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 ρ-invariant, we obtain new real-valued homology cobordism invariants ρn for closed (4k-1)-dimensional manifolds. For 3-dimensional manifolds, we show that {ρn|n ∈ N} is a linearly independent set and for each n ≥ 0, the image of ρn 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 Fm(n) of the m-component (string) link concordance group, called the (n)-solvable filtration. They also define a grope filtration Gmn. We show that ρn vanishes for (n+1)-solvable links. Using this, and the nontriviality of ρn, we show that for each m ≥ 2, the successive quotients of the (n)-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 (n)-solvable filtration are known to be infinite. However, for knots, it is unknown if these quotients have infinite rank when n≥3. © 2008 Mathematical Sciences Publishers.