  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 $h(\delta_0)=\pm
\delta_1$.
