A primary aim of this project is to use topological methods to study which open subsets of complex surfaces are Stein. Preliminary results show that up to topological isotopy, Stein open subsets are as common as possible. That is, if U satisfies the basic necessary condition (it is homeomorphic to an open handlebody with handles of index at most 2), then it is topologically isotopic (not necessarily smoothly or ambiently) to a Stein open subset. The project is revealing much more detailed structure: a 2-complex topological spine K for U that is smoothly embedded except for one point on each 2-cell, and a neighborhood system of K consisting of homeomorphic Stein surfaces indexed by a Cantor set and frequently realizing uncountably many diffeomorphism types. There is much control over minimal genera of homology classes of the Stein neighborhoods. The project investigates these phenomena using techniques from topological 4-manifold theory (Casson handles and reimbedding), and also applies these methods to study pseudoconvex smooth and topological embeddings of 3-manifolds into complex surfaces. A second thrust of the proposal concerns the topology of closed, symplectic 4-manifolds, such as the recently discovered simply connected 4-manifolds with small Euler characteristics. For example, what range of signature and Euler characteristic is realized for a fixed fundamental group? Which symplectic manifolds decompose completely after connected sum with a single CP^2? Can one find general bounds on the number of CP^2 summands required to decompose symplectic manifolds? These questions can be studied via Kirby calculus and Lefschetz fibrations. One can also drop the symplectic structures and ask the above questions for irreducible smooth 4-manifolds.
The project concerns the classification of 3- and 4-manifolds and their contact, symplectic and Stein structures. An n-manifold is a space locally indistinguishable from Euclidean n-space. For example the space in which we live and the universe (space-time) are 3- and 4-manifolds, respectively. Contact and symplectic structures first arose in classical physics (optics and Hamiltonian mechanics, respectively), but they are also intimately connected with more recent physics such as string theory, as well as various branches of mathematics such as algebraic geometry, topology and dynamical systems. Stein manifolds are symplectic manifolds endowed with additional structure that is naturally expressed via contact topology. They also have various equivalent definitions in terms of complex analysis - for example, Stein manifolds are essentially the same as complex submanifolds of complex n-space. Complex analysts have recognized Stein manifolds as fundamental objects of study for most of the past century, but much about them remains unknown. The present project investigates which open subsets of a complex manifold are Stein manifolds. This question (like many in topology) is most subtle in the context of 4-manifolds. In this setting, the project has already shown that every subset satisfying the most basic requirements can be deformed into a Stein subset, although the deformation cannot usually be made smooth. Such unsmoothable deformations can only be accessed via esoteric techniques from topological 4-manifold theory. Since these techniques are quite alien to complex analysts, the research is in some sense interdisciplinary, and the results are completely unlike anything obtainable by traditional analytic methods. For example, the resulting Stein manifolds frequently come in uncountably many types, all topologically equivalent but smoothly distinct. In addition to Stein manifolds, the project deals with 4-manifolds and symplectic and contact manifolds. Most of what is known about the classifications of these objects has been discovered in the past few decades. While the fields are developing rapidly, the final classification schemes remain elusive.
