Heegaard splitting and surgery of 3-manifolds

PI计划使用融合、外科、分枝曲面和分层的方法来研究三维流形的拓扑，特别是三维流形的Heegaard分裂。该项目的第一个目标是构建一个与Rank猜想相反的双曲反例。研究的第二部分是利用分枝曲面和分层来研究可约三维流形中纽结和链环上的Dehn手术。其中一个目标是证明布线猜想的主要部分。项目的第三部分是研究Heegaard拆分的各个方面。这项拟议的研究与其他研究人员的几个活跃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的工具来实现这些目标。三重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。这些物体在物理和其他自然科学中的许多背景下自然产生，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将一个复杂的三维流形沿着二维曲面切割成一组更简单的三维碎片。例如，Heegaard分裂就是这样的分解。利用这个想法，人们还可以通过对众所周知的流形进行手术来构建有用的新的三维流形。PI计划使用Heegaard分裂和外科手术来研究3-流形。这项研究针对低维拓扑和纽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，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.