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