Knot concordance and von Neumann p-invariants

We present new results, announced in [T], on the classical knot concordance group W. We establish the nontriviality at all levels of the (n)-solvable filtration ...subset of F (n) subset of ... subset of F (1) subset of F (0) subset of C introduced in [COT]]. 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 [COT]], 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 [COT]].