Noncommutativity in low-dimensional topology
by
Tim Cochran
Rice University

I will survey recent work of S. Harvey, T. Kim, K.Orr, P. Teichner and myself on applications of noncommutative algebra to problems in low-dimensional topology. Many äbelian" algebraic invariants used in low-dimensional topology have "noncommutative" generalizations which can be much more discriminating. One fundamental such object is the homology of certain solvable covering spaces of the topological space in question. These are modules over noncommutative rings which generalize the Alexander modules. Another is the Cheeger-Gromov-Atiyah von Neumann rho invariant associated to an arbitrary 3- manifold together with a map to an arbitrary group. These generalize various signature invariants. A third is the bordism class of such pairs. The applications to date are to knot theory and 3-manifolds, specifically to knot concordance, knot genera, fibering 3-manifolds, Thurston norm, and to obstructing symplectic structures on 4-manifolds of the form S^1xM^3.

Date received: March 4, 2002