On the Topology of Foliations with a First Integral

The main objective of this article is to study the topology of the fibers of a generic rational function of the type $F^p/G^q$ in the projective space of dimension two. We will prove that the action of the monodromy group on a single Lefschetz vanishing cycle $\delta$ generates the first homology group of a generic fiber of $F^p/G^q$. In particular, we will prove that for any two Lefschetz vanishing cycles $\delta_0$ and $\delta_1$ in a regular compact fiber of $F^p/G^q$, there exists a monodromy $h$ such that $h(\delta_0)=\pm \delta_1$.