Journal of G¨okova Geometry Topology
Volume 2 (2008) 83106
Every 4-Manifold is BLF
Selman Akbulut and C¸agri Karakurt
Abstract. Here we show that every compact smooth 4-manifold X has a structure
of a Broken Lefschetz Fibration (BLF in short). Furthermore, if b+
2 (X) > 0 then it
also has a Broken Lefschetz Pencil structure (BLP) with nonempty base locus. This
improves a theorem of Auroux, Donaldson and Katzarkov, and our proof is topological
(i.e. uses 4-dimensional handlebody theory).
In order to state our theorem in a precise form, we first need to recall some basic
definitions and fix our conventions.
Let X be an oriented 4-manifold and be an oriented surface. We say that a map
: X has a Lefschetz singularity at p X, if after choosing orientation preserving
, 0) (X, p) and C , is represented by the map (z, w) zw. If
is represented by the map (z, w) z ¯w we call p an achiral Lefschetz singularity of .
We say p X is a base point of , if it is represented by the map (z, w) z/w. We