Proof of the Extended Reeb's Theorem for dimensions less than 7 Reeb's Theorem states that every compact smooth manifold which admits a Morse funtion with exactly two critical points is homeomorphic to n-sphere.
I have heard an extension of this theorem for dimensions less than 7, as homeomorphism replaced by diffeomorphism and couldn't found any proof yet.
One might say it is a direct consequence of Smooth Poincare Conjecture for dimensions less than 7 which states the equivalency of being homeomorphic to standart n-sphere and being diffeomorphic to standart n-sphere.
However, since the reason I am asking this question is that this theorem does constitutes a half of the proof of the theorem which states that exotic 4-spheres, if any, have at least 4 Morse critical points, I wonder if there exist a Poincare Conjecture independent proof of this extended Reeb's Theorem, since Smooth Poincare Conjecture in dimension 4 is still an open problem.
 A: This question amounts showing that for all integer $1\leqslant n\leqslant 6$, the twisted $n$-sphere group (denoted by $\Gamma_n$) is trivial, where $\Gamma_n$ fits in the following short exact sequence:
$$\pi_0\operatorname{Diff}^+(D^n)\to\pi_0\operatorname{Diff}^+(S^{n-1})\to\Gamma_n\to 0,$$
where $S^{n-1}$ is seen as the boundary of $D^n$ and the first map is induced by the restriction, then $\Gamma_n$ basically measures the obstruction for a diffeomorphism of $S^{n-1}$ to extend to a diffeomorphism of $D^n$.
Let me explain why this is true. Let $M^n$ be a closed manifold and $f\colon M\to\mathbb{R}$ be a Morse function on $M$ with two critical points. Since $M$ is compact, the critical points of $f$ are a minimum $x_-$ and a maximum $x_+$, thus the stable manifold of $x_-$ and the unstable manifold of $x_+$ (for any Riemannian metric on $M)$ are $n$-dimensional disks. The proof of Reeb's theorem now boils down to showing that $M$ is obtained by gluing these disks along their boundaries, but this gluing map is an orientation preserving diffeomorphism of $S^{n-1}$. If is isotopic to the identity, then $M$ is diffeomorphic to the standard $n$-sphere.
It is easy to show that $\Gamma_1$ and $\Gamma_2$ are trivial (the extension is explicit and straightforward), however, the higher dimensional cases are more subtle:

*

*for $n=3$, this is a result due independently to Munkres, 1960 (the proof is elementary and uses the Poincaré-Bendixson theorem) and Smale, 1959,

*for $n=4$, this is a hard result due to Cerf, 1968 (see also this wonderful contact geometric proof),

*for $n=5$ or $6$, this is a result due to Kervaire and Milnor, 1963.

