Low-area Floer theory and non-displaceability

Floer Theory and Low Dimensional Topology

The new 3and 4-manifold invariants recently constructed by Ozsváth and Szabó are based on a Floer theory associated with Heegaard diagrams. The following notes try to give an accessible introduction to their work. In the first part we begin by outlining traditional Morse theory, using the Heegaard diagram of a 3-manifold as an example. We then describe Witten’s approach to Morse theory and how ...

Gauged Floer Theory of Toric Moment Fibers

We investigate the small area limit of the gauged Lagrangian Floer cohomology of Frauenfelder [17]. The resulting cohomology theory, which we call quasimap Floer cohomology, is an obstruction to displaceability of Lagrangians in the symplectic quotient. We use the theory to reproduce the results of FukayaOh-Ohta-Ono [21], [19] and Cho-Oh [12] on non-displaceability of moment fibers of not-neces...

Monodromy in Hamiltonian Floer Theory

Schwarz showed that when a closed symplectic manifold (M,ω) is symplectically aspherical (i.e. the symplectic form and the first Chern class vanish on π2(M)) then the spectral invariants, which are initially defined on the universal cover of the Hamiltonian group, descend to the Hamiltonian group Ham(M,ω). In this note we describe less stringent conditions on the Chern class and quantum homolog...

Development in symplectic Floer theory

In the middle of the 1980s, Floer initiated a new theory, which is now called the Floer theory. Since then the theory has been developed in various ways. In this article we report some recent progress in Floer theory in symplectic geometry. For example, we give an outline of a proof of the flux conjecture, which states that the Hamiltonian diffeomorphism group is C1-closed in the group of sympl...

Floer Homology and low-dimensional topology