Let $S$ be a Riemann surface, it is well known that there are two ways to compute the Euler number of $S$.
The first approach is using Morse theory. Given a Morse function $f: S \to \mathbb{R}$, the Euler number $$\chi(S)= \sum_{p \in Crit(f) } (-1)^{ind p}, $$ where $indp $ is the index of $p$, that is the number of negative eigenvalues of the Hessian $Hess f $ at $p$.
The other approach is using meromorphic 1-form. Given a meromorphic 1-form $\alpha$ over $S$, then the Euler number is equal to the number of poles of $\alpha$ minus the number of zeros of $\alpha$, counted with multiplicity. That is $$\chi(S) = \# \alpha^{-1}(\infty)-\#\alpha^{-1}(0). $$
From the view point of Morse theory, the topology of $S$ only change at the critical points. On the other hand, $\alpha$ has non-vanishing winding number only at the zero or pole. I was wondering if there any relationship between this two approach?
More precisely, given a Morse function $f$, can we always find a meromorphic $1$-form $\alpha$ such that $\alpha$ has poles and zeros only at the critical points of $f$? Or conversely, given a meromorphic $1$-form $\alpha$, can we find a Morse function $f$ such that the critical points appear at the zeros or poles of $\alpha$?