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 $\mathcal{C}$,

$$\cdots \subseteq {\mathcal{ℱ}}_n\subseteq \cdots \subseteq {\mathcal{ℱ}}_1\subseteq {\mathcal{ℱ}}_{0.5}\subseteq {\mathcal{ℱ}}_0\subseteq \mathcal{C}.$$

The filtration is important because of its strong connection to the classification of topological $4$–manifolds. Here we introduce new techniques for studying $\mathcal{C}$ and use them to prove that, for each $n\in {\mathbb{N}}_0$, the group ${\mathcal{ℱ}}_n∕{\mathcal{ℱ}}_{n.5}$ 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.