Volume 12, issue 1 (2008)

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

Tim D Cochran and Shelly Harvey

Geometry & Topology 12 (2008) 199–232

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 H2. In 2003 the second author introduced a new characteristic series, GH(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.

derived series, gropes, link concordance, homology equivalence
Mathematical Subject Classification 2000
Primary: 57M07
Secondary: 20J06, 55P60
Received: 17 December 2006
Revised: 6 August 2007
Accepted: 23 October 2007
Published: 8 February 2008
Proposed: Peter Teichner
Seconded: Walter Neumann, Rob Kirby
Tim D Cochran
Department of Mathematics
Rice University
Houston, TX 77005-1892
Shelly Harvey
Department of Mathematics
Rice University
Houston, TX 77005-1892