Constructing Stein manifolds after Eliashberg

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.