Homology and derived series of groups II: Dwyer's Theorem

We give new information about the relationship between the low-dimensional homology of a space and the derived series of its fundamental group. Applications are given to detecting when a set of elements of a group generates a subgroup "large enough" to map onto a nonabelian free solvable group, and to concordance and grope cobordism of classical links. We also greatly generalize several key homological results employed in recent work of Cochran-Orr-Teichner in the context of classical knot concordance. In 1963 J Stallings established a strong relationship between the low-dimensional homology of a group and its lower central series quotients. In 1975 W Dwyer extended Stallings' theorem by weakening the hypothesis on H-2. In 2003 the second author introduced a new characteristic series, G(H)((n)), associated to the derived series, called the torsion-free derived series. The authors previously established a precise analogue, for the torsion-free derived series, of Stallings' theorem. Here our main result is the analogue of Dwyer's theorem for the torsion-free derived series. We also prove a version of Dwyer's theorem for the rational lower central series. We apply these to give new results on the Cochran-Orr-Teichner filtration of the classical link concordance group.