Morse theory for strongly indefinite functionals

Abstract:

For strongly indefinite functionals, i.e. functionals all of whose critical points have infinite Morse index and co-index, no classical Morse theory can possibly exist due to the fact that attaching an infinite dimensional cell does not produce any change in the topology of sublevel sets. In this talk I will show how to instead construct a Morse complex for a suitable class of such functionals (including e.g. the Lorentzian energy functional, and the Hamiltonian action in cotangent bundles). The Morse complex is generated by critical points, and the boundary operator counts the number (modulo two) of gradient flow lines connecting pairs of critical points whose (suitably defined) relative Morse indices differ by one. In contrast to Floer theory, such gradient flow lines are obtained as genuine intersection between stable and unstable manifolds, which (despite being infinite dimensional) turn out to have finite dimensional intersection with good compactness properties. Transversality is achieved by generically perturbing the negative gradient vector field within a class of vector fields preserving all good compactness properties. In the particular case of the Hamiltonian action in cotangent bundles, the resulting Morse homology is isomorphic to the Floer homology of T*M as well as to the singular homology of the free loop space of the base manifold M. Nevertheless, the Morse complex approach has potentially several advantages over Floer homology which will be discussed if time permits. This is joint work with Alberto Abbondandolo and Maciej Starostka.