Homology Cobordism Invariants and the Cochran-Orr-Teichner Filtration of the Link Concordance Group

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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 rhoₙ for closed (4k-1)-dimensional manifolds. For 3-dimensional manifolds, we show that rhoₙ is a linearly independent set and for each n, the image of rhoₙ 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 F(n)ᵐ of the m-component (string) link concordance group, called the (n)-solvable filtration. They also define a grope filtration Gₙᵐ. We show that rhoₙ vanishes for (n+1)-solvable links. Using this, and the non-triviality of rhoₙ, we show that for each m>1, 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>2.