I've taken a course on Morse theory a couple of years ago, but I have no idea about how to solve the following problem. Could you give some hints?
Problem: Let $M$ be a smooth manifold homeomorphic but not diffeomorphic to the $4$-dimensional sphere $S^4$. Show that every Morse function on $M$ has at least $4$ critical points.
Many thanks in advance.
Recall $\chi(S^4) = 2$. Let $\beta_i$ be the number of critical points of index $i$ (for $i = 0, 1, 2, 3, 4$). Then $\beta_0 - \beta_1 + \beta_2 - \beta_3 + \beta_4 = 2$. Note that $\beta_0$ and $\beta_4$ are $\geq 1$ (since $f$ must have a minimum and a maximum). Suppose for now that $\beta_0 = \beta_4 = 1$. Then $\beta_1 + \beta_3 = \beta_2$. It is then easily seen that the only way to have $< 4$ critical points is if $\beta_1 = \beta_2 = \beta_3 = 0$, but that would make $M$ diffeomorphic to $S^4$ since $f$ would have only two critical points and $4$ is small enough.
Now let's deal with the case $\beta_0 > 1$ (if this is not true, but $\beta_4 > 1$ instead, switch from $f$ to $-f$). But then clearly there must be a critical point of index between 1 and 3 or the Euler characteristic would be too big. So we have at least $4$ critical points (at least two of index 0, at least one of index 4, and at least something in between).