This question comes up when I am working on an exercise finding the Heegaard genus of the 3-torus $S^1\times S^1\times S^1$.
By definition of Heegaard genus, it is the minimal possible genus of the boundary of the two handlebodies that decompose a closed oriented 3-manifold. In fact, we can find a self-indexing Morse function of the 3-torus given by $f(\theta_1,\theta_2,\theta_3)=\frac{1}{2}(3+\sin\theta_1+\sin\theta_2+\sin\theta_3)$, where $\theta_i$ are denoting the parameters of each of the circle.
From the Morse function given above, we see that there are 3 critical points of index 1 and 2 respectively. This concluded the Morse function came up from a Heegaard decomposition of genus 3.
Now, my question is: Is my argument sufficient in concluding that the Heegaard genus of the 3-torus is 3?
I don't think it is sufficient as we haven't disprove the possibility of Heegaard genus 2. If so, what should be the argument required to disprove such possibility?