Symplectic, Contact and Low-dimensional Topology

University Of Texas At Austin, Austin TX

Investigators

Abstract

Abstract Award: DMS-0102922 Principal Investigator: Robert E. Gompf This project focuses on constructing symplectic, contact and 4-dimensional manifolds. The Principal Investigator has defined a topological structure called a "hyperpencil" on an even-dimensional closed manifold, generalizing a linear system of curves on a complex algebraic manifold. He has shown that any hyperpencil canonically determines a symplectic structure on the manifold. In dimensions less than 8, this correspondence maps the set of hyperpencils onto a dense subset of all symplectic forms on the manifold (up to isotopy and scale). A similar statement seems likely in higher dimensions, and would provide a complete topological characterization of those manifolds admitting symplectic structures. If the fibers of this correspondence can be topologically specified, the result will be a purely topological description of a dense subset of all symplectic structures on all closed manifolds. In addition to this investigation, Stein surfaces and their contact 3-manifold boundaries will be studied by methods such as Legendrian Kirby calculus. The topology of exotic R^4's and other smooth 4-manifolds will also be investigated. An n-manifold is a space that in small regions looks just like n-dimensional Euclidean space. Points, curves and surfaces are manifolds of dimensions 0,1 and 2, respectively. The space in which we live is a 3-manifold, and the universe (space-time) is a 4-manifold. Surprisingly, 3- and 4-manifolds are much less well understood than their higher-dimensional counterparts, although major progress has been made in recent years. Symplectic and contact structures on manifolds were discovered through classical physics (Hamiltonian mechanics and optics, respectively), but they are now seen to be important in such diverse areas as quantum physics, complex analysis, differential geometry and topology. For example, a rigid pendulum swinging in 3-dimensional space determines a symplectic manifold. The bob moves on a spherical surface, so its position is specified by 2 variables (latitude and longitude on this sphere). To completely specify the state of the system, one must also include the momentum of the bob. At any position, its momentum is tangent to the sphere and thus specified by two more variables. Hence, the set of states of the system is a 4-manifold (locally specified by 4 variables). This 4-manifold has a special symplectic structure whose role is to link each position variable to the corresponding momentum variable. This linkage ultimately results in Hamilton's equations describing the motion of the pendulum under the influence of a specified force field. There are many other situations in which manifolds and their symplectic and contact structures arise. While our understanding of such structures has advanced enormously in recent years, some of the most basic questions remain to be answered. Which manifolds have symplectic or contact structures, and how many such structures are there on a given manifold? How many 4-manifolds (if any) are there satisfying a given description? Questions such as these form the basis of this project.