#### Volume 12, issue 1 (2008)

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

### Shelly L Harvey

Geometry & Topology 12 (2008) 387–430
##### 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 $\left\{{\rho }_{n}|n\in ℕ\right\}$ 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 ${\mathsc{ℱ}}_{\left(n\right)}^{m}$ of the $m$–component (string) link concordance group, called the $\left(n\right)$–solvable filtration. They also define a grope filtration ${\mathsc{G}}_{n}^{m}$. 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 $n\ge 3$.

##### Keywords
link concordance, derived series, homology cobordism
Primary: 57M27
Secondary: 20F14