Transversality in Morse theory for the (perturbed) geodesic action functional - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T23:42:42Z http://mathoverflow.net/feeds/question/55129 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55129/transversality-in-morse-theory-for-the-perturbed-geodesic-action-functional Transversality in Morse theory for the (perturbed) geodesic action functional Orbicular 2011-02-11T14:43:47Z 2011-03-01T14:00:11Z <p>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.</p> <p>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.</p> <p>There seem to be two obvious ways to realize Morse-Smale transversality in this setting:<br> 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".<br> 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.</p> <p>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.</p> http://mathoverflow.net/questions/55129/transversality-in-morse-theory-for-the-perturbed-geodesic-action-functional/57003#57003 Answer by Stephan Mescher for Transversality in Morse theory for the (perturbed) geodesic action functional Stephan Mescher 2011-03-01T14:00:11Z 2011-03-01T14:00:11Z <p>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. </p> <p>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".</p> <p>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:</p> <p><a href="http://arxiv.org/abs/0907.1976" rel="nofollow">Abbdondandolo-Schwarz</a></p>