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，它们是在2000年左右发现的，此后引起了大量的研究活动。特别地，Khovanov同调和Heegaard Floer同调都被用来证明拓扑Milnor猜想和开四球上奇异光滑结构的存在性。以前，这样的结果只能通过规范理论获得。虽然Khovanov同调是通过量子群的表示论所激发的一种构造来组合定义的，但Heegaard Floer同调是通过微分方程解的模空间来解析定义的。在2008年左右，Heegaard Floer同调被推广到一个非空边界的三流形的不变量，称为“有边Floer同调”，它可以用来组合计算Heegaard Floer同调，只要给出一个三流形分解成合适的较小块。本项目的主要目标是研究缠结的加边Floer同调和Khovanov同调之间的关系。比较这两种理论，可以更清楚地了解Khovanov同调的几何内容，从而使Khovanov同调更适合于实际应用。此外，加边弗洛尔同调和霍瓦诺夫同调之间的关系将为辛几何和表示论之间可能更普遍的联系提供一个例子。该项目的其他目标是发展新的接触三维流形的同调理论，并分析n-cables的knot的Khovanov同调群的性质。数学家长期以来一直对分类局部三维（如我们的物理宇宙）或局部四维（如四维时空）的拓扑空间感兴趣。与这种空间的分类问题相关的是对嵌入在给定三维空间中的打结环进行分类的问题。在过去的20年里，数学家们利用来自数学和数学物理学几个不同领域（特别是辛几何、量子场论、弦理论和圈量子引力）的思想，开发出了对纽结和低维空间进行分类的强大新工具。在这些新工具中最值得注意的是Khovanov同源性和Heegaard Floer同源性。这个建议的目的是研究Khovanov同调和Heegaard Floer同调的某些推广之间的联系，并利用这些联系来研究纽结理论问题。数学纽结理论在生物医学研究中有应用，它被用来研究在细胞分裂过程中负责解开DNA链的过程。因此，该项目不仅从理论角度来看很重要，而且其在生物医学科学中的潜在应用也很重要。