# Proof of Theorem 2.1.11 of book "Morse Theory and Floer Homology" by Audin and Damian

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).

So here is the outline of the proof.

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:

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.

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 $$V^{\alpha+\epsilon}\supset F^{-1}(]-\infty, \alpha+\epsilon])$$).