About existence of Morse functions

Question

Let's consider 4-manifold $M$, $\partial M = \partial M_1 + \partial M_2 = S^1 \times S^2 + \mathbb{RP}^3$. Is it true that there exist a Morse function $$f\colon M^4 \to [0,1],\quad f^{-1}(0) = \partial M_1 = S^1 \times S^2,\ f^{-1}(1) = \partial M_2 = \mathbb{RP}^3$$ with only one critical point of index 2.

Answer 1

No, this is never true.

Attach a copy of $S^1 \times D^3$ to $M$ to get a 4-manifold with one boundary component, $\mathbb{RP}^3$. If $M$ has a Morse function with one critical point of index two, this new manifold can be given a Morse function with three critical points with indices 0, 1 and 2. This means it is a rational ball, so the first and second Betti numbers are zero. The boundary of a rational ball has to have first homology of order $k^2$ for some $k$ (you can show this by messing about with exact sequences) and so it cannot be $\mathbb{RP}^3$.