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

三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。研究它们的一种几何方法是沿着二维表面将复杂的三流形切割成更简单的三维块。例如，所谓的Heegaard分裂就是这样一种分解。人们也可以使用一种被称为三歧管手术的技术，用不同的几何块替换一个几何块。在这个项目中，PI计划使用heegard分裂和手术来研究关于三流形的各种问题。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。在这个项目中，PI计划研究三流形的拓扑结构。研究的第一部分是研究一个长期存在的关于Heegaard属和1度图的猜想。这一猜想可以翻译为研究heeggaard属在特殊手术下的变化，PI计划研究这种手术。PI还计划研究某些合并三流形的Heegaard属的几个相关问题。该项目的第二部分是探索一种证明贝尔热猜想的新方法。这种方法的灵感来自PI的观察，如果有一个特定的相交模式，那么可以执行一个稳定，然后一个不稳定，改变一个Heegaard分裂的结外部（在镜头空间）成子午原始分裂。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.