Abstract:

If one considers the set of m-component based links in R³ with a 4-dimensional equivalence relationship on it, called concordance, one can form a group called the link concordance group, C^m. Questions in concordance are extremely important in 4-dimensions and are related to one remaining version of the 4-dimensional Poincare Conjecture (the smooth 4-dimensional Poincare Conjecture).  It is well known that the link concordance group contains the isotopy class of pure braid with m strands, P_m. That is, two braids are concordant if and only if they are isotopic!  In the late 90's Tim Cochran, Kent Orr, and Peter Teichner defined a filtration of the knot/link concordance group called the n-solvable filtration. This filtration gives a way to approximate whether a link is trivial in the group. We discuss the relationship between pure braids and the n-solvable filtration as well as various other more geometrically defined filtrations coming from gropes and Whitney towers. This is joint work with Aru Ray and Jung Hwan Park.