I am interested in Morse homology on the loop space of a given compact (Riemannian) manifold. A small perturbation renders the geodesic action ("energy") functional Morse. Now I am interested in the Morse-Smale property, i.e. for any critical points x and y the unstable manifold of x intersects the stable manifold of y transversally.

Could anyone please provide a reference that a generic choice of metric on the loop space yields the Morse-Smale property? (Notice that the correct choice of perturbations of the metric is part of the problem.) I have difficulties finding an appropriate reference for this.

There seem to be two obvious ways to realize Morse-Smale transversality in this setting:
1. The abstract way: Here one considers a given Hilbert manifold with a metric. The space of perturbations consist of (some class of) metrics which are uniformly equivalent to the given one. This is for instance the approach followed by Abbondandolo/Majer: "Lectures on Morse homology for infinite-dimensional manifolds". The problem with this reference is that their space of perturbations is too big - the space in question is not separable. In particular the Sard-Smale theorem, which is crucial in this setting, cannot be applied. I have difficulties in writing down a separable Banach space of perturbations which is still enough to provide surjectivity of the linearized "master section".
2. The concrete setting: Obviously, it is not enough to consider metrics on the loop space which come from metrics on the base manifold. I do not know whether it suffices to consider metrics on the loop space which come from metrics on the base times $S^1.$ My problem is that the "master section" involves the gradient (w.r.t. the induced metric on the loop space) of the perturbed energy functional in question. I have no clue how to obtain a useful formula for its linearization.

So, could anyone please give me a hint about solving 1. or 2.? It is also possible that pursuing the paths 1. or 2. might not be a clever idea, in which case I would appreciate any advice.


1 Answer 1


Instead of considering the Morse homology of the energy functional my making it Morse-Smale, it might be easier for you (and geometrically more natural) to view the geodesic energy as a Morse-Bott functional, whose critical points appear in $S^1$-families.

The appropriate setting of Morse homology for Morse-Bott functions was worked out by Urs Frauenfelder in the Appendix of "The Arnold-Givental conjecture and moment Floer homology".

The Morse(-Bott) homology of the geodesic functional was considered by Abbondandolo and Schwarz in section 4 of "Estimates and Computations in Rabinowitz-Floer homology", see here:



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