Noncommutative Algebraic Invariants in Low-Dimensional Topology

William Marsh Rice University, Houston TX

Investigators

Abstract

Noncommutative Algebraic Invariants in Low-Dimensional Topology This 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 f oundations 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.