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I am learning Morse-Bott-Floer theory and found the following cool paper

http://de.arxiv.org/abs/1310.5080

by P. Albers and D Hein. In order to prove a cup-length estimate on the number of critical points of a perturbation of a Morse-Bott function on a smooth finite dimensional manifold the paper makes use of a perturbation result. I would like to know in what generality the result holds and if anyone knows any references. The ideal result I can imagine is something along the following lines.

Suppose $\mathcal{M}:=s^{-1}(0_L)$ is the 0-set of a Fredholm section $s\in \Gamma (L\to M)$ where $M$ is an infinite dimensional Banach manifold together with an infinite dimensional Banach bundle $L\to M$. Suppose moreover that $\mathcal{M}\subset M$ is compact. Then there exists a small abstract perturbation $\tilde{s}\in \Gamma (L\to M)$ of $s$ such that $\tilde{s}$ is transverse to the 0-section $0_L\subset L$, is Fredholm with $index(s)=index(\tilde{s})$ and satisfies the condition that $\tilde{\mathcal{M}}:=\tilde{s}^{-1}(0_L)$ is compact.

Does anyone know which additional assumptions I have to make for the above result to hold, or if it holds in the stated form? Both references and ideas are welcome

Thanks in advance.

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    $\begingroup$ Could you add some more tags to describe your question? You can have up to 5 tags, and there are tags corresponding to the arXiv classes (like sg.symplectic-geometry corresponding to the linked article). $\endgroup$
    – Joonas Ilmavirta
    Commented May 27, 2015 at 14:16
  • $\begingroup$ You need not feel uncomfortable. In the stated generality the result is not obvious. To obtain such a result you need to have some additional information about the Banach spaces involved, and some additional information on the original section $s$ so that you can get a bit of compactness. $\endgroup$
    – Liviu Nicolaescu
    Commented May 27, 2015 at 14:54
  • $\begingroup$ Thanks for pointing out the tags, I have added some more now. $\endgroup$
    – MBIS
    Commented May 27, 2015 at 15:54
  • $\begingroup$ @LiviuNicolaescu I changed the question a little, since in fact the paper actually only uses the result in a specific application where additional assumptions are satisfied. However it is not clear to me exactly what makes it work in their paper. Do you have any suggestions as to which assumptions on $s$ and the Banach spaces involved I would have to make for the result to hold? $\endgroup$
    – MBIS
    Commented May 27, 2015 at 18:57
  • $\begingroup$ The tricky part is the compactness of the zero set. This requires some additional conditions on the original section and the nature of perturbation. The transversality is a consequence of Sard's theorem. $\endgroup$
    – Liviu Nicolaescu
    Commented May 27, 2015 at 22:25

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