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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?

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  • $\begingroup$ 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. $\endgroup$
    – Will Jagy
    Commented May 23, 2017 at 2:59
  • $\begingroup$ @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. $\endgroup$
    – Matthew Kvalheim
    Commented May 23, 2017 at 3:15
  • $\begingroup$ 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. $\endgroup$
    – Will Jagy
    Commented May 23, 2017 at 3:25
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    $\begingroup$ @WillJagy I'm looking now. So far I haven't found anything useful, but that's a good suggestion. $\endgroup$
    – Matthew Kvalheim
    Commented May 23, 2017 at 3:35

2 Answers 2

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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!

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    $\begingroup$ Ah, I see. It looks like the fault was mine. Your comments are very helpful for me. Thank you very much. $\endgroup$
    – Matthew Kvalheim
    Commented May 24, 2017 at 22:27
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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.

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    $\begingroup$ 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." $\endgroup$
    – Matthew Kvalheim
    Commented May 24, 2017 at 20:08
  • $\begingroup$ 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? $\endgroup$
    – Matthew Kvalheim
    Commented Feb 14, 2021 at 22:30
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    $\begingroup$ 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. $\endgroup$
    – Gael Meigniez
    Commented Feb 16, 2021 at 7:01

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