Symplectic, Contact and Low-Dimensional Topology

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9802533 Gompf The focus of this project is on constructing symplectic, contact, and 4-dimensional manifolds. The principal investigator is attempting to construct a symplectic structure on any manifold endowed with a Lefschetz pencil (a kind of singular fibration originally discovered in algebraic geometry). He has already solved the problem in dimension 4; a solution in all dimensions, together with recent work of Donaldson, will give a complete topological characterization of those manifolds admitting symplectic structures: They are precisely those that admit Lefschetz pencils. (A key part of the problem is suitably to define Lefschetz pencils in this generality.) A related topic concerns Stein surfaces (affine analytic varieties of complex dimension 2). There are many fundamental existence questions here, for example, what 3-manifolds arise as boundaries of Stein surfaces, how many Stein surfaces have a given 3-manifold as boundary, and how many contact structures are induced on a given 3-manifold by such Stein surfaces? These questions are amenable to the principal investigator's current techniques of describing Stein surfaces as handlebodies on Legendrian link diagrams in connected sums of copies of S1 x S2. The principal investigator is also currently writing a graduate text on 4-manifolds and Kirby calculus with A. Stipsicz and is advising two PhD students and an NSF Postdoctoral Fellow. An n-manifold is a space that on small scales looks like Euclidean space of dimension n. For example, the space in which we live is a 3-manifold, and the universe (space-time) is a 4-manifold. As such, manifolds play a central role in mathematics, physics, and to some extent, various other sciences. Symplectic and contact structures first appeared in classical physics (Hamiltonian dynamics) but have since become important in pure mathematics (geometry, topology, analysis) as well. It is a basic unsolved problem to determine which manifolds can have s ymplectic or contact structures. This project aims for a complete answer in the symplectic case and for additional understanding of the contact case. The project should also shed light on basic unsolved problems regarding the possible shapes of 3- and 4-manifolds. ***