Where can I find this Morse Theory result?

Question

I'm reading through chapter 1 of Curve and Surface Reconstruction: Algorithm and Mathematical Analysis. There's lemma 1.1 where in the proof apparently a result of Morse Theory is used, and I'd like to study the related theory so I can understand the result.

I quote only the relevant bit:

We claim the function $h$ has a critical point in $Int (B \cap \Sigma)$ other than $m$ where $B$ becomes tangent to $\Sigma$. If not, as we shrink $B$ centrally the level set $bd (B \cap \Sigma)$ does not change topology until it reaches the minimum $m$ when it vanishes. This follows from Morse theory of smooth functions over smooth manifolds.

This relation between minimum and change of topology is exactly what I'm looking for.

Is there a specific theorem I can study to understand rigorously that statememt?

The only reference mentioned in the book is : Milnor (1963) Morse Theory.

I started reading through it but I couldn't find the specific result.

Could you help me please?

Answer 1

This follows from (the proof of) Theorem 3.1 in Milnor's Morse Theory, which I paraphrase part of below.

Let $M$ be a smooth manifold and let $f:M\to\mathbb{R}$ be a smooth function. Let $a,b\in \mathbb{R}$ with $a<b$ such that $f^{-1}([a,b])$ is compact and contains no critical points of $f$. Then there is a diffeomorphism $f^{-1}((-\infty,a])\to f^{-1}((-\infty,b])$ which restricts to a diffeomorphism $f^{-1}(\{a\})\to f^{-1}(\{b\})$ [this latter part is not stated explicitly by Milnor but is clear from his proof].

In other words, the level sets of a smooth function on a compact manifold do not change topology unless you pass through a critical point.