#### Volume 13, issue 3 (2009)

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Knot concordance and higher-order Blanchfield duality

### Tim D Cochran, Shelly Harvey and Constance Leidy

Geometry & Topology 13 (2009) 1419–1482
##### Abstract

In 1997, T Cochran, K Orr, and P Teichner [Ann. of Math. (2) 157 (2003) 433-519] defined a filtration of the classical knot concordance group $\mathsc{C}$,

$\cdots \subseteq {\mathsc{ℱ}}_{n}\subseteq \cdots \subseteq {\mathsc{ℱ}}_{1}\subseteq {\mathsc{ℱ}}_{0.5}\subseteq {\mathsc{ℱ}}_{0}\subseteq \mathsc{C}.$

The filtration is important because of its strong connection to the classification of topological $4$–manifolds. Here we introduce new techniques for studying $\mathsc{C}$ and use them to prove that, for each $n\in {ℕ}_{0}$, the group ${\mathsc{ℱ}}_{n}∕{\mathsc{ℱ}}_{n.5}$ has infinite rank. We establish the same result for the corresponding filtration of the smooth concordance group. We also resolve a long-standing question as to whether certain natural families of knots, first considered by Casson–Gordon and Gilmer, contain slice knots.

##### Keywords
concordance, (n)-solvable, knot, slice knot, Blanchfield form, von Neumann signature
Primary: 57M25
Secondary: 57M10