Classical Knot Concordance and Problems in Four-Manifold Theory

两个纽结被定义为协调的，如果一个纽结与另一个纽结的镜像的连通和是切片的：也就是说，如果连通和限定了四个球中嵌入的圆盘。等价类的集合形成了所谓的纽结和谐群，最早研究于20世纪60年代初。我们知道这个群是可数的，交换的，并且它映射到Levine代数协调群上，这个群同构于无穷阶2阶和4阶循环群的无穷直和。这个映射的核是无限生成的。该提案的一个主要目标是继续研究和谐组以及纽结的三维性质(如对称性和亏格)与和谐组中它们的性质之间的关系。具体主题包括了解Levine映射的核的代数结构(特别是由Alexander多项式一的纽结生成的核的一部分)，协调群中的挠率，以及Levine同态的分裂。为了进行这项研究，首席研究员将开发和应用来自于Oszvath和Szabo定义的Heegaard Floer同调理论的方法。在该方案下的其他研究主题涉及4-流形的基本群与它们的拓扑结构之间的相互作用：例如，将继续研究群的Hausmann-Weinberger不变量、以该群为基本群的闭4-流形的最小Euler特征及其推广。从三维的角度来看，关于结的问题已经研究了一百多年。最近，人们认识到，四维空间的结构，即四维流形，与可以与该四维流形相关联的经典纽结的性质有关。这一建议的一个主要焦点是研究纽结的那些与四维拓扑相关的性质；这些性质中的许多被封装在称为纽结的和谐群的东西中。关于这个调和组的问题提供了测试案例和从一般四流形理论中进一步发展技术的动机。此外，对和谐性的研究在经典的三维纽结理论领域带来了新的和迷人的问题，例如与纽结对称性和亏格有关的问题。

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.