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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 (4k-1)–dimensional manifolds. For 3–dimensional manifolds,
we show that {ρn ∣ n in N} is a linearly
independent set and for each n ≥ 0, the image of ρn
is an infinitely generated and dense subset of R.
In their seminal work on knot concordance, T Cochran, K Orr
and P Teichner define a filtration F(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.
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