If one considers the set of m-component based links in R^3
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 important in for classification
questions in topological and smooth 4-manifolds 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.