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1 April 2007 Knot concordance and von Neumann ρ -invariants
Tim D. Cochran, Peter Teichner
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Duke Math. J. 137(2): 337-379 (1 April 2007). DOI: 10.1215/S0012-7094-07-13723-2


We present new results, announced in [T], on the classical knot concordance group $\mathscr{C}$. We establish the nontriviality at all levels of the $(n)$-solvable filtration $$\cdots \subseteq \mathscr{F}_{n} \subseteq \cdots \subseteq \mathscr{F}_1 \subseteq\mathscr{F}_{0} \subseteq\mathscr{C}$$ introduced in [COT1]. Recall that this filtration is significant due to its intimate connection to tower constructions arising in work of A. Casson and M. Freedman on the topological classification problem for $4$-manifolds and due to the fact that all previously known concordance invariants are reflected in the first few terms in the filtration. In [COT1], nontriviality at the first new level $n=3$ was established. Here, we prove the nontriviality of the filtration for all $n$, hence giving the ultimate justification to the theory.

A broad range of techniques is employed in our proof, including cut-and-paste topology and analytical estimates. We use the Cheeger-Gromov estimate for von Neumann $\rho$-invariants, a deep analytic result. We also introduce a number of new algebraic arguments involving noncommutative localization and Blanchfield forms. We have attempted to make this article accessible to readers with only passing knowledge of [COT1].


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Tim D. Cochran. Peter Teichner. "Knot concordance and von Neumann ρ -invariants." Duke Math. J. 137 (2) 337 - 379, 1 April 2007.


Published: 1 April 2007
First available in Project Euclid: 28 March 2007

zbMATH: 1186.57006
MathSciNet: MR2309149
Digital Object Identifier: 10.1215/S0012-7094-07-13723-2

Primary: 57M25 , 57N70

Rights: Copyright © 2007 Duke University Press


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Vol.137 • No. 2 • 1 April 2007
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