Floer Homology, the Maslov Index And Periodic Orbits of Hamiltonian Equations

Abstract

This chapter discusses Floer homology, the Maslov index, and periodic orbits of Hamiltonian equations. It is well-known that the structure of all solutions of a Hamiltonian equation can in general be very intricate. Therefore, one looks first for very special solutions, and a natural task is to find periodic phenomena. This leads to a boundary value problem characterized by a variational principle. The variational principle is used in the proof of the Arnold conjecture, which leads to the concept of Floer homology. A periodic solution is called nondegenerate if 1 is not a Floquet multiplier. An analogous homotopy invariance property for the Floer homology groups can be used to show that they are nontrivial.