Suppose a meromorphic function $f(z)$ has two poles, with residues $1$ and $\gamma$, respectively. Then the topology of the Riemann surface of the anti-derivative of $f(z)$ depends on whether or not $\gamma$ is irrational. More generally, the topology of a meromorphic function with $n$ poles with residues $\gamma_1,\gamma_2,\cdots,\gamma_n$ depends on the linearity of $\gamma_1,\cdots,\gamma_n$ over $\mathbb{Q}$. Has anyone considered this phenomenon? Are there relationships between the (co)homology groups of the covering and the residues? Could one attempt to prove the irrationality (or rationality) of a given complex number by considering the residues of the poles of a meromorphic function in this way? (In such a case, one would need other ways of extracting topological information about the given meromorphic function.)