We establish several new results about both the
–solvable
filtration of the set of link concordance classes and the
–solvable
filtration of the string link concordance group,
. The set of
–solvable
–component string
links is denoted by .
We first establish a relationship between Milnor’s invariants and
links, , with certain
restrictions on the –manifold
bounded by , the
zero-framed surgery of
on . Using this relationship,
we can relate –solvability
of a link (or string link) with its Milnor’s
–invariants. Specifically, we
show that if a link is –solvable,
then its Milnor’s invariants vanish for lengths up to
.
Previously, there were no known results about the “other half” of the filtration, namely
.
We establish the effect of the Bing doubling operator on
–solvability and using
this, we show that
is nontrivial for links (and string links) with sufficiently many
components. Moreover, we show that these quotients contain an infinite
cyclic subgroup. We also show that links and string links modulo
–solvability
is a nonabelian group. We show that we can relate other
filtrations with Milnor’s invariants. We show that if a link is
–positive,
then its Milnor’s invariants will also vanish for lengths up to
. Lastly, we
prove that the grope filtration of the set of link concordance classes is not the same as the
–solvable
filtration.
Keywords
Milnor's invariants, solvable filtration, link concordance
