The Role of the Legendre Transform in the Study of the Floer Complex of Cotangent Bundles

Abstract

Consider a classical Hamiltonian H on the cotangent bundle T * M of a closed orientable manifold M , and let L : T M → R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Φ from the Morse complex of the Lagrangian action functional which is associated to L to the Floer complex which is determined by H. In this paper we give an explicit construction of a homotopy inverse Ψ of Φ. Contrary to other previously defined maps going in the same direction, Ψ is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder which on the boundary satisfy "half" of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a heuristic argument which, independently on any Fredholm and compactness analysis, explains why the spaces of maps which are used in the definition of Φ and Ψ are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our heuristic arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pull-back of the second Stiefel-Whitney class of T M on 2-tori in M .

The regularized free fall I -- Index computations

“…As in the ode case [AS15] also in the present delay equation situation both functionals are related through the maps ι and π (Lemma 5.3) in the form…”

Floer Homologies, with Applications

“…The proof we sketched here is due to the first author and Schwarz, see [3] and [7]. The orientation issue which requires the use of local coefficients when the second Stiefel-Whitney class does not vanish on tori had been overlooked in [194,167,3] and was discovered by Kragh in [122], see also [123], and corrected in [6,7]. See also [8] for another approach to this isomorphism and [51] for an extension of these ideas towards homotopy.…”

The Palais–Smale condition for the Hamiltonian action on a mixed regularity space of loops in cotangent bundles and applications

“…Floer homology can be defined also for noncompact symplectic manifolds which are suitably convex at infinity. In this case, the theory requires the use of Hamiltonians having a suitable behavior at infinity and is a genuine infinite dimensional homology theory: for instance, the Floer homology of T * M, the total space of the cotangent bundle of a closed manifold M, is isomorphic to the singular homology of the free loop space of M, see [3,5,32]. On particular symplectic manifolds however, a Morse theory for the Hamiltonian action functional A H can be obtained by more classical methods.…”