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I am reading the book "Morse theory and Floer Homology " by Audin and Damian and I am stuck understanding the proof of this theorem. (Sorry I dont know the exact name of it that is why I just put its number in the book).

enter image description here

So here is the outline of the proof. enter image description here

I understand the idea and how the use of Theorem 2.1.7 helps. The first thing that I dont get here, is that in the second point of the outline, the author says that $V^{\alpha + \epsilon}$ is just the sublevel set of $F$ for $\alpha + \epsilon$. Thus, as far as I know that would mean that $$ F^{-1}((-\infty, \alpha + \epsilon]) = V^{\alpha + \epsilon}$$. But if this is the case then why do we need to use a theorem to find a deformation retract between them. If they are the same set, woudln't the deformation retract be "trivial"?

So, in the proof of the theorem, the author states this: enter image description here

Here I understand the construction of F and all the calculation involving its derivative and so on, but I am quite lost in why it follows directly that $F^{-1}((-\infty, \alpha + \epsilon])$ is a deformation retract of $V^{\alpha + \epsilon}$. I understand that by Theorem 2.1.7, $F$ has all the requisites so that we have a deformation retract between $F^{-1}((-\infty, \alpha + \epsilon])$ and $F^{-1}((-\infty, \alpha - \epsilon])$, but I do not see how we can conclude a deformation retract to $V^{\alpha + \epsilon)}$.

I have been stuck trying to understand this for a while. Any help would be appreciated and if it is a small detail I am not seeing, then forgive me, I am starting to learn this stuff.

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Sometimes Audin and Damian have unclear bits. I think in the outline of the proof the first part is referring to $F^{-1}(]-\infty, \alpha+\epsilon])$ as the union of the two hatched parts in Fig. 2.5. It should be a $+\epsilon$ instead of a $-\epsilon$.

In the second part of the outline the right statement is that $F^{-1}(]-\infty, \alpha+\epsilon])$ is a retract of some $V^{\alpha+\epsilon + \epsilon'} = F^{-1}(]-\infty, \alpha+\epsilon + \epsilon'])$. That is, by considering a slightly higher regular value where you're outside the Morse chart, we get $f$ and $F$ to agree again, and their sublevel sets are back to being the same. Now 2.1.7 gives a deformation retraction from $F^{-1}(]-\infty, \alpha+\epsilon + \epsilon']) = V^{\alpha+\epsilon + \epsilon'}$ to $F^{-1}(]-\infty, \alpha+\epsilon])$, since there are only regular values from $\alpha+\epsilon$ to $\alpha+\epsilon + \epsilon'$.

In particular, this gives by restriction a deformation retraction from $V^{\alpha+\epsilon}$ to $F^{-1}(]-\infty, \alpha+\epsilon])$ (skipping the initial deformation from $V^{\alpha+\epsilon + \epsilon'}$ to $V^{\alpha+\epsilon}$ and starting from $V^{\alpha+\epsilon}\supset F^{-1}(]-\infty, \alpha+\epsilon])$).

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