Geometry and Topology Seminar

I will define a novel invariant, called Heegaard Floer symplectic cohomology (HFSH), which serves as a closed string analog of the higher-dimensional Heegaard Floer homology. This invariant can also be regarded as a deformation of a k-th symmetric version of symplectic cohomology given by counting curves of higher genus. Alternatively, one may view it as a Floer invariant associated with a problem of Hamiltonian motion of multiple identical particles.  I will also introduce a multiloop Morse complex of a manifold, which is supposed to be a Morse-theoretic counterpart of HFSH. At last, I will show that HFSH of a cotangent bundle T^*M is isomorphic to the cohomology of the multiloop complex of M. This result generalizes Viterbo's isomorphism theorem to the case of multiple particles.