Connections between Homology Theories for Knots and Three-Manifolds

结和三流形的同调理论之间的联系

• 批准号: 1111680
• 负责人: Stephan Martin Wehrli
• 金额: $ 11.02万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1111680&HistoricalAwards=false
• 关键词: Connections; between; Homology; Theories; Knots

Khovanov homology and Heegaard Floer homology are powerful invariants for knots and three-manifolds, which were discovered around the year 2000, and which have since stirred a tremendous amount of research activity. In particular, both Khovanov homology and Heegaard Floer homology have been used to give new proofs of the topological Milnor conjecture and of the existence of exotic smooth structures on the open four-ball. Previously, such results had only been accessible via gauge theory. While Khovanov homology is defined combinatorially, via a construction which is motivated by the representation theory of quantum groups, Heegaard Floer homology is defined analytically, through moduli spaces of solutions of differential equations. Around 2008, Heegaard Floer homology was extended to an invariant for three-manifolds with non-empty boundary, called "bordered Floer homology", which can be used to compute Heegaard Floer homology combinatorially whenever a decomposition of a three-manifold into suitable smaller pieces is given. A main goal of this project is to study the relationship between bordered Floer homology and Khovanov homology for tangles. Comparing these two theories will expectedly shed more light on the geometric content of Khovanov homology and thus make Khovanov homology more suited to applications. Moreover, the envisioned relationship between bordered Floer homology and Khovanov homology will provide an example of a perhaps more general connection between symplectic geometry and representation theory. Other goals of this project are to develop new homology theories for contact three-manifolds, and to analyze the properties of Khovanov homology groups of n-cables of knots.Mathematicians have long been interested in classifying topological spaces that are locally three-dimensional (like our physical universe) or locally four-dimensional (like four-dimensional space-time). Related to the problem of classifying such spaces is the problem of classifying knotted loops embedded in a given three-dimensional space. Over the past two decades, mathematicians have used ideas coming from several different areas of mathematics and mathematical physics (in particular from symplectic geometry, quantum field theory, string theory, and loop quantum gravity) to develop powerful new tools for classifying knots and low-dimensional spaces. The most notable ones among these new tools are Khovanov homology and Heegaard Floer homology. This proposal aims to investigate the connections between certain generalizations of Khovanov homology and Heegaard Floer homology, and to use these connections to study knot theoretical problems. Mathematical knot theory has applications in biomedical research, where it is used to study the processes that are responsible for unraveling DNA strands during cell division. Thus, this project is important not only from a theoretical perspective, but also for its potential applications to biomedical sciences.

Khovanov 同调和 Heegaard Floer 同调是纽结和三流形的强大不变量，它们于 2000 年左右被发现，自此引发了大量的研究活动。特别是，Khovanov 同调和 Heegaard Floer 同调都被用来给出拓扑 Milnor 猜想和开放四球上奇异光滑结构存在的新证明。此前，此类结果只能通过规范理论获得。霍瓦诺夫同调是通过量子群表示论所激发的构造以组合方式定义的，而赫加德弗洛尔同调则是通过微分方程解的模空间以分析方式定义的。 2008年左右，Heegaard Floer同调被扩展为具有非空边界的三流形的不变量，称为“有边界Floer同调”，每当将三流形分解为合适的较小块时，它可以组合地计算Heegaard Floer同调被给出。该项目的主要目标是研究缠结的有界 Floer 同源性和 Khovanov 同源性之间的关系。比较这两种理论有望更好地阐明霍瓦诺夫同调的几何内容，从而使霍瓦诺夫同调更适合应用。此外，有界弗洛尔同调和霍万诺夫同调之间的设想关系将提供辛几何和表示论之间也许更普遍的联系的一个例子。该项目的其他目标是开发新的接触三流形同调理论，并分析 n 线结的霍瓦诺夫同调群的性质。数学家长期以来对局部三维拓扑空间的分类感兴趣（例如我们的物理宇宙）或局部四维（如四维时空）。与对此类空间进行分类的问题相关的是对嵌入给定三维空间中的打结环进行分类的问题。在过去的二十年里，数学家们利用来自数学和数学物理几个不同领域的思想（特别是辛几何、量子场论、弦理论和圈量子引力）来开发强大的新工具来对结和低能粒子进行分类。维度空间。这些新工具中最引人注目的是 Khovanov 同源性和 Heegaard Floer 同源性。该提案旨在研究 Khovanov 同调和 Heegaard Floer 同调的某些概括之间的联系，并利用这些联系来研究纽结理论问题。数学结理论在生物医学研究中得到应用，用于研究细胞分裂过程中 DNA 链解开的过程。因此，该项目不仅从理论角度来看很重要，而且在生物医学科学中的潜在应用也很重要。