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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 adoi:10.2140/gt.2009.13.1419 fatcat:h6kdflihbjhujhxvst3fop43qi