Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group

Abstract

For any group G, we define a new characteristic series related to the derived series, that we call the torsion-free derived series of G. Using this series and the Cheeger–Gromov $\rho$–invariant, we obtain new real-valued homology cobordism invariants ${\rho }_n$ for closed $\left(4k-1\right)$–dimensional manifolds. For $3$–dimensional manifolds, we show that $\left\{{\rho }_n|n\in \mathbb{N}\right\}$ is a linearly independent set and for each $n\ge 0$, the image of ${\rho }_n$ is an infinitely generated and dense subset of $ℝ$.

In their seminal work on knot concordance, T Cochran, K Orr and P Teichner define a filtration ${\mathcal{ℱ}}_{\left(n\right)}^m$ of the $m$–component (string) link concordance group, called the $\left(n\right)$–solvable filtration. They also define a grope filtration ${\mathcal{G}}_n^m$. We show that ${\rho }_n$ vanishes for $\left(n+1\right)$–solvable links. Using this, and the nontriviality of ${\rho }_n$, we show that for each $m\ge 2$, the successive quotients of the $\left(n\right)$–solvable filtration of the link concordance group contain an infinitely generated subgroup. We also establish a similar result for the grope filtration. We remark that for knots ($m=1$), the successive quotients of the $\left(n\right)$–solvable filtration are known to be infinite. However, for knots, it is unknown if these quotients have infinite rank when $n\ge 3$.