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I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have same number of $i$ index critical points for each $i\in \mathbb N$, then whether I can conclude they are homeomorphic or not.

It is clear that they are of same dimension since the maximum of $f$ and $g$ are of same index. And we know that if a closed manifold has exactly two critical points then it is homeomorphic with $S^n$.

But still I don't expect an positive answer for my doubt. So to find a contradiction I was thinking about handle decomposition of some closed manifold. And now I got stuck with this doubt that suppose I have a handle decomposition of a closed manifold $X$, then can I construct a Morse function $f$ on $X$ such that corresponding to this Morse function I get the same handle representation?? This is kind of converse of Morse handle presentation theorem. So far I could not able to construct such a function.

And also I could not able to find any counter example for my initial doubts.

It'll be very helpful if someone clarify my these two doubts.

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2 Answers 2

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In looking for counter-examples it helps to notice that $M$ and $N$ have CW structures with the same numbers of $i$ cells, and hence they have the same Euler characteristic.

It turns out that there is a Morse function on the Klein bottle with critical points of index $0, 1, 1, 2$, so together with the standard height function on the 2-torus we get a counter-example. This is explained at the end of these notes by Michael Landry.

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  • $\begingroup$ this is nice, and can you tell anything about my second doubt? $\endgroup$ Jun 24, 2016 at 12:02
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    $\begingroup$ And conversely, if two closed manifolds of the same dimension have the same Euler characteristic, then they have Morse functions with the same number of critical points at each index. Just start with two arbitrary Morse functions and then stabilize via canceling pairs on both sides. $\endgroup$ Jun 24, 2016 at 17:24
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In answer to your second question, yes, given a handle decomposition it is possible to find a Morse function whose corresponding handle decomposition is the same as the one you started with. I think there's a proof, via "gradient-like vector fields", in Milnor's book on Morse theory (though I don't have a copy in front of me now).

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