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G&T Monographs |
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Knot concordance and higher-order Blanchfield duality
Tim D Cochran, Shelly Harvey and Constance Leidy
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Geometry & Topology 13 (2009)
1419–1482
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Abstract
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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,
 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 in 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.
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Keywords
concordance, (n)-solvable, knot, slice
knot, Blanchfield form, von Neumann signature
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Mathematical Subject Classification
Primary: 57M25
Secondary: 57M10
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Publication
Received: 10 September 2008
Accepted: 1 December 2008
Published: 19 February 2009
Proposed: Peter Teichner
Seconded: Cameron Gordon, Tom Goodwillie
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