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

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AbstractAward: DMS-0104275Principal Investigator: Tim CochranThis project develops a new area of noncommutative algebraictopology and its applications to low-dimensional topology. Thesuccess of algebraic topology in knot theory, for example, has,until recently centered around abelian invariants, that is tosay, invariants associated to the universal abelian coveringspace of the knot or link exterior. These invariants are theAlexander module, which is the first homology of this cover as amodule over a commutative Laurent polynomial ring, and theBlanchfield pairing. These determine the S-equivalence class ofthe knot as well as various other invariants. From theperspective of the knot group G, the Alexander module is simplyG'/G". Hence any behavior associated to G" will be invisible tothese abelian invariants. We remedy this deficiency by studyingthe quotients of successive terms of the higher derived series,or, put another way, study modules associated to more generalsolvable covering spaces. These are modules over noncommutativerings and thus are difficult to work with. We use techniques fromnoncommutative algebra and C* algebras to define invariants. Wefind , for each integer n, an entire theory which parallels theAlexander module and Blanchfield form and signatures. There areapplications to estimating genus, detecting fibered knots and3-manifolds, new invariants of concordance and representations ofmapping class groups.The advent of quantum mechanics led scientists to manyparadoxical, but now accepted, facts about our universe. Inparticular, there came the realization that "commutativemathematics" was inadequate to describe our physicalworld. Recall that 2 times 3 equals 3 times 2 is the commutativelaw of multiplication of numbers. Quantum mechanics showed thatphysical quantities are not mere numbers but more like arrays ormatrices of numbers. Since multiplication of matrices is notcommutative, this explains and models noncommutative phenomena atthe most fundamental levels of the physical world. This projectstudies the shape of 3 and 4-dimensional spaces by using newnoncommutative mathematics arising from algebra.