## Geometry & Topology

### Knot concordance and higher-order Blanchfield duality

#### 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 $C$,

$⋯ ⊆ ℱ n ⊆ ⋯ ⊆ ℱ 1 ⊆ ℱ 0 . 5 ⊆ ℱ 0 ⊆ C .$

The filtration is important because of its strong connection to the classification of topological $4$–manifolds. Here we introduce new techniques for studying $C$ and use them to prove that, for each $n∈ℕ0$, the group $ℱn∕ℱn.5$ 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.

#### Article information

Source
Geom. Topol., Volume 13, Number 3 (2009), 1419-1482.

Dates
Accepted: 1 December 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800250

Digital Object Identifier
doi:10.2140/gt.2009.13.1419

Mathematical Reviews number (MathSciNet)
MR2496049

Zentralblatt MATH identifier
1175.57004

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M10: Covering spaces

#### Citation

Cochran, Tim D; Harvey, Shelly; Leidy, Constance. Knot concordance and higher-order Blanchfield duality. Geom. Topol. 13 (2009), no. 3, 1419--1482. doi:10.2140/gt.2009.13.1419. https://projecteuclid.org/euclid.gt/1513800250

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