Knot concordance and higher-order Blanchfield duality

Abstract

In 1997, T Cochran, K Orr, and P Teichner [Ann. of Math. (2) 157 (2003) 433519] 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 longstanding question as to whether certain natural families of knots, first considered by Casson-Gordon and Gilmer, contain slice knots. © 2009 Mathematical Sciences Publishers.