Noncommutative Algebraic Invariants in Low-Dimensional Topology

低维拓扑中的非交换代数不变量这个项目将发现低维流形中某些高度非交换的代数不变量的拓扑意义。如果X是一个拓扑空间，G是它的基本群，则与G的任一正规子群H相联系的是X的一个覆盖空间，它的同调群是商群G/H的积分群环上的模。当G/H是可交换的时，这些模在代数拓扑学应用于拓扑学问题中起到了中心作用。例如，如果X是S_3中圆的环的外环L，H是换位子群，则这些模称为L的Alexander模。本课题研究了更一般情况下的这些“高阶”模，特别是当子群H是G的派生系列的元素时，由此得到了推广Alexander模的模族。虽然这些都是非交换环上的模，但它们与Alexander模有许多重要的共同性质。如果X是流形，那么在这些模上还定义了厄米特形式和链接形式，从而给出了额外的结构。本项目研究这些结构及其应用。借助于非对易代数和泛函分析的技巧，我们观察到纽结理论、三维流形、四维流形和曲面同胚中的新的非对易现象。特别是，该项目将在纽结的拓扑协调类的群中发现更多的结构；在所有同构类的纽结的么半群中找到结构；将研究这个么半群，并将研究涉及到“摸索”的某些等价关系；将找到关于三维流形的叶的深度的新信息，并将找到新的三维流形的不变量，以及映射类群。随着量子力学的到来，二十世纪末的科学家们越来越意识到，描述宇宙的结构将需要非对易数学。在乘法中，2乘3=3乘2。但现在已知粒子的行为更像矩阵，并且矩阵乘法是不可交换的，即AB通常不等于BA。然而，直到最近，即使在数学本身的领域，交换代数和线性技巧也发挥了更大的作用，原因很简单，因为非交换代数非常困难。为了理解4维时空、3维空间和弦理论的更精细的结构，有必要理解非对易代数结构的全部作用。该项目为非对易代数拓扑学在研究三维和四维流形以及纽结理论中的应用奠定了数学基础。此外，由于国际和平研究所的研究助理中有很高比例是美国妇女，而且妇女在研究数学领域的代表性不足，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.