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 (4k-1)-dimensional manifolds. For 3-dimensional manifolds, we show that {rho_n} is a linearly independent set and for each n, 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 F_(n)^m of the m-component (string) link concordance group, called the (n)-solvable filtration. They also define a grope filtration G_n^m. We show that rho_n vanishes for (n+1)-solvable links. Using this, and the non-triviality of rho_n, 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.