Does the "easy" part of the Morse lemma follow from Ehresmann's fibration theorem?

Question

Corollary 2.3 of these notes on Morse theory seem to suggest the "easy" part of the Morse lemma is a corollary of Ehresmann's fibration theorem.

That is, if $f:X\to \mathbb R$ is a proper smooth map without critical values in $[a,a+\varepsilon]$ then there's a diffeomorphism $f^{-1}(-\infty,a]\cong f^{-1}(-\infty,a+\varepsilon]$.

I don't understand how to deduce this from Ehresmann's fibration theorem.

The proof I know constructs a normalized gradient field on $X$ which is supported on a compact neighborhood of $f^{-1}[a,a+\varepsilon]$ and flows along it.

Answer 1

As you say I am not sure the other answer covers the detail you want. If $f$ is proper, then the set of critical values is closed (exercise) so if there are no critical values in $[a, a+\epsilon]$, then nor are there in $[a-\delta, a+\epsilon]$ for $\delta$ small. Now you know from Ehressman that there is a diffeomorphism

$$g: X_{[a-\delta, a+\epsilon]} \cong [a-\delta, a+\epsilon] \times X_a,$$ so that under this diffeomorphism $f(g^{-1}(t,x)) = t$; that is, $f$ is taken to projection onto the first factor. Here I write $X_S = f^{-1}(S)$ as notation I prefer.

We will use this to construct the desired diffeomorphism. To do so, pick a diffeomorphism $$h: [a-\delta, a] \to [a-\delta, a+\epsilon]$$ which is the identity near $a-\delta$. Of course, this induces a diffeomorphism $$H: X_{[a-\delta, a] \to [a-\delta, a+\epsilon]}$$ which is the identity near $X_{a-\delta}$; this induced diffeo comes from the diffeo $g$ above (and its restriction to $X_{[a-\delta, a]}$ as well).

Then the desired diffeomorphism $$F: X_{(-\infty, a]} \to X_{(-\infty, a+\epsilon]}$$ by saying that $F$ is the identity on $X_{(-\infty, a-\delta]}$, and on $X_{[a-\delta, a]}$, $F = H$, the function defined above.

The idea is that we are flowing backwards from $a+\epsilon$ to $a$, but we use $h$ to "slow down the flow" past $a$ so that by $a-\delta$ we have stopped flowing completely.

Answer 2

If there is no critical points in $[a,a+\epsilon], f:f^{-1}([a,a+\epsilon])\rightarrow [a,a+\epsilon]$ is a submersion. If follows from Ehresmann that it is a fibration. Since $[a,a+\epsilon]$ is contractible this fibration is trivial, we deduce that $f^{-1}([a,a+\epsilon])=f^{-1}(a)\times [a,+\epsilon]$which enables to construct an homotopy (via a deformation retract) between $f^{-1}((-\infty,a])$ and $f^{1}(a,a+\epsilon])$.