Classical Knot Concordance and Problems in Four-Manifold Theory

经典结索引和四流形理论中的问题

• 批准号: 0406934
• 负责人: Charles Livingston
• 金额: --
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0406934&HistoricalAwards=false
• 关键词: Classical; Knot; Concordance; Problems; Four

Two knots are defined to be concordant if the connected sum of one with the mirror of the other is slice: that is, if the connected sum bounds an embedded disk in the four-ball. The set of equivalence classes forms what is called the concordance group of knots, first studied in the early 1960s. It is known that this group is countable, abelian, and that it maps onto Levine's algebraic concordance group, which is isomorphic to an infinite direct sum of cyclic groups of infinite order and of order 2 and 4. The kernel of this map is infinitely generated. A main goal of the proposal is the continued study of the concordance group and relationships between three-dimensional properties of knots, such as symmetry and genus, and their properties in the concordance group. Specific topics include understanding the algebraic structure of the kernel of Levine's map (in particular the part of the kernel generated by knots of Alexander polynomial one), torsion in the concordance group, and the splitting of Levine's homomorphism. To pursue this study the principal investigator will develop and apply methods coming from the theory of Heegaard Floer homology as defined by Oszvath and Szabo. Other topics of study under the proposal relate to the interplay between the fundamental groups of 4-manifolds and their topological structure: for instance, a continued investigation of the Hausmann-Weinberger invariant of a group, the minimal Euler characteristic of a closed 4-manifold having that group as its fundamental group, and its generalizations, will be pursued.The goal of this proposal is the study of knots. From a three-dimensional perspective, questions about knots have been studied for over a hundred years. More recently it has been recognized that the structure of a four-dimensional space, a four-manifold, is connected to properties of classical knots that can be associated to that four-manifold. A main focus of this proposal is the study of those properties of knots that are of relevant to four-dimensional topology; many of these properties are encapsulated in something called the concordance group of knots. Questions regarding this concordance group offer test cases and motivation for the further development of techniques from general four-manifold theory. In addition, the study of concordance brings to the fore new and fascinating questions in the realm of classical three-dimensional knot theory, relating for instance to questions of knot symmetry and genus.

如果一个结点与另一个结点的镜像的连通和为切片，则两个结点被定义为一致结点：即，如果连通和限定了四球中嵌入的磁盘。等价类的集合形成了所谓的结的和谐群，在20世纪60年代初首次被研究。已知这个群是可数的、阿贝的，并且映射到Levine的代数谐和群上，而Levine的代数谐和群同构于无限阶循环群和2、4阶循环群的无限直和。这个映射的核是无限生成的。该提案的主要目标是继续研究和谐群和结的三维性质之间的关系，如对称和属，以及它们在和谐群中的性质。具体的主题包括理解Levine映射核的代数结构（特别是由Alexander多项式1的结点生成的核的部分），调和群中的扭转，以及Levine同态的分裂。为了开展这项研究，首席研究员将开发和应用来自Oszvath和Szabo定义的Heegaard flower同源理论的方法。其他研究课题涉及到4流形的基本群与其拓扑结构之间的相互作用：例如，继续研究群的Hausmann-Weinberger不变量，以该群为其基本群的封闭4流形的最小欧拉特征及其推广，将被追求。本提案的目的是研究结。从三维角度来看，关于结的问题已经研究了一百多年。最近人们认识到，一个四维空间的结构，一个四流形，是与经典结点的性质相联系的，这些性质可以与这个四流形相联系。本提案的主要焦点是研究与四维拓扑相关的结的性质；这些性质中的许多都被封装在一种叫做“结的和谐群”的东西中。关于这个一致性组的问题为从一般四流形理论进一步发展技术提供了测试用例和动力。此外，一致性的研究带来了新的和迷人的问题，在经典三维结理论的领域，有关的问题，例如，结的对称性和属。