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I'm trying to prove Reeb's theorem as stated in Milnor's Morse Theory.

Suppose we have an $n$-manifold $M$ together with a smooth function $f$ with exactly two critical points (both non-degenerate), then $M$ is homeomorphic to $S_n$.

But at the end of the proof he said it's easy to show it is homeomorphic. I have problem to show that. Could you help me? I am a starter in morse theory.

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First off, you want $M$ to be compact. Then the two critical points correspond to a maximum and minimum. Denote the minimum $p$ and maximum $q$. Let $f(M) = [0,1]$. For small enough $\epsilon >0$, $f^{-1}[1-\epsilon, 1]$ and $f^{-1}[0,\epsilon]$ are $n$-disks due to Morse's lemma. That is, pick coordinates $(x_1, \ldots, x_n)$ in a neighborhood $U$ around $p$ so that $$f = x_1^2 + \ldots +x_n^2$$ around $p$ and similarly coordinates $(y_1, \ldots, y_n)$ in a neighborhood $V$ around $q$ so that $$f = 1-y_1^2 - \ldots -y_n^2$$ around $q$. The sublevel sets $M^{\epsilon}$ and $M^{1-\epsilon}$ are diffeomorphic since there aren't any other critical points besides $p$ and $q$. Now, we have that $M$ is the union of two $n$-disks.

Now, the basic idea is that $S^{n}$ is just two $n$-disks glued via the identity map on $S^{n-1}$ while our manifold has the two $n$-disks glued via $\varphi$. If we can construct a homeomorphism between these spaces, then we are done.

Let $D_1$ be the disk we got from looking at $f^{-1}[1-\epsilon,1]$ and $D_2$ the disk from $f^{-1}[0,\epsilon]$ (which is in fact diffeomorphic to $M^{1-\epsilon}$). Let's construct the homeomorphism $$g: D_1 \cup_{\text{Id}} D_2 \to D_1 \cup_{\varphi} D_2$$ explicitly via: $ g(x) = \left\{ \begin{array}{l l} x & \quad \text{ for $x \in D_1$}\\ 0 & \quad \text{ for $x=0$ (recall $0\in D_2$)}\\ \|x\| \varphi\left(\frac{x}{\|x\|}\right) & \quad \text{ for $x \in D_2 - \{0\}$} \end{array} \right. $

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  • $\begingroup$ How do we know that every point of the preimage will be contained in the $n$-disk? When you say that, for example, $f^{-1}[0,\varepsilon]$ is an $n$-disk, do you mean after applying a coordinate chart? $\endgroup$
    – INQUISITOR
    Dec 3, 2020 at 13:55
  • $\begingroup$ Also, what is $\varphi$? $\endgroup$
    – INQUISITOR
    Dec 3, 2020 at 20:08

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