Symplectic Geometry and Floer Homology

Summary. Here are the revised notes for lectures held at the 13th Brazilian
Topology Meeting in Belo Horizonte (July 2002). The purpose is to give an
introduction to symplectic Floer homology and, in a simple case, a sketch of
proof of the Arnold conjecture. This conjecture gives a lower bound for the
number of fixed points of a Hamiltonian diffeomorphism in terms of the sum of
the Betti numbers. Floer theory is a sort of infinite dimensional Morse theory
on a loop space. The Morse index is replaced by the Maslov-Conley-Zehnder
index. Some results about the Maslov cycle in the linear symplectic group are
gathered in an appendix.