Constructing Stein manifolds after Eliashberg

TL;DR

This paper provides a comprehensive overview of the existence and construction of Stein manifolds across all dimensions, highlighting topological methods to classify and generate diverse Stein structures, including exotic R^4's.

Contribution

It extends Eliashberg's characterization by removing complex dimension 2 restrictions using Freedman theory and demonstrates how to construct uncountably many Stein structures and exotic R^4's.

Findings

Uncountably many diffeomorphism types of Stein open subsets of C^2.

Topological isotopy can produce diverse Stein domains.

Any Morse function can be subdivided into a handlebody with the same maximal index.

Abstract

A unified summary is given of the existence theory of Stein manifolds in all dimensions, based on published and pending literature. Eliashberg's characterization of manifolds admitting Stein structures requires an extra delicate hypothesis in complex dimension 2, which can be eliminated by passing to the topological setting and invoking Freedman theory. The situation is quite similar if one asks which open subsets of a fixed complex manifold can be made Stein by an isotopy. As an application of these theorems, one can construct uncountably many diffeomorphism types of exotic R^4's realized as Stein open subsets of C^2 (i.e. domains of holomorphy). More generally, every domain of holomorphy in C^2 is topologically isotopic to other such domains realizing uncountably many diffeomorphism types. Any tame n-complex in a complex n-manifold can be isotoped to become a nested intersection of Stein open subsets, provided the isotopy is topological when n=2. In the latter case, the Stein neighborhoods are homeomorphic, but frequently realize uncountably many diffeomorphism types. It is also proved that every exhausting Morse function can be subdivided to yield a locally finite handlebody of the same maximal index, both in the context of smooth n-manifolds and for Stein surfaces.