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