Non-Commutative Algebraic Phenomena in the Topology of Three- and Four-dimensional Spaces

William Marsh Rice University, Houston TX

Investigators

Abstract

Abstract Award: DMS-0104275 Principal Investigator: Tim Cochran This project develops a new area of noncommutative algebraic topology and its applications to low-dimensional topology. The success of algebraic topology in knot theory, for example, has, until recently centered around abelian invariants, that is to say, invariants associated to the universal abelian covering space of the knot or link exterior. These invariants are the Alexander module, which is the first homology of this cover as a module over a commutative Laurent polynomial ring, and the Blanchfield pairing. These determine the S-equivalence class of the knot as well as various other invariants. From the perspective of the knot group G, the Alexander module is simply G'/G". Hence any behavior associated to G" will be invisible to these abelian invariants. We remedy this deficiency by studying the quotients of successive terms of the higher derived series, or, put another way, study modules associated to more general solvable covering spaces. These are modules over noncommutative rings and thus are difficult to work with. We use techniques from noncommutative algebra and C* algebras to define invariants. We find , for each integer n, an entire theory which parallels the Alexander module and Blanchfield form and signatures. There are applications to estimating genus, detecting fibered knots and 3-manifolds, new invariants of concordance and representations of mapping class groups. The advent of quantum mechanics led scientists to many paradoxical, but now accepted, facts about our universe. In particular, there came the realization that "commutative mathematics" was inadequate to describe our physical world. Recall that 2 times 3 equals 3 times 2 is the commutative law of multiplication of numbers. Quantum mechanics showed that physical quantities are not mere numbers but more like arrays or matrices of numbers. Since multiplication of matrices is not commutative, this explains and models noncommutative phenomena at the most fundamental levels of the physical world. This project studies the shape of 3 and 4-dimensional spaces by using new noncommutative mathematics arising from algebra.