Preprint A69/2001
On the Topology of Foliations with a First Integral

Hossein Movasati

**Keywords: **
Lefschetz pencil and vanishing cycle - homology groups - fiber bundl

The main objective of this article is to study the topology of the fibers of a generic rational function of the type f=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 generates the first homology group of a generic fiber of f.
In particular, we will prove that for any two Lefschetz vanishing cycles in
a regular compact fiber of f, there exists a monodromy sending one of the vanishing cycles to other.