We discuss several applications of Seiberg-Witten theory in conjunction with an embedding theorem (proved elsewhere) for complex 2-dimensional Stein manifolds with boundary. We show that a closed, real 2-dimensional surface smoothly embedded in the interior of such a manifold satisfies an adjunction inequality, regardless of the sign of its self-intersection. This inequality gives constraints on the minimum genus of a smooth surface representing a given 2-homology class. We also discuss consequences for the contact structures existing on the boundaries of these Stein manifolds. We prove a slice version of the Bennequin-Eliashberg inequality for holomorphically fillable contact structures, and we show that there exist families of homology 3-spheres with arbitrarily large numbers of homotopic, nonisomorphic tight contact structures, Another result we mention is that the canonical class of a complex 2-dimensional Stein manifold with boundary is invariant under self-diffeomorphisms fixing the boundary.

Stein 4-manifolds with boundary and contact structures

LISCA, PAOLO;
1998-01-01

Abstract

We discuss several applications of Seiberg-Witten theory in conjunction with an embedding theorem (proved elsewhere) for complex 2-dimensional Stein manifolds with boundary. We show that a closed, real 2-dimensional surface smoothly embedded in the interior of such a manifold satisfies an adjunction inequality, regardless of the sign of its self-intersection. This inequality gives constraints on the minimum genus of a smooth surface representing a given 2-homology class. We also discuss consequences for the contact structures existing on the boundaries of these Stein manifolds. We prove a slice version of the Bennequin-Eliashberg inequality for holomorphically fillable contact structures, and we show that there exist families of homology 3-spheres with arbitrarily large numbers of homotopic, nonisomorphic tight contact structures, Another result we mention is that the canonical class of a complex 2-dimensional Stein manifold with boundary is invariant under self-diffeomorphisms fixing the boundary.
1998
Lisca, Paolo; Matic, G.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/47718
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