Heegaard splitting of 3-manifolds

PI计划研究3-流形的拓扑，特别是3-流形的Heegaard分裂。项目的第一部分是研究3-流形的Heegaard亏格与其基本群的秩间的关系。这项研究是建立在PI最近成功地构建了Rank对Genus猜想的双曲反例的基础上的。研究的第二个目的是回答三维流形拓扑中一个长期存在的关于Heegaard亏格和一次映射的问题。这个问题是问是否有可能有一个从一个三维流形到另一个三维流形的具有较大Heegaard亏格的一次映射。这个问题与Poincare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-球面上的隧道节数的一个猜想。这位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。三重流形是以我们所居住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沿着二维表面将复杂的三维流形切割成两个更简单的三维部件，称为手形。这种分解称为Heegaard分裂。PI计划使用Heegaard分裂来研究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 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.