Noncommutative Low-Dimensional Topology

非交换低维拓扑

• 批准号: 0805867
• 负责人: Constance Leidy
• 金额: $ 12.03万
• 依托单位: Wesleyan University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805867&HistoricalAwards=false
• 关键词: Noncommutative; Low; Dimensional; Topology

A fundamental goal in low-dimensional topology is the classification of manifolds. The fundamental group of the manifold is an important algebraic invariant, but is rarely abelian. On the other hand, since the homology groups of the manifold are abelian, they can be classified easily.Unfortunately, a lot of the rich structure of the fundamental group is lost when considering the homology groups. Homology with local coefficients associates modules to the manifold. Since modules are abelian groups, distinguishing them is still tenable. Furthermore, since these are often modules over noncommutative rings, some of the rich structure of the fundamental group is retained. These homology modules, as well as linking forms defined on them, have been used to define algebraic invariants of knot complements, 3-manifolds, and algebraic curve complements. These have led to new results about knot concordance, obstructions to the existence of symplectic structures on the product of a 3-manifold with the circle, and restrictions on which groups can be realized as the fundamental group of an algebraic curve complement. The goal of this project is to use these noncommutative techniques to find answers to questions related to three particular areas of low-dimensional topology: knot Floer homology, the topology of algebraic curve complements, and the structure of the knot concordance group.The goal of this project is to use non-commutative algebra to better understand knotted curves and surfaces. The study of knotted curves and surfaces has important applications in biology and physics. For example, DNA strands are naturally knotted, but must unknot in order to replicate. Also understanding knotted surfaces is fundamental to understanding the shape of the universe. Many mathematicians have used algebra to better understand how curves and surfaces can knot, however the type of algebra considered is usually commutative. By employing non-commutative algebras, we can get a more refined understanding for how curves and surfaces can knot.

低维拓扑的一个基本目标是流形的分类。流形的基本群是一个重要的代数不变量，但很少是阿贝尔的。另一方面，由于流形的同调群是交换群，因此可以很容易地对它们进行分类。不幸的是，在考虑同调群时，基本群的许多丰富结构都丢失了。与局部系数的同源将模块与流形相关联。由于模块是阿贝尔群，区分它们仍然是站得住脚的。此外，由于这些通常是非交换环上的模，因此保留了基本群的一些丰富结构。这些同源模块以及在它们上定义的连接形式已用于定义结补、3-流形和代数曲线补的代数不变量。这些导致了关于结一致性、对 3-流形与圆的乘积上辛结构存在性的阻碍以及对哪些群可以实现为代数曲线补的基本群的限制的新结果。该项目的目标是使用这些非交换技术来寻找与低维拓扑的三个特定领域相关的问题的答案：结弗洛尔同调、代数曲线补的拓扑以及结协调群的结构。该项目的目标是使用非交换代数更好地理解结曲线和曲面。打结曲线和曲面的研究在生物学和物理学中具有重要的应用。例如，DNA 链自然打结，但必须解开才能复制。此外，理解打结表面对于理解宇宙的形状至关重要。许多数学家使用代数来更好地理解曲线和曲面如何结，但是所考虑的代数类型通常是可交换的。通过使用非交换代数，我们可以更精确地理解曲线和曲面如何结。