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We construct many examples of nonslice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the CassonGordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the L-theory of skew fields associated to certain universal groups. . | Annals of Mathematics Knot concordance Whitney towers and 2-signatures By Tim D. Cochran Kent E. Orr and Peter Teichner Annals of Mathematics 157 2003 433 519 Knot concordance Whitney towers and L2-signatures By Tim D. Cochran Kent E. Orr and Peter Teichner Abstract We construct many examples of nonslice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants including the Casson-Gordon invariants. As a first step we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the L-theory of skew fields associated to certain universal groups. Finally we use the dimension theory of von Neumann algebras to define an L2-signature and use this to detect the first unknown step in our obstruction theory. Contents 1. Introduction 1.1. Some history -solvability and Whitney towers 1.2. Linking forms intersection forms and solvable representations of knot groups 1.3. L2-signatures 1.4. Paper outline and acknowledgements 2. Higher order Alexander modules and Blanchfield linking forms 3. Higher order linking forms and solvable representations of the knot group 4. Linking forms and Witt invariants as obstructions to solvability 5. L2-signatures 6. Non-slice knots with vanishing Casson-Gordon invariants 7. n -surfaces gropes and Whitney towers 8. H1 -bordisms 9. Casson-Gordon invariants and solvability of knots References All authors were supported by MSRI and NSF. The third author was also supported by a fellowship from the Miller foundation UC Berkeley. 434 TIM D. COCHRAN KENT E. ORR AND PETER TEICHNER 1. Introduction This paper begins a detailed investigation into the group of topological concordance classes of knotted circles in the 3-sphere. Recall that a knot K is topologically .
