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In Salamon's notes on Floer homology, it's claimed that under some non-degenerancy assumptions the operator $$D:= \partial_s+J_0\partial_t+S(s,t): W^{1,p}(\mathbb{R}\times S^1,\mathbb{R}^{2n})\rightarrow L^p(\mathbb{R}\times S^1,\mathbb{R}^{2n})$$ is Fredholm for $1<p<\infty$, and moreover its Fredholm index is given by index $$D=\mu_{CZ}(\Psi^{+})-\mu_{CZ}(\Psi^{-}).$$

Now I was trying to see why this index formula holds up. In the lectures notes the author claims that this formula is true for $p=2$, where we are dealing with Hilbert spaces, by using the spectral flow of the family of operators $J_0\partial_t+S(s,t)$. However nothing else is said for the case where $p\neq 2$. It's clear that in the end this Fredholm index won't depend on the choice of $p$, but a priori we would need to prove that this is true so that we get the desired result just by proving the case $p=2$. Is this what is happening in this situation ? We can show that $D$ has the same Fredholm index for any $p$ without using the Conley-Zehnder index ?

Any insight is appreciated, thanks in advance.

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  • $\begingroup$ Chapter 8, specifically section 8.8 of Audin-Damian's book on Morse Theory and Floer Homology might be helpful. In general, whenever I didn't understand something in Salamon's notes, Audin-Damian has been an excellent reference. $\endgroup$
    – Soham
    Commented Nov 19, 2023 at 19:40

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The idea is to show that the kernel & cokernel consist of smooth sections, and thus are independent of $p$. Since the Fredholm index is the difference between the dimensions of these, it doesn't depend on $p$.

This is proved, admittedly in the case without punctures, in an appendix of McDuff & Salamon's J-holomorphic curve book.

Another good reference, this time with punctures, is from Chris Wendl's lecture notes/book draft. The direct link to the PDF file is here: https://www.mathematik.hu-berlin.de/~wendl/Sommer2020/SFT/lecturenotes.pdf

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  • $\begingroup$ So the fact that the kernel and Cokernel are smooth sections I think will follow from elliptic bootstrapping .By this I mean that every element in those sets will be of class $C$\infty}, I am not sure this is what you mean. But why does the fact that they consist of smooth sections imply that it is independent of $p$? $\endgroup$
    – Someone
    Commented Oct 11, 2021 at 8:51
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    $\begingroup$ @Something: I should have written "smooth and of exponential decay". Since the sections are then smooth and decaying exponentially, they are in each of the Sobolev spaces you consider. I can't give you an immediate reference, but I'd look at Wendl's book. I'm editing my answer to include a link to it. $\endgroup$
    – Sam Lisi
    Commented Oct 11, 2021 at 22:13

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