Is there any work on Morse-Bott homology for infinite-dimensional manifolds (e.g. Hilbert manifolds). I am particularly interested in the case where we have a locally trivial fiber bundle and the Morse-Bott function is the function on the total space which is a pulled back Morse function on the base.
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1$\begingroup$ Orbicular, can you be a little more specific? Is the base of the bundle finite-dimensional? What's a Morse-Bott function, for you? (Finite-dim critical manifolds? Finitely many negative eigenvalues of the Hessian?) Depending on the specifics, references might be Bott's "Stable homotopy of the classical groups" or Cohen-Jones-Segal's "Floer's infinite-dimensional Morse theory and homotopy theory". $\endgroup$– Tim PerutzJun 11, 2010 at 13:41
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$\begingroup$ I am interested in the path space fibration (free loop space over the base with fiber the based loop space) and the loop space of a locally trivial fibre bundle (fibre and base closed and finite-dimensional) $\endgroup$– OrbicularJun 11, 2010 at 19:23
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$\begingroup$ Did you have a look at Klingenberg's 'Lectures on Closed Geodesics'? He uses Morse-Bott theory for free loop spaces. $\endgroup$– Lennart MeierJun 11, 2010 at 20:44
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$\begingroup$ @Lennart: Correct me if I'm wrong, but I always thought Klingenberg was dealing with Morse theory for the energy functional on the free loop space. Time-shifting the "time" coordinate of your closed geodesic gives you another one, yielding S^1-families of critical points (in the nondegenerate case). But I am specifically interested in the case where the fibers are not of finite dimension, as for example in the fibration based loop space -> free loop space ->> base manifold. $\endgroup$– OrbicularJun 12, 2010 at 9:02
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$\begingroup$ The literal answer to your question ("is there work...?") is "lots", including the loop-space examples you discussed with Lennart, and various Floer-theoretic examples. But there's no overarching theory, because of the subtleties of gradient flows in infinite dimensions (see e.g. Jost's "Riemannian geometry and geometric analysis"), so you may want to focus your question further. $\endgroup$– Tim PerutzJun 13, 2010 at 1:01
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See Atiyah and Bott's papers on Yang-Mills for Riemann surfaces.
The essence of it is that the Yang-Mills functional acts like a
perfect Morse functional after quotienting by gauge transformations.
So, look at it before the quotient to get your Morse-Bott function.
answered Jun 22, 2010 at 23:51