Heegaard splitting and surgery of 3-manifolds

PI计划使用合并、手术、分支表面和层压等方法研究3-流形的拓扑结构，特别是3-流形的均匀分裂。该项目的第一个目标是为秩猜想构造一个双曲反例。本研究的第二部分是利用分支表面和薄片研究可约3流形中结点和连杆的Dehn手术。一个目标是证明布线猜想的主要部分。项目的第三部分是对Heegaard分裂的各个方面进行研究。在拟议的研究和其他研究人员正在进行的几个项目之间有许多有趣的联系。PI计划探索这些联系并开发新工具来实现这些目标。三维流形是以我们生活的三维空间为模型的物体。甜甜圈和空间宇宙都是3流形的例子。这些对象在物理和其他自然科学的许多情况下自然出现，并模拟了许多有趣的现象。研究三维流形的一种几何方法是沿着二维曲面将复杂的三维流形切割成简单的三维碎片的集合。例如，heegard分裂就是这样的分解。利用这一思想，我们还可以通过对熟知的3-流形进行手术来构造有用的新3-流形。PI计划使用heegard分裂和手术来研究3-流形。这项研究的目标是低维拓扑和结理论中的几个核心问题，这对其他科学研究领域有潜在的影响，比如DNA的拓扑结构。

The PI plans to study the topology of 3-manifolds, especially Heegaard splitting of 3-manifolds, using methods of amalgamation, surgery, branched surface and lamination. The first goal of the project is to construct a hyperbolic counterexample to the Rank Conjecture. The second part of the proposed research is to use branched surfaces and laminations to study Dehn surgery on knots and links in reducible 3-manifolds. One goal is to prove a major part of the Cabling Conjecture. The third part of the project is to study various aspects of Heegaard splitting. There are many interesting connections between the proposed research and several active programs of other researchers. The PI plans to explore these connections and develop new tools to achieve these goals.Three-manifolds are objects modeled on the 3-dimensional space that we are living in. A donut and the spatial universe are both examples of 3-manifolds. These objects arise naturally in many contexts in physical and other natural sciences and model many interesting phenomena. A geometric way of studying 3-manifolds is to cut a complicated 3-manifold into a collection of simpler 3-dimensional pieces along 2-dimensional surfaces. For example a Heegaard splitting is such a decomposition. Using this idea, one can also construct useful new 3-manifolds by doing surgery on well-understood ones. The PI plans to study 3-manifolds using Heegaard splitting and surgery. The research targets several central questions in low-dimensional topology and knot theory, which has potential impact on other areas of scientific investigations, such as the topological structures of DNA.