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)例外Dehn填充，(2)桥数，Heegaard属和Dehn手术，以及(3)l -空间和左序性。(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种填充物是塞弗特纤维的情况下，并且填充物之间的距离至少为5。项目(2)将考虑由Dehn手术在三球结上得到的流形，目的是找到流形Heegaard格的上界和对偶结的桥数。(3)的重点是关于素有理同调三球是l空间的假设，当且仅当其基群中的一个不是左序的。三维拓扑学研究的是三维流形的大尺度结构，这些抽象空间在局部与我们生活的普通空间相似。由于佩雷尔曼在大约十年前证明了几何化猜想，我们现在对这些空间的定性性质有了一个很好的认识。然而，三维拓扑学是非常丰富的，有许多不同种类的数学发挥作用，提出的研究将寻求进一步我们对这个主题的各个方面的理解。具体来说，(1)将研究三流形的几何在具有特定的简单空间时何时退化，(2)是探索三流形和结之间基本关系的一部分，(3)将研究三维拓扑的两个非常不同的方面之间存在的神秘联系，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.