Topology of 3-Manifolds

Princeton University, Princeton NJ

Investigators

Abstract

Proposal: DMS-0071852 PI: David Gabai Abstract: During the last 100 years there has been tremendous interest in understanding the topology and geometry of 3-dimensional manifolds. In part, because a 3-manifold is (roughly speaking) a mathematical object modeled on our familiar 3-dimensional environment. A 3-manifold has the property that sufficiently small subsets can be parametrized by 3 coordinates. The problem of characterizing the geometry and topology of surfaces, i.e., manifolds of dimension 2, was completely answered nearly a century ago. It was determined that the only closed orientable connected surfaces are the sphere, torus, and the multi-holed tori. Furthermore, such surfaces can be given geometries of constant curvature, +1 for the sphere, 0 for the torus, and -1 for the other surfaces. Although great progress has been made during the last century, the situation for dimension-3 is not nearly so well understood. Nevertheless a conjectural picture for the structure of 3-manifolds emerged almost 25 years ago (by Thurston) and there has been much theoretical and experimental evidence supporting his conjecture. The hyperbolic 3-manifolds, i.e. the manifolds of constant -1 curvature, play a central role in this picture. In this proposal the PI in collaboration with various other mathematicians, plan to investigate the structure of hyperbolic manifolds and the structure of manifolds which are conjecturally hyperbolic. In particular they will study the structure of the diffeomorphism group of hyperbolic 3-manifolds, and the structure of hyperbolic 3-manifolds of low volume. They will also address whether or not a closed aspherical, atoroidal, 3-manifold has a Gromov negatively curved fundamental group. A positive resolution would imply that a 3-manifold which is conjecturally hyperbolic, has at least the coarse algebraic structure of a hyperbolic 3-manifold.