A Morse complex for the Hamiltonian action in cotangent bundles

Abstract: Critical points having infinite Morse index and co-index are invisible to homotopy theory, since attaching an infinite dimensional cell does not produce any change in the topology of sublevel sets. Therefore, no classical Morse theory can possibly exist for strongly indefinite functionals (i.e. functionals whose all critical points have infinite Morse index and co-index). In this talk, we will briefly explain how to instead construct a Morse complex for certain classes of strongly indefinite functionals on a Hilbert manifold by looking at the intersection between stable and unstable manifolds of critical points whose difference of (suitably defined) relative indices is one. As a concrete example, we will consider the case of the Hamiltonian action functional defined by a smooth time-periodic Hamiltonian $H: S^1 \times T^*Q \to \mathbb R$, where $T^*Q$ is the cotangent bundle of a closed manifold $Q$. As one expects, in this case the resulting Morse homology is isomorphic to the Floer homology of $T^*Q$, however the Morse complex approach has several advantages over Floer homology which will be discussed if time permits. This is joint work with Alberto Abbondandolo and Maciej Starostka.

algebraic geometrydifferential geometrysymplectic geometry

Audience: researchers in the topic

Geometria em Lisboa (IST)

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