#### Volume 14, issue 5 (2014)

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 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
The $(n)$–solvable filtration of link concordance and Milnor's invariants

### Carolyn Otto

Algebraic & Geometric Topology 14 (2014) 2627–2654
##### 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, ${\mathsc{C}}^{m}$. The set of $\left(n\right)$–solvable $m$–component string links is denoted by ${\mathsc{ℱ}}_{n}^{m}$. 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 $\stackrel{̄}{\mu }$–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 ${\mathsc{ℱ}}_{n.5}^{m}∕{\mathsc{ℱ}}_{n+1}^{m}$. We establish the effect of the Bing doubling operator on $\left(n\right)$–solvability and using this, we show that ${\mathsc{ℱ}}_{n.5}^{m}∕{\mathsc{ℱ}}_{n+1}^{m}$ 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.

##### Keywords
Milnor's invariants, solvable filtration, link concordance
Primary: 57M25