Three Dimensional Manifold Topology

DMS-0204428 Xingru Zhang The study of compact irreducible 3-manifolds splits naturally into cases of finite fundamental groups and infinite fundamental groups. In the case of infinite fundamental groups, the virtual Haken conjecture of Waldhausen has been serving as a guiding open problem, because virtual Haken 3-manifolds possess similar nice properties as Haken 3-manifolds, such as topological rigidity, residually finite fundamental groups and geometric decomposition in Thurston's sense. Concerning 3-manifolds with finite fundamental groups, the Poincare conjecture is perhaps the most fundamental open problem. The well known Property-P conjecture may be considered as a special case of the Poincare conjecture. Xingru Zhang proposes to continue his investigation of the virtual Haken conjecture and the Property-P conjecture, along with some closely related problems, such as embedded or immersed essential surfaces in 3-manifolds, various exceptional Dehn surgeries on hyperbolic knots, and representations of 3-manifold groups. Three dimensional manifold topology, including the knot theory, has been one of the most active research areas in topology over the last twenty-five years. This is a rich, beautiful and challenging area where topology meshes up harmonically with algebra and geometry. For instance, if a compact 3-manifold without boundary admits a complete hyperbolic metric, then the topology, the fundamental group and the hyperbolic metric of the manifold mutually determine each other. In general, it is fundamental to know that to what extent the topology of a compact 3-manifold is determined by the fundamental group of the manifold, and that whether the interior of a compact 3-manifold admits one of the eight standard complete metrics under the condition that the manifold contains no essential 2-spheres or 2-tori. In this proposal the PI plans to continue his investigation in this direction.