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Are there any examples of closed manifolds with the property that the minimal number of critical points (possibly degenerate) of a smooth function on this manifold is strictly bigger than the stable Morse number of this manifold?

Please pay attention to the fact that critical points can be degenerate, i.e. there is no assumption that the function is Morse (in other words, the comparison is between two numbers, one coming from smooth functions (NOT MORSE in general), and another coming from Morse functions on the stabilizations of this manifold).

Finally, note that Morse number of a manifold is greater or equal than the stable Morse number of a manifold. Also, Morse number of a manifold is greater of equal than the minimal number of critical points (possibly degenerate) of a smooth function on this manifold.

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    $\begingroup$ What's the "stable morse number" of a manifold? When I google it I only get the paper mentioned by Chris Gerig, and it's behind a paywall. $\endgroup$ May 5, 2016 at 3:19
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    $\begingroup$ Actually, this paper ("On the stable Morse number of a closed manifold") gives examples for minimal critical number of generic smooth functions. The "stable Morse number" is the minimum number of critical points of a Morse function $f:M\times\mathbb{R}^n\to\mathbb{R}$ (with an added technical assumption that $f$ is "almost quadratic at infinity") ranging over all such functions and all $n\in\mathbb{N}$. $\endgroup$ May 5, 2016 at 6:18
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    $\begingroup$ I am now confused in view of what @ChrisGerig wrote. Do you (Alex) mean you want the minimal number of critical points to be strictly smaller than the stable Morse number instead? $\endgroup$ May 5, 2016 at 13:18
  • $\begingroup$ @WillieWong This could happen, because stabilisation will not increase the Morse number. Moreover, there are manifolds where the $\text{cup length}+1$ equals the Morse number. $\endgroup$ May 5, 2016 at 15:26

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