Dehn Surgery and Related Topics in 3-Dimensional Topology

PI将研究三维拓扑中的各种问题。将讨论的具体主题是：(1)特殊的Dehn填充，(2)桥数，Heegaard亏格与Dehn手术，以及(3)L-空间与左序。(1)的背景是希望对所有具有两个非双曲Dehn填充的双曲三维流形进行分类；目前的重点将放在其中一个填充是Seifert纤维且填充之间的距离至少为5的情况。项目(2)将考虑通过Dehn运算对三个球面上的纽结进行Dehn运算得到的流形，目的是找到流形的Heegaard亏格和对偶纽结的桥数的上界。(3)的重点是猜想：素有理同调三维球面是L空间当且仅当其基本群不是左可序的。三维拓扑学是研究三维流形的大规模结构，抽象空间局部类似于我们生活的普通空间。由于Perelman在大约十年前证明了几何猜想，我们现在对这类空间的性质有了一个非常好的了解。然而，三维拓扑学是极其丰富的，有许多不同类型的数学发挥作用，拟议的研究将寻求加深我们对该学科的各个方面的理解。具体地说，(1)将研究三维流形的几何何时因附加了某个简单空间而退化，(2)是探索三维流形与纽结之间基本关系的一部分，以及(3)将研究三维拓扑的两个非常不同的方面之间似乎存在的神秘联系，一个是解析的，另一个是代数的。

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.