A stronger form of the adjunction inequality is proved for immersed real surfaces in non simply-connected Stein surfaces. The result is applied to the geometry of Stein domains and analytic continuation on complex surfaces. | Turk J Math 27 (2003) , 161 – 172. ¨ ITAK ˙ c TUB Adjunction inequality and coverings of Stein surfaces Stefan Nemirovski Abstract A stronger form of the adjunction inequality is proved for immersed real surfaces in non simply-connected Stein surfaces. The result is applied to the geometry of Stein domains and analytic continuation on complex surfaces. 1. Introduction A complex manifold X is Stein if it admits a strictly plurisubharmonic exhaustion function ϕ : X → R. (A C 2 -smooth function on a complex manifold is called strictly ahler form.) Every connected component of a regular plurisubharmonic if ddcϕ is a K¨ sublevel set {x ∈ X | ϕ(x) 2. Hence, it is homeomorphic to a Stein complex surface by the results of Gompf and Eliashberg (see [8, Ch. 11]). 162 NEMIROVSKI Corollary . Suppose that Y is an open 3-manifold such that Y × R is diffeomorphic to a Stein complex surface. Then every embedded 2-sphere in Y bounds a homotopy 3-ball. Proof. If S ⊂ Y is an embedded 2-sphere in Y , then Σ = S × {0} ⊂ Y × R is an embedded 2-sphere whose self-intersection index is zero. In particular, Σ cannot satisfy the adjunction inequality and hence its homotopy class is trivial by Theorem . Since the inclusion Y × {0} ⊂ Y × R is a homotopy equivalence, it follows that S is nullhomotopic in Y . It is a standard result that a null-homotopic embedded 2-sphere in a 3-manifold bounds a homotopy ball (see, for instance, [11, Prop. ]). Example . Let M be a closed orientable three-manifold and M (n) the open threemanifold obtained by removing n ≥ 1 points from M . If the smooth manifold M (n) × R admits a Stein complex structure, then n = 1 and M is a homotopy 3-sphere. Indeed, let us apply the previous corollary to the boundary of a small ball about one of the punctures in M . It follows that this 2-sphere bounds a homotopy ball in M (n) , which is only possible if n = 1 and M is a homotopy sphere. In other words, if the three-dimensional Poincar´e conjecture holds .
