Floer Homology: Spaces of Trajectories
In this chapter we construct the Floer complex for a generic Hamiltonian on a symplectic manifold satisfying a condition on the second homotopy group. The complex is generated by the periodic orbits of fixed Maslov index (a nondegeneracy assumption ensures that they are isolated). We define the differential of the complex using the flow of the gradient of the action functional and we prove that this is indeed a complex. This involves a delicate gluing argument.