Cecilia Karlsson : Legendrian contact homology III
In this third talk we will define Legendrian contact homology for Legendrian submanifolds in the 1-jet space of a smooth manifold M. Again, this will be the homology of a DGA generated by the double points of the Legendrian under the Lagrangian projection. The differential is defined by a count of punctured pseudo-holomorphic disks in the cotangent bundle of M, with boundary on the projected Legendrian. To prove that this indeed gives a differential we will use the theory of Fredholm operators from functional analysis. I will also say something about Floer theories in general. In particular, one of the main difficulties when defining Floer theories via pseudo-holomorphic curve techniques is to achieve transversality for the dbar-operator. There has been a development of several different machineries to solve these problems, for examle Polyfolds by Hofer et al., and Pardon's work on Virtual fundamental cycles. In our case, however, it is enough to perturb either the Legendrian submanifold or the almost complex structure.