Noncommutative Algebraic Invariants in Low-Dimensional Topology

低维拓扑中的非交换代数不变量本课题将发现低维流形中某些高度非交换代数不变量的拓扑意义。如果X是拓扑空间，G是其基群，则与G的任何正规子群H相关联的是X的一个覆盖空间，其同调群是商群G/H的整群环上的模。当G/H可交换时，这些模块在代数拓扑应用于拓扑问题中发挥了核心作用。例如，如果X是S3中环环L的外环，H是交换子群，则这些模称为L的Alexander模。本课题研究了这些“高阶”模在更一般的情况下，特别是当子群H是g的派生级数的一个元素时，从而得到了推广Alexander模的模族。虽然它们是非交换环上的模，但它们与Alexander模具有许多重要的特性。如果X是流形，那么在这些模块上还定义了厄米形式和连接形式，给出了额外的结构。本项目研究这些结构及其应用。利用非交换代数和泛函分析技术，在结论、三维流形、四维流形和曲面同胚中发现了新的非交换现象。特别是该项目将发现更多的结构在组的拓扑一致性类的结；在所有同位素类结的单群中找到构造；将研究这类单模的某些涉及“gropes”的等价关系；会发现关于3流形叶化深度的新信息，会发现3流形的新不变量，以及映射类群。随着量子力学的出现，20世纪后期的科学家们越来越意识到，描述宇宙的结构将需要非交换数学。在数字乘法中，2乘以3 = 3乘以2。但现在已知粒子的行为更像矩阵，而矩阵乘法是不可交换的，即AB一般不等于BA。然而，直到最近，即使在数学本身的领域，交换代数和线性技术也发挥了更大的作用，仅仅是因为非交换代数非常困难。为了理解四维时空、三维空间和弦理论的精细结构，有必要了解非交换代数结构的全部作用。本项目为非交换代数拓扑在三维和四维流形研究和结理论中的应用奠定了数学基础。此外，由于PI的研究助理中有很高比例是美国女性，并且由于女性在研究数学领域的代表性不足，因此该项目将有助于提高美国的科学潜力。

Noncommutative Algebraic Invariants in Low-Dimensional TopologyThis project will discover the topological significance of certain highly noncommutative algebraic invariants of low-dimensional manifolds. If X is a topological space and G is its fundamental group, then associated to any normal subgroup H of G is a covering space of X whose homology groups are modules over the integral group ring of the quotient group G/H. When G/H is commutative, these modules have played a central role in the applications of algebraic topology to the problems of topology. For example, if X is the exterior of a link L of circles in S3 and H is the commutator subgroup, then these modules are called Alexander modules of L. This project investigates these "higher-order" modules in more general situations, especially where the subgroup H is an element of the derived series of G. Families of modules that generalize the Alexander module are thus obtained. Although these are modules over noncommutative rings, they share many important properties with the Alexander module. If X is a manifold then there are also Hermitian forms and linking forms defined on these modules, giving additional structure. This project investigates these structures and their applications. With the help of techniques of noncommutative algebra and functional analysis, one observes new noncommutative phenomena in knot theory, 3-dimensional manifolds, 4-dimensional manifolds and in surface homeomorphisms. In particular the project will find more structure in the group of topological concordance classes of knots; find structure in the monoid of all isotopy classes of knots; will investigate this monoid modulo certain equivalence relations involving "gropes"; will find new information about the depth of foliations of 3-manifolds, and will find new invariants of 3-manifolds, and mapping class groups.With the advent of quantum mechanics, scientists in the late twentieth century have become increasingly aware that describing the structure of the universe will necessitate noncommutative mathematics. In multiplying numbers, 2 times 3 = 3 times 2. But particles are now known to behave more like matrices, and matrix multiplication is not commutative, i.e. AB is not in general equal to BA. Yet, until recently, even in the field of mathematics itself commutative algebra and linear techniques have played the greater role, simply because noncommutative algebra is very difficult. To understand the finer structure of 4-dimensional space-time, of 3-dimensional space and of string theory, it will be necessary to understand the full role of noncommutative algebraic structures. This project lays the mathematical foundations for the use of noncommutative algebraic topology in the study of 3 and 4-dimensional manifolds and in knot theory. In addition, since a high percentage of the research assistants of the PI are U.S. women, and since women are under-represented in the field of research mathematics, this project will contribute to the increase in the scientific potential of the United States.