Creating Stein surfaces by topological isotopy

Abstract

We combine Freedman’s topology with Eliashberg’s holomorphic theory to construct Stein neighborhood systems in complex surfaces, and use these to study various notions of convexity and concavity. Every tame, topologically embedded $2$-complex $K$ in a complex surface, after $C^0$-small topological ambient isotopy, is the intersection of an uncountable nested family of Stein regular neighborhoods that are all topologically ambiently isotopic rel $K$, but frequently realize uncountably many diffeomorphism types. These arise from the Cantor set levels of a topological mapping cylinder. The boundaries of the neighborhoods are $3$-manifolds that are only topologically embedded, but still satisfy a notion of pseudoconvexity. Such $3$-manifolds share some basic properties of hypersurfaces that are strictly pseudoconvex in the usual smooth sense, but they are far more common. The complementary notion of topological pseudoconcavity is realized by uncountably many diffeomorphism types homeomorphic to $\mathbb{R}^4$.