Homology and Derived Series of Groups Ii: Dwyer’s Theorem

We give new information about the relationship between the lowdimensional 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 link concordance. 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 lowdimensional homology of a group and its lower central series quotients. In 1975 W. Dwyer extended Stallings’ theorem by weakening the hypothesis on H2. The naive analogues of these theorems for the derived series are false. In 2003 the second author introduced a new characteristic series, G (n) H , 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. For historical completeness we prove a version of Dwyer’s theorem for the rational lower central series.