Knot Concordance and Higher-Order Blanchfield Duality

Abstract

In 1997, T. Cochran, K. Orr, and P. Teichner defined a filtration {F_n} 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 natural number n, the abelian group F_n/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.

Corrected Figure in Example 8.4, Added Remark 5.11 pointing out an important strengthening of Theorem 5.9 that is needed in a subsequent paper

References in corpus (3)

• Homology and Derived p-Series of Groups
• Link concordance and generalized doubling operators
• Knot concordance and Blanchfield duality

Cited by in corpus (26)

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• Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance
• Derivatives of Knots and Second-order Signatures
• The (n)-solvable filtration of the link concordance group and Milnor's invariants
• Concordance of links with identical Alexander invariants
• Hidden torsion, 3-manifolds, and homology cobordism
• On the knot Floer filtration of the concordance group
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• Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders
• Splittings of von Neumann rho-invariants of knots
• A survey of the homology cobordism group
• An explicit computation of the Blanchfield pairing for arbitrary links
• A second order algebraic knot concordance group
• Non-concordant links with homology cobordant zero framed surgery manifolds
• Classification of Links Up to 0-Solvability
• Whitney towers, Gropes and Casson-Gordon style invariants of links
• Linear independence of cables in the knot concordance group
• Covering link calculus and iterated Bing doubles
• Amenable signatures, algebraic solutions, and filtrations of the knot concordance group
• Action of the Mazur pattern up to topological concordance
• Rational knot concordance and homology cobordism