On the topology of compact Stein surfaces

Abstract

The main result of this paper is that any compact Stein surface with boundary embeds naturally into a symplectic Lefschetz fibration over S2. Also along the way we obtain the following results. (1) There exists a minimal elliptic fibration over D2, which is not Stein. (2) The circle bundle over a genus n ≥ 2 surface with Euler number e = −1 admits at least n + 1 mutually nonhomeomorphic simply-connected Stein fillings. (3) Any surface bundle over S1, whose fiber is a closed surface of genus n ≥ 1 can be embedded into a closed symplectic 4-manifold, splittingthe symplectic 4-manifoldinto two pieces both of which have positive b2+⁠. (4) Every closed, oriented, connected 3-manifold has a weakly symplectically fillable double cover, branched along a 2-component link.