Topology of 3-Manifolds

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Proposal: DMS-0071852PI: David GabaiAbstract:During the last 100 years there has been tremendous interest inunderstanding the topology and geometry of 3-dimensional manifolds. Inpart, because a 3-manifold is (roughly speaking) a mathematical objectmodeled on our familiar 3-dimensional environment. A 3-manifold has theproperty that sufficiently small subsets can be parametrized by 3coordinates.The problem of characterizing the geometry and topology of surfaces, i.e.,manifolds of dimension 2, was completely answered nearly a century ago. Itwas determined that the only closed orientable connected surfaces are thesphere, torus, and the multi-holed tori. Furthermore, such surfaces canbe given geometries of constant curvature, +1 for the sphere, 0 for thetorus, and -1 for the other surfaces.Although great progress has been made during the last century, thesituation for dimension-3 is not nearly so well understood. Nevertheless aconjectural picture for the structure of 3-manifolds emerged almost 25years ago (by Thurston) and there has been much theoretical andexperimental evidence supporting his conjecture. The hyperbolic3-manifolds, i.e. the manifolds of constant -1 curvature, play a centralrole in this picture. In this proposal the PI in collaboration withvarious other mathematicians, plan to investigate the structure ofhyperbolic manifolds and the structure of manifolds which are conjecturallyhyperbolic. In particular they will study the structure of thediffeomorphism group of hyperbolic 3-manifolds, and the structure ofhyperbolic 3-manifolds of low volume. They will also address whether or nota closed aspherical, atoroidal, 3-manifold has a Gromov negatively curvedfundamental group. A positive resolution would imply that a 3-manifoldwhich is conjecturally hyperbolic, has at least the coarse algebraicstructure of a hyperbolic 3-manifold.