 
Summary: LEFSCHETZ FIBRATIONS ON COMPACT STEIN SURFACES
SELMAN AKBULUT AND BURAK OZBAGCI
Abstract. Let M be a compact Stein surface with boundary. We show that M
admits infinitely many pairwise nonequivalent positive allowable Lefschetz fibrations
over D2
with bounded fibers.
0. Introduction
The existence of a positive allowable Lefschetz fibration on a compact Stein surface
with boundary was established by Loi and Piergallini [LP] using branched covering
techniques. We give an alternative simple proof of this fact and construct explicitly
the vanishing cycles of the Lefschetz fibration, obtaining a direct identification of com
pact Stein surfaces with positive allowable Lefschetz fibrations over D2
. In the process
we associate to every compact Stein surface infinitely many pairwise nonequivalent
such Lefschetz fibrations.
We would like to thank Lee Rudolph, Yasha Eliashberg, Emmanuel Giroux and Ko
Honda for useful discussion about the contact geometry literature.
1. Preliminaries
1.1. Mapping class groups. Let F be a compact, oriented and connected surface
with boundary. Let Diff+
