Degree-One Maps, Surgery, and Heegaard Genus of 3-Manifolds

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三种流形的例子。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。研究它们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更简单的三维部分。例如，所谓的Heegaard分裂就是这样的分解。人们还可以在三个流形上使用一种称为外科手术的技术，用不同的几何部分取代几何部分。在这个项目中，PI计划使用Heegaard分裂和外科手术来研究三个流形上的各种问题。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。研究的第一部分是研究关于Heegaard亏格和一次映射的一个长期存在的猜想。这一猜想可以转化为一项关于Heegaard属在一种特殊类型的手术下如何变化的研究，PI计划研究这种手术。PI还计划研究某些融合三流形的Heegaard亏格的几个相关问题。该项目的第二部分是探索一种证明Berge猜想的新方法。这种方法的灵感来自于PI的观察，即如果存在某个相交图案，则可以执行稳定化，然后执行非稳定化，将外部结点(在镜头空间中)的Heegaard分裂改变为经络原始分裂。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。

Three-manifolds are objects modeled on the three-dimensional space that we are living in. A doughnut and the spatial universe are both examples of three-manifolds. These objects arise naturally in many contexts in physical and other natural sciences and model many interesting phenomena. A geometric way of studying them is to cut a complicated three-manifold into simpler three-dimensional pieces along two-dimensional surfaces. For example, a so-called Heegaard splitting is such a decomposition. One can also use a technique known as surgery on a three-manifold that replaces a geometric piece with a different one. In this project, the PI plans to study various questions on three-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.In this project, the PI plans to study topology of three-manifolds. The first part of the research is to study a long-standing conjecture concerning Heegaard genus and degree-one map. The conjecture can be translated into a study of how Heegaard genus changes under a special type of surgeries and the PI plans to study such surgeries. The PI also plans to study several related questions on Heegaard genus of certain amalgamated three-manifolds. The second part of the project is to explore a new approach to proving the Berge Conjecture. This approach is inspired by the PI's observation that if there is a certain intersection pattern, then one can perform a stabilization and then an unstabilization, changing a Heegaard splitting of a knot exterior (in a lens space) into a meridionally primitive splitting. The PI plans to develop new tools and use techniques from his previous work to achieve these goals.