Dehn Surgery and Related Topics in 3-Dimensional Topology

Dehn 手术和 3 维拓扑中的相关主题

• 批准号: 1309021
• 负责人: Cameron Gordon
• 金额: $ 14.52万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309021&HistoricalAwards=false
• 关键词: Dehn; Surgery; Related; Topics; Dimensional

The PI will investigate various problems in three-dimensional topology. Specific topics that will be addressed are: (1) Exceptional Dehn Filling, (2) Bridge Number, Heegaard Genus and Dehn Surgery, and (3) L-Spaces and Left-Orderability. The context of (1) is the desire to classify all hyperbolic three-manifolds with two non-hyperbolic Dehn fillings; the present emphasis will be on the case where one of the fillings is Seifert fibered and the distance between the fillings is at least 5. Project (2) will consider manifolds obtained by Dehn surgery on knots in the three-sphere, the aim being to find upper bounds on the Heegaard genus of the manifold and the bridge number of the dual knot. The focus of (3) is the conjecture that a prime rational homology three-sphere is an L-space if and only of its fundamental group is not left-orderable.Three-dimensional topology is the study of the large-scale structure of three-manifolds, abstract spaces that are locally like the ordinary space in which we live. As a result of Perelman's proof of the Geometrization Conjecture about ten years ago, we now have a very good picture of the qualitative nature of such spaces. Nevertheless, three-dimensional topology is extremely rich, with many different kinds of mathematics playing a role, and the proposed research will seek to further our understanding of various aspects of the subject. Specifically, (1) will study when the geometry of a three-manifold degenerates on having a certain simple space attached to it, (2) is part of the exploration of the fundamental relationship between three-manifolds and knots, and (3) will investigate a mysterious connection that appears to exist between two very different aspects of three-dimensional topology, one analytical, the other algebraic.

PI将研究三维拓扑中的各种问题。将涉及的具体主题是：（1）异常Dehn填充，（2）桥数，Heegaard属和Dehn手术，（3）L-空间和左可序性。（1）的背景是希望对所有具有两个非双曲Dehn填充的双曲三流形进行分类;目前的重点将放在其中一个填充是塞弗特纤维且填充之间的距离至少为5的情况上。项目（2）将考虑通过Dehn手术在三球中的结上获得的流形，目的是找到流形的Heegaard亏格和对偶结的桥数的上界。（3）的重点是猜想一个素有理同调三球面是一个L-空间当且仅当它的基本群不是左序的。三维拓扑学是研究三维流形的大尺度结构的，三维流形是局部类似于我们所生活的普通空间的抽象空间。由于佩雷尔曼的证明的几何化猜想约十年前，我们现在有一个很好的图片定性性质的空间。尽管如此，三维拓扑学是非常丰富的，许多不同种类的数学发挥了作用，拟议的研究将寻求进一步我们对该主题的各个方面的理解。具体地说，（1）将研究当一个三维流形的几何在有一个特定的简单空间连接到它时退化，（2）是探索三维流形和结点之间的基本关系的一部分，（3）将研究一种神秘的联系，这种联系似乎存在于三维拓扑的两个非常不同的方面之间，一个是解析的，另一个是代数的。