# 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?

• He says the gradient of $V$ is never zero off the origin. Seems to me $V^{-1}(c)$ is compact, and the map to (and onto) a small sphere around the origin given by following the gradient is an ODE system, differentiable in initial conditions. May 23, 2017 at 2:59
• @WillJagy I agree about the gradient being nonzero away from the origin. How do you know that $\nabla V$ isn't always tangent somewhere to arbitrarily small spheres? If your argument does work, I'm confused about why Wilson proved went to the trouble of proving Thm 1.2 on the third page. May 23, 2017 at 3:15
• good point. Managed to download the correct article this time. I guess I don't know. Have you looked up later citations to this article on MathSciNet? Personally, I think codimension one is a helpful thing. May 23, 2017 at 3:25
• @WillJagy I'm looking now. So far I haven't found anything useful, but that's a good suggestion. May 23, 2017 at 3:35

## 2 Answers

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!

• Ah, I see. It looks like the fault was mine. Your comments are very helpful for me. Thank you very much. May 24, 2017 at 22:27

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.

• Many thanks for your answer. It's quite a coincidence to receive an answer from you since I currently happen to be reading your paper "Submersions, Fibrations, and Bundles." May 24, 2017 at 20:08
• Regarding the case $n=5$ being open: I just discovered that, in the 2008 paper "On Brockett's Necessary Condition for Stabilizability and the Topology of Liapunov Functions on $\Bbb{R}^n$" by C. I. Byrnes, it seems to be claimed in his Theorem 3.1 that diffeomorphism holds even for $n=5$. (See also his Theorem 4.1 and Lemma 4.2 for similar claims). Based on what I have learned from your answers here, I am suspicious of what Byrnes claims. Do you think Byrnes is mistaken? Feb 14, 2021 at 22:30
• I don't understand Byrnes' claims and arguments, and especially not on line 10 of page 341. Again, Freedman's beautiful result is topological, and implies no diffeomorphism between the level set and the 4-sphere, only a homeomorphism. Feb 16, 2021 at 7:01