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.