On the Topology of Foliations with a First Integral

Details On the Topology of Foliations with a First Integral On the Topology of Foliations with a First Integral

2002-06-07

arxiv.org/abs/math/0206075v1

Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 3, 305-336

Mathematics

Geometric Topology

31 pages

Scientific paper

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

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