Minimal number of Morse critical points

I am interested in the minimal number of critical points of a Morse function on a closed manifold or a proper Morse function on a manifold with boundary; I will call this the Morse function of the manifold. A priori, this number is a smooth invariant. A lower bound on this number is obtained by the minimal number of generators for integral homology. I know that Smale proved that if $M^n, n \ge 5,$ is simply-connected, then the Morse number is precisely this number of generators for integral homology. So in the simply-connected case, this number is an invariant of the homotopy type.

In the non-simply-connected case (still $n \ge 5$), the minimal number of critical points may be bigger than the number of generators for homology (for example, it should also generate $\pi_1$ which may not vanish even if first homology vanishes). My question is whether this number (for $n \ge 5$ and in the non-simply-connected case) depends just on the homotopy type or whether there are homotopy equivalent manifolds with different Morse numbers? What about homeomorphic manifolds with different Morse numbers? Can it be arbitrarily large for manifolds within a fixed homotopy/homeomorphism class?

Edit: I would also be happy knowing the answer for non-trivial h-cobordisms. I know that Whitehead group of $\mathbb{Z}[\pi_1(M)]$ is involved in the s-cobordism theorem; this group depends only on $\pi_1(M)$ and hence only the homotopy type of $M$. But I think an s-cobordism is determined by an element of this Whitehead group. So I think my geometric question above reduces to the algebraic question of whether there are elements of the group that need to be represented by different size (or arbitrarily large) matrices.

• There are contractible compact 4-manifolds with non-trivial fundamental group at infinity. These cannot have a proper Morse function with a single min (obviously), but are homotopy equivalent to the 4-ball. The first example was constructed by Mazur (see en.wikipedia.org/wiki/Mazur_manifold). The typical examples of these have a Morse function (handle decompositon) with one critical point (handle) of index 0, one of index 1 and one of index 2 . – PVAL-inactive Oct 3 '17 at 18:09
• @PVAL-inactive Thanks for this example. Do you know any examples in high dimensions based on Whitehead torsion? – user39598 Oct 4 '17 at 18:26