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Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group

Shelly L Harvey

Geometry & Topology 12 (2008) 387–430

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 ρ–invariant, we obtain new real-valued homology cobordism invariants ρn for closed (4k1)–dimensional manifolds. For 3–dimensional manifolds, we show that {ρn|n } is a linearly independent set and for each n 0, the image of ρn is an infinitely generated and dense subset of .

In their seminal work on knot concordance, T Cochran, K Orr and P Teichner define a filtration (n)m of the m–component (string) link concordance group, called the (n)–solvable filtration. They also define a grope filtration Gnm. We show that ρn vanishes for (n+1)–solvable links. Using this, and the nontriviality of ρn, we show that for each m 2, 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 3.

link concordance, derived series, homology cobordism
Mathematical Subject Classification 2000
Primary: 57M27
Secondary: 20F14
Received: 9 April 2007
Revised: 2 September 2007
Accepted: 15 November 2007
Published: 12 March 2008
Proposed: Peter Teichner
Seconded: Peter Ozsvath, Ron Fintushel
Shelly L Harvey
Department of Mathematics
Rice University
PO Box 1892, MS 136
Houston TX 77005-1892