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A NOTE ON LEFSCHETZ FIBRATIONS ON COMPACT STEIN 4-MANIFOLDS
 

Summary: A NOTE ON LEFSCHETZ FIBRATIONS ON COMPACT STEIN
4-MANIFOLDS
SELMAN AKBULUT AND M. FIRAT ARIKAN
Abstract. Loi-Piergallini and Akbulut-Ozbagci showed that every compact Stein sur-
face admits a Lefschetz fibration over the disk D2
with bounded fibers. In this note we
give a more intrinsic alternative proof of this result.
1. Introduction
In [AO] (also [LP] and [P]) it was proven that every compact Stein surface admits a
positive allowable Lefschetz fibration over D2
with bounded fibers (PALF in short), and
conversely in [AO] it was shown that every 4-dimensional positive Lefschetz fibration over
D2
with bounded fibers is a Stein surface. The proof of [AO] uses the fact that every
torus link is fibered in S3
. Here we prove this by using a more intrinsic different approach,
namely by an algorithmic use of positive stabilizations. This new approach is more closely
related Giroux's proof of constructing open books to contact manifolds via "contact cell
decomposition" [Gi]. The algorithm in [A] constructs compatible open books for contact
structures on 3-manifolds using their surgery representations. The algorithm here is for

  

Source: Akbulut, Selman - Department of Mathematics, Michigan State University

 

Collections: Mathematics