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Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post Morse theory in TOP and PL categories? for references).

It is well known that a TOP (resp. DIFF) closed $n$-manifold that admits a TOP (resp. DIFF) Morse function having exactly two non-degenerate critical points must be homeomorphic to the $n$-sphere.

In DIFF this is due to G. Reeb, and it remains true even if the critical points are degenerate as proven by Milnor and Rosen (Theorem 1, p.124 of [Mil2007]).

In TOP it is due to Kuiper, [Kui1961]; and my first question is: Does the result remain true even if the points are allowed to be degenerate?

Working in DIFF, let $M$ be an $n$-manifold and $f$ a Morse function on $M$. Then if $[a,b]$ is an interval of regular values of $f$, it is not difficult to show that $f^{-1}(a)$ is homeomorphic to $f^{-1}(b)$ (e.g. Milnor's Morse Theory, p. 12). However, is this true in the topological case?

[Kui1961] N.H. Kuiper, A continuous function with two critical points, Bull. Amer. Math. Soc., 67(1961), 281-285.

[Mil2007] J. Milnor, Collected Papers of John Milnor: Vol. III, Differential Topology, AMS, 2007.

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Is there a good example to think about with degenerate critical points and yet only two of them? I'm having trouble picturing how this could happen. Thanks! –  Patricia Hersh May 24 '12 at 5:37
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Sorry, I don't have a good example. The height function, when restricted to e.g., the unit 2-sphere in $\mathbb{R}^{3}$ has exactly two non-degenerate critical points (both in the DIFF and TOP senses). In DIFF, it is possible to smoothly deform the sphere slightly in such a way that the height function on the new manifold still has exactly two critical points, but now the Hessians will vanish, thus giving us degenerate ones (there is an illustrative picture in p.127 of Milnor's work cited above). In TOP this will not work, I will think about it more. –  Victor May 24 '12 at 16:11
    
Thanks! It's been taking me some time to track down this Milnor volume, but I should finally get to look at this volume and see this picture tomorrow. I ended up ordering it -- right now I have some funding to buy more math books, and this certainly sounded like a good one. Good luck in thinking more about TOP! –  Patricia Hersh May 28 '12 at 12:43
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It will be worth it! Milnor's works are examples of great mathematics, and of great mathematical writing –  Victor May 30 '12 at 5:30

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