Given a closed 3-manifold $M$, a self-indexing Morse function $f:M\to[0,3]$ produces a Heegaard splitting by : $$M=f^{-1}([0,3/2])\cup f^{-1}([3/2,3]).$$ (This question discussed a bit on the topic.)
Now, we define :
Definition : Let $M$ be a closed 3-manifold. The Morse number $\mathfrak{m}(M)$ of $M$ is the minimal number of critical points of Morse functions. The Heegard number $\mathfrak{h}(M)$ of $M$ is the smallest genus of any Heegard splitting of $M$.
We have a theorem that Milnor used to create exotic spheres :
Theorem : (Reeb) Let $M$ be a closed $n$-manifold such that $\mathfrak{m}(M)=2$. Then $M$ is a topological $n$-sphere.
Moreover, by a result from Alexander, we have that $\mathbb{S}^3$ is the only manifold whose Heegaard number is $0$, that is we have, for a closed 3-manifold : $$\mathfrak{h}(M)=0\iff\mathfrak{m}(M)=2\iff M\cong\mathbb{S}^3.$$
This suggests that the two may be related. My questions are the following :
Are $\mathfrak{m}(M)$ and $\mathfrak{h}(M)$ related ?
Given a self-indexing Morse function $f:M\to[0,3]$, is the genus of the associated Heegaard splitting related to the number of critical points of $f$ ?
About the first question, I am wondering whether there is a somewhat-nice formula, like a linear relation $\mathfrak{m}(M)=A\mathfrak{h}(M)+B$ ? About the second question, I found no literature about this construction for a Heegaard splitting, the one I knew about was using Moise's triangulation theorem.
As a side note, about the first question, if a linear relation existed, taking $M=\mathbb{S}^3$ or $\mathbb{S}^1\times\mathbb{S}^2$ would yield (unless I made mistakes in my computations) $\mathfrak{m}(M)=2[\mathfrak{h}(M)+1]$. However, I didn't manage to compute the two numbers for a 3-torus $\mathbb{T}^3$ to check whether there is a contradiction. In any case, a polynomial relation $\mathfrak{m}(M)=P(\mathfrak{h}(M))$ would be respectable !
Edit :
I've been reading some stuff here and there, and I have noticed :
Given a self-indexing Morse function $f:M\to[0,3]$, assuming that there are only one index zero and one index three critical points, then the genus of the associated Heegaard splitting is $g=\# f^{-1}(1)$ the number of index one critical points.
It is always possible to construct a Morse function satisfying the hypotheses of the previous bullet.
However, my concern is that the function constructed this way is not necessarily minimal (in the sense that its number of critical points is not necessarily the Morse number of the manifold). Moreover, this construction is certainly not natural. At last, I am not sure that given a self-indexing Morse function $f:M\to[0,3]$, the construction of the new Morse function $\tilde{f}:M\to[0,3]$ with $\# f^{-1}(0)=\# f^{-1}(3)=1$ preserves the total number of critical points -- I haven't yet looked at the very details of this construction (see Milnor, Lectures on the h-cobordism theorem).
In any case, this partially answers my second question, at least in that special setting where we assume only one index zero/three critical point.
It would be interesting to see what you think about this !