Low Dimensional Topology and Hyperbolic Geometry

Princeton University, Princeton NJ

Investigators

Abstract

The principal investigator plans to investigate, in collaboration with others, several central issues in hyperbolic geometry and low dimensional topology. He will study the structure of low volume complete orientable hyperbolic 3-manifolds. He also plans to develop topological obstructions to geodesizing, via isotopy, simple curves in complete hyperbolic 3-manifolds and to investigate whether or not geometrically simply connected Schoenflies 4-balls are diffeomorphic to the standard 4-ball. Three-manifold topology is the study of objects which appear locally like the standard three dimensional space in which we live and two-manifold topology is the study of surfaces. The topology of two-dimensional manifolds, like the sphere and torus, was completely understood over 100 years ago. The geometry of two-dimensional manifolds is also very well understood. Despite the tremendous progress of the last 30 years some of the most basic issues of three-dimensional geometry and topology still must be understood. While we have experimental evidence predicting the low volume hyperbolic three-manifolds, we still do not even know the lowest volume closed ones. Given a hyperbolic three-manifold, say as a three-ball with face identifications, there is strikingly little understanding of how to pick out the shortest length paths. The last project is about understanding one of the most basic types of four-dimensional spaces.