Welcome to MathOverflow

Question

5

4

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.

flag	asked Jun 11 2010 at 11:34 Orbicular 1,794●8●18

1

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". – Tim Perutz Jun 11 2010 at 13:41

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) – Orbicular Jun 11 2010 at 19:23

Did you have a look at Klingenberg's 'Lectures on Closed Geodesics'? He uses Morse-Bott theory for free loop spaces. – Lennart Meier Jun 11 2010 at 20:44

@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. – Orbicular Jun 12 2010 at 9:02

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. – Tim Perutz Jun 13 2010 at 1:01

show 1 more comment

Richard Montgomery · Answer 1 · 2010-06-22 23:51:11Z UTC

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.

Morse-Bott homology for infinite-dimensional manifolds

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.