Knot concordance, Whitney towers and L^2 signatures

Details Knot concordance, Whitney towers and L^2 signatures Knot concordance, Whitney towers and L^2 signatures

1999-08-22

arxiv.org/abs/math/9908117v2

Ann. of Math. (2), Vol. 157 (2003), no. 2, 433--519

Mathematics

Geometric Topology

87 pages published version

Scientific paper

We construct many examples of non-slice 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 {\em universal} groups. Finally, we use the dimension theory of von Neumann algebras to define an L^2 signature and use this to detect the first unknown step in our obstruction theory.

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