Heegaard splitting of 3-manifolds

PI计划研究3-流形的拓扑结构，特别是3-流形的分裂。项目的第一部分是研究3流形的Heegaard格与其基群的秩之间的关系。这项研究是建立在PI最近成功构建秩对属猜想的双曲反例的基础上的。提出的研究的第二个目标是回答一个长期存在的问题，在3流形拓扑有关Heegaard属和一次映射。这个问题问的是是否可能有一个从一个3流形到另一个具有更大Heegaard属的3流形的1度映射。这个问题与庞加莱猜想以及3流形拓扑中的其他几个重要问题有关。项目的第三部分是解决Morimoto和Moriah关于3球隧道结点数的猜想。PI计划开发新的工具并使用他以前工作中的技术来实现这些目标。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一对简单的三维块，称为手体。这种分解叫做Heegaard分裂。PI计划使用heegard分裂来研究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 topology of 3-manifolds, especially Heegaard splittings of 3-manifolds. The first part of the project is to investigate the relation between the Heegaard genus of a 3-manifold and the rank of its fundamental group. This research is built on the PI's recent success in constructing a hyperbolic counterexample to the Rank versus Genus Conjecture. The second goal of the proposed research is to answer a long-standing question in 3-manifold topology concerning Heegaard genus and a degree-one map. This question asks whether it is possible to have a degree-one map from a 3-manifold to another 3-manifold with larger Heegaard genus. This question is related to the Poincare Conjecture as well as a few other important questions in 3-manifold topology. The third part of the project is to solve a conjecture of Morimoto and Moriah on tunnel number of knots in the 3-sphere. The PI plans to develop new tools and use techniques from his previous work 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 pair of simpler 3-dimensional pieces called handebodies along a 2-dimensional surface. This decomposition is called a Heegaard splitting. The PI plans to study 3-manifolds using Heegaard splittings. 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.