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Summary: arXiv:submit/0018799[math.GT]7Apr2010
LEFSCHETZ FIBRATIONS ON COMPACT STEIN MANIFOLDS
SELMAN AKBULUT AND M. FIRAT ARIKAN
Abstract. Here we prove that a compact Stein manifold W2n+2
of dimension 2n+2 > 4
admits a Lefschetz fibration over the disk D2
with Stein fibers, such that the monodromy
of the fibration is a symplectomorphism induced by compositions of "generalized Dehn
twists" along imbedded n-spheres on the generic fiber. Also, the open book on the
boundary W, which is determined by the fibration, is compatible with the contact
structure induced by the Stein structure. This generalizes the Stein surface case of
n = 1, previously proven by Loi-Piergallini and Akbulut-Ozbagci.
1. Introduction
In [AO] (see also [LP]), it is proved that every compact Stein surface admits a positive
allowable Lefschetz fibration over D2
with bounded fibers (PALF in short), and conversely
every 4-dimensional positive Lefschetz fibration over D2
with bounded fibers is a Stein
surface. Here we prove the following, which can be thought as a generalization of this
results to higher dimensions:
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