Homology and Derived Series of Groups II: Dwyer's Theorem

Abstract

We give new information about the relationship between the low-dimensional homology of a group and its derived series. This yields information about how the low-dimensional homology of a topological space constrains 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 non-abelian free solvable group, and to concordance and grope cobordism of 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 the second homology groups. The naive analogues of these theorems for the derived series are false. In 2003 the second author introduced a new characteristic series, 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.Comment: 26 pages. In this version, we have included a new proof of part of the main theorem. The new proof is somewhat simpler and stays entirely in the world of group homology and homological algebra rather than using Eilenberg-Mac Lane spaces. Other minor corrections. This is the final version to appear in Geometry & Topolog