Mihai Damian

From Floer to Morse

We prove that in the case of a nondegenerate Hamiltonian that does not depend on time and is sufficiently small, when we are able to define the Morse complex and the Floer complex, the two coincide.

The Arnold Conjecture and the Floer Equation

We state the Arnold conjecture, which gives a lower bound for the number of fixed points of certain Hamiltonian diffeomorphisms. We then identify these fixed points with periodic orbits of Hamiltonian systems and with critical points of the “action functional” a function on the space of the contractible loops on the symplectic manifold, as well as the differential equation defining the flow of the gradient of this functional, called the Floer equation. This is a partial derivatives equation because it involves both the loop’s variable and that of the gradient’s flow. We begin studying the space of solutions of this equation by showing a compactness property.

Linearization and Transversality

We show that the spaces of solutions of the Floer equation connecting two periodic orbits is, after a perturbation of the Hamiltonian if necessary, a manifold of dimension the difference of the two indices. For this, we use the analogue of Sard’s theorem in infinite dimension. This requires the assumption that the tangent maps are Fredholm operators. We prove this and compute the index of these operators, which is the dimension of the space of solutions.

Morse Homology, Applications

We give a few computations and a few applications of Morse homology: the Knnneth formula for products, the Poincare duality, the Euler characteristic (and the Morse inequalities), the link with the connectedness and the simple connectedness of the manifold in question. We also show the functoriality of Morse homology and present a long exact sequence for a pair consisting of a manifold and a submanifold.

The Geometry of the Symplectic Group, the Maslov Index

We define a version of the Maslov index, which will play the role of the index of the critical points in the Floer complex.

What You Need to Know About Symplectic Geometry

This is an introduction to the basic notions of symplectic geometry, symplectic forms and manifolds, Hamiltonian vector fields, almost complex structures.

Floer Homology: Invariance

The object of this chapter is to show that the Floer homology does not depend on the chosen Hamiltonian and almost complex structure, provided the pair is regular in an appropriate sense.

The Morse Complex

In this chapter, to a compact manifold endowed with a Morse function a pseudo-gradient field satisfying the Smale condition, we associate a complex. It is generated by the critical points of the function, and the differential is defined by the trajectories connecting critical points. We prove that this is indeed a complex, and that its homology does not depend on the actual function and vector field we used.

The Elliptic Regularity of the Floer Operator

In this chapter we state and prove a few “elliptic regularity” statements we used to assert the regularity of the solutions of the Floer equation.

The Lemmas on the Second Derivative of the Floer Operator and Other Technicalities

In this chapter we prove a number of technical results used in this book. All the particulars are given in detail, as they will not fail to elucidate certain complex, but essential, passages of the previous chapters.