Knot Concordance

We prove the nontriviality at all levels of the filtration of the classical topological knot concordance group C · · · ⊆ Fn ⊆ · · · ⊆ F1 ⊆ F0 ⊆ C. defined in [COT]. This filtration is significant because not only is it strongly connected to Whitney tower constructions of Casson and Freedman, but all previously-known concordance invariants are related to the first few terms in the filtration. In [COT] we proved nontriviality at the first new level n = 3 by using von Neumann ρ-invariants of the 3-manifolds obtained by surgery on the knots. For larger n we use the Cheeger-Gromov estimate for such ρ-invariants, as well as some rather involved algebraic arguments using our noncommutative Blanchfield forms. In addition, we consider a closely related filtration, {Gn}, of C defined in terms of Gropes in the 4-ball. We show that this filtration is also non-trivial for all n > 2.