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.