Symplectic, Contact and Low-dimensional Topology

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AbstractAward: DMS-0102922Principal Investigator: Robert E. GompfThis project focuses on constructing symplectic, contact and4-dimensional manifolds. The Principal Investigator has defineda topological structure called a "hyperpencil" on aneven-dimensional closed manifold, generalizing a linear system ofcurves on a complex algebraic manifold. He has shown that anyhyperpencil canonically determines a symplectic structure on themanifold. In dimensions less than 8, this correspondence mapsthe set of hyperpencils onto a dense subset of all symplecticforms on the manifold (up to isotopy and scale). A similarstatement seems likely in higher dimensions, and would provide acomplete topological characterization of those manifoldsadmitting symplectic structures. If the fibers of thiscorrespondence can be topologically specified, the result will bea purely topological description of a dense subset of allsymplectic structures on all closed manifolds. In addition tothis investigation, Stein surfaces and their contact 3-manifoldboundaries will be studied by methods such as Legendrian Kirbycalculus. The topology of exotic R^4's and other smooth4-manifolds will also be investigated.An n-manifold is a space that in small regions looks just liken-dimensional Euclidean space. Points, curves and surfaces aremanifolds of dimensions 0,1 and 2, respectively. The space inwhich we live is a 3-manifold, and the universe (space-time) is a4-manifold. Surprisingly, 3- and 4-manifolds are much less wellunderstood than their higher-dimensional counterparts, althoughmajor progress has been made in recent years. Symplectic andcontact structures on manifolds were discovered through classicalphysics (Hamiltonian mechanics and optics, respectively), butthey are now seen to be important in such diverse areas asquantum physics, complex analysis, differential geometry andtopology. For example, a rigid pendulum swinging in3-dimensional space determines a symplectic manifold. The bobmoves on a spherical surface, so its position is specified by 2variables (latitude and longitude on this sphere). To completelyspecify the state of the system, one must also include themomentum of the bob. At any position, its momentum is tangent tothe sphere and thus specified by two more variables. Hence, theset of states of the system is a 4-manifold (locally specified by4 variables). This 4-manifold has a special symplectic structurewhose role is to link each position variable to the correspondingmomentum variable. This linkage ultimately results in Hamilton'sequations describing the motion of the pendulum under theinfluence of a specified force field. There are many othersituations in which manifolds and their symplectic and contactstructures arise. While our understanding of such structures hasadvanced enormously in recent years, some of the most basicquestions remain to be answered. Which manifolds have symplecticor contact structures, and how many such structures are there ona given manifold? How many 4-manifolds (if any) are theresatisfying a given description? Questions such as these form thebasis of this project.