Nielsen theory, Floer homology and a generalisation of the Poincare-Birkhoff theorem

TL;DR

This paper explores the relationship between Nielsen fixed point theory and symplectic Floer homology for surface symplectomorphisms, proposing a generalization of classical fixed point theorems and calculating related symplectic invariants.

Contribution

It establishes a connection between Nielsen theory and Floer homology, and introduces a generalized fixed point theorem extending classical results.

Findings

Calculation of Seidel's symplectic Floer homology for various mapping classes

Description of symplectic zeta functions and asymptotic invariants

Proposal of a generalized Poincare-Birkhoff fixed point theorem

Abstract

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltotic symplectic invariant. A generalisation of the Poincare- Birkhoff fixed point theorem and Arnold conjecture is proposed.