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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060

Knot concordance and higher-order Blanchfield duality

Tim D Cochran, Shelly Harvey and Constance Leidy

Geometry & Topology 13 (2009) 1419–1482

DOI: 10.2140/gt.2009.13.1419

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,

••• ⊆ <b>F</b>n ⊆ ••• ⊆ <b>F</b>1 ⊆ <b>F</b>0.5 ⊆ <b>F</b>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 in N0, the group FnFn.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

Mathematical Subject Classification

Primary: 57M25

Secondary: 57M10

References
Publication

Received: 10 September 2008
Accepted: 1 December 2008
Published: 19 February 2009
Proposed: Peter Teichner
Seconded: Cameron Gordon, Tom Goodwillie

Authors
Tim D Cochran
Department of Mathematics
Rice University
Houston, Texas 77005-1892
http://math.rice.edu/~cochran
Shelly Harvey
Department of Mathematics
Rice University
Houston, Texas 77005-1892
http://math.rice.edu/~shelly
Constance Leidy
Wesleyan University
Wesleyan Station
Middletown, CT 06459
http://cleidy.web.wesleyan.edu