The (n)-solvable filtration of the link concordance group and Milnor's invariants

TL;DR

This paper explores the (n)-solvable filtration of link and string link concordance groups, establishing new relationships with Milnor's invariants, and demonstrating the nontriviality and structure of certain filtration quotients.

Contribution

It introduces new connections between Milnor's invariants and (n)-solvability, and shows that certain filtration quotients are nontrivial with infinite cyclic subgroups.

Findings

Milnor's invariants vanish for links with (n)-solvability up to length 2^{n+2}-1

F_{n.5}^m/F_{n+1}^m is nontrivial and contains an infinite cyclic subgroup

Links modulo (1)-solvability form a nonabelian group

Abstract

We establish several new results about both the (n)-solvable filtration, F_n^m, of the set of link concordance classes and the (n)-solvable filtration of the string link concordance group. We first establish a relationship between Milnor's invariants and links, L, with certain restrictions on the 4-manifold bounded by M_L. Using this relationship, we can relate (n)-solvability of a link (or string link) with its Milnor's invariants. Specifically, we show that if a link is (n)-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 F_{n.5}^m/F_{n+1}^m. We establish the effect of the Bing doubling operator on (n)-solvability and using this, we show that F_{n.5}^m/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 (1)-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, G_n^m, of the set of link concordance classes is not the same as the (n)-solvable filtration.