The $(n)$–solvable filtration of link concordance and Milnor's invariants

Abstract

We establish several new results about both the $\left(n\right)$–solvable filtration of the set of link concordance classes and the $\left(n\right)$–solvable filtration of the string link concordance group, ${\mathcal{C}}^m$. The set of $\left(n\right)$–solvable $m$–component string links is denoted by ${\mathcal{ℱ}}_n^m$. We first establish a relationship between Milnor’s invariants and links, $L$, with certain restrictions on the $4$–manifold bounded by $M_L$, the zero-framed surgery of $S^3$ on $L$. Using this relationship, we can relate $\left(n\right)$–solvability of a link (or string link) with its Milnor’s $\bar{\mu }$–invariants. Specifically, we show that if a link is $\left(n\right)$–solvable, then its Milnor’s invariants vanish for lengths up to $2^{n+2}-1$. Previously, there were no known results about the “other half” of the filtration, namely ${\mathcal{ℱ}}_{n.5}^m∕{\mathcal{ℱ}}_{n+1}^m$. We establish the effect of the Bing doubling operator on $\left(n\right)$–solvability and using this, we show that ${\mathcal{ℱ}}_{n.5}^m∕{\mathcal{ℱ}}_{n+1}^m$ 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 $\left(1\right)$–solvability is a nonabelian group. We show that we can relate other filtrations with Milnor’s invariants. We show that if a link is $n$–positive, then its Milnor’s invariants will also vanish for lengths up to $2^{n+2}-1$. Lastly, we prove that the grope filtration of the set of link concordance classes is not the same as the $\left(n\right)$–solvable filtration.