Knot concordance and von Neumann ρ -invariants

Abstract

We present new results, announced in [T], on the classical knot concordance group $\mathscr{C}$. We establish the nontriviality at all levels of the $(n)$-solvable filtration $$\cdots \subseteq \mathscr{F}_{n} \subseteq \cdots \subseteq \mathscr{F}_1 \subseteq\mathscr{F}_{0} \subseteq\mathscr{C}$$ introduced in [COT1]. Recall that this filtration is significant due to its intimate connection to tower constructions arising in work of A. Casson and M. Freedman on the topological classification problem for $4$-manifolds and due to the fact that all previously known concordance invariants are reflected in the first few terms in the filtration. In [COT1], nontriviality at the first new level $n=3$ was established. Here, we prove the nontriviality of the filtration for all $n$, hence giving the ultimate justification to the theory. A broad range of techniques is employed in our proof, including cut-and-paste topology and analytical estimates. We use the Cheeger-Gromov estimate for von Neumann $\rho$-invariants, a deep analytic result. We also introduce a number of new algebraic arguments involving noncommutative localization and Blanchfield forms. We have attempted to make this article accessible to readers with only passing knowledge of [COT1].