Periodic symplectic cohomologies and obstructions to exact Lagrangian immersions

Abstract

Periodic Symplectic Cohomologies and Obstructions to Exact Lagrangian Immersions Jingyu Zhao Given a Liouville manifold (M, θ), symplectic cohomology is defined as the Hamiltonian Floer homology for the symplectic action functional on the free loop space LM := Maps(S,M). In this thesis, we propose two versions of periodic S-equivariant homology or S-equivariant Tate homology for the natural S-action on the free loop space LM . The first version, denoted as PSH∗(M), is called periodic symplectic cohomology. We prove that there is a localization theorem or a fix point property for PSH∗(M). The second version PSH∗(M) is called the completed periodic symplectic cohomology which satisfies Goodwillie’s excision isomorphism. Inspired by the work of Seidel and Solomon on the existence of dilations on symplectic cohomology, we formulate the notion of Liouville manifolds admitting higher dilations using Goodwillie’s excision isomorphism on the completed periodic symplectic cohomology PSH∗(M). As an application, we derive obstructions to existence of certain exact Lagrangian immersions in Liouville manifolds admitting higher dilations.