  In 1997, T. Cochran, K. Orr, and P. Teichner defined a filtration {F_n} 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
natural number n, the abelian group F_n/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.
