On Wilson's claim that Lyapunov function level sets are not exotic spheres In Wilson's paper "The structure of the level surfaces of a Lyapunov function," he states in Corollary 1.3 that the level sets of a smooth Lyapunov function are diffeomorphic to a standard sphere. (The Lyapunov function is for a globally asymptotically stable equilibrium point for a flow on $\mathbb{R}^n$.)
To prove this for a Lyapunov function $V:\mathbb{R}^n\to \mathbb{R}$ and $c > 0 $, he uses the flow to show that there are diffeomorphisms $V^{-1}(c) \approx \mathbb{R}^n\setminus \{0\} \approx S^{n-1}\times \mathbb{R}$. Thus $V^{-1}(c)$ is a homotopy sphere, and since $c$ is a regular value of $V$ it follows from the generalized Poincaré conjecture in Top that $V^{-1}(c)$ is homeomorphic to a sphere. 
Wilson claims that $V^{-1}(c)$ is diffeomorphic to a sphere. Why is this true?
He also makes this comment on the third page: "Our spheres will always have the standard differentiable structure, since it is induced by the embedding in $\mathbb{R}^n$." But all exotic spheres also embed into some $\mathbb{R}^N$ by Whitney's theorem -- so I am not sure what Wilson means. Could this be due to the fact that $V^{-1}(c) \subset \mathbb{R}^n$ is codimension-1? 
 A: For $n\le 4$, the level hypersurface $V^{-1}(c)$ is diffeomorphic to $S^{n-1}$ because there are no exotic spheres of dimension $1$, $2$ or $3$.
For $n\ge 6$, $V^{-1}(c)$ is diffeomorphic to $S^{n-1}$ by Smale's h-cobordism theorem: a h-cobordism
between $V^{-1}(c)$ and $S^{n-1}$ is given by the domain bounded by $V^{-1}(c)$
and a small standard sphere centered at your equilibrium point.
For $n=5$, as far as I know, the question remains open today.
A: Matthew. I had a look at Wilson's paper; he is of course rigourous; he says that


*

*$V^{-1}(c)$ is homotopy-equivalent to $S^{n-1}$ for every $n$;

*$V^{-1}(c)$ is diffeomorphic to $S^{n-1}$ for every $n\neq 4, 5$


which corresponds to what was known in 1967. Nowadays, one can say a little more:


*

*$V^{-1}(c)$ is also diffeomorphic to $S^{n-1}$ for $n=4$ (thanks to Perelman's proof of the Poincare conjecture);

*
*

*$V^{-1}(c)$ is homeomorphic to $S^{n-1}$ for every $n$ (thanks to Friedman's topological h-cobordism theorem).



Good reading!
