 
Summary: ON THE TOPOLOGY OF COMPACT STEIN SURFACES
SELMAN AKBULUT AND BURAK OZBAGCI
Abstract. In this paper we obtain the following results: (1) Any compact Stein
surface with boundary embeds naturally into a symplectic Lefschetz fibration over
S2
. (2) There exists a minimal elliptic fibration over D2
, which is not Stein. (3) The
circle bundle over a genus n 2 surface with euler number e = 1 admits at least
n + 1 mutually nonhomeomorphic simplyconnected Stein fillings. (4) Any surface
bundle over S1
, whose fiber is a closed surface of genus n 1 can be embedded into
a closed symplectic 4manifold, splitting the symplectic 4manifold into two pieces
both of which have positive b+
2 . (5) Every closed, oriented connected 3manifold has
a weakly symplectically fillable double cover, branched along a 2component link.
0. Introduction
In [AO] (see also [LP]), it was proved that every compact Stein surface admits a
PALF (positive allowable Lefschetz fibration over D2
with bounded regular fibers)
and conversely every PALF is Stein. In this paper we first prove that any compact
