Math 758 — Spring 2002 Topics in Topology

This course is an introduction to the currently developing field of finite type (or Vassiliev) invariants of low dimensional objects. The theory of finite type invariants supports a rich and beautiful structure, but so far it has remained essentially independent from the rest of topology. A recent preprint of Garoufalidis and Rozansky, however, starts to draw a connection with classical algebraic topology, and it is the ultimate goal of the course to understand this paper. First, we will have to understand the so-called Kontsevich integral, which is a powerful invariant of knots and links, taking values in power series of trivalent graphs. Secondly, we have to understand the generalization of the Kontsevich integral to knots in homology 3-spheres. The paper of Garoufalidis and Rozansky starts to give an algebraic topological explanation of this last invariant, similar in spirit to the algebraic topological explanation of the Alexander polynomial as a homological invariant of the infinite cyclic cover of a knot complement.