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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 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 $L^2$-signature and use this to detect the first unknown step in our obstruction theory.
Tim D. Cochran 1 ; Kent E. Orr 2 ; Peter Teichner 
@article{10_4007_annals_2003_157_433, author = {Tim D. Cochran and Kent E. Orr and Peter Teichner}, title = {Knot concordance, {Whitney} towers and $L^2$-signatures}, journal = {Annals of mathematics}, pages = {433--519}, publisher = {mathdoc}, volume = {157}, number = {2}, year = {2003}, doi = {10.4007/annals.2003.157.433}, mrnumber = {1973052}, zbl = {1044.57001}, language = {en}, url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.433/} }
TY - JOUR AU - Tim D. Cochran AU - Kent E. Orr AU - Peter Teichner TI - Knot concordance, Whitney towers and $L^2$-signatures JO - Annals of mathematics PY - 2003 SP - 433 EP - 519 VL - 157 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.433/ DO - 10.4007/annals.2003.157.433 LA - en ID - 10_4007_annals_2003_157_433 ER -
%0 Journal Article %A Tim D. Cochran %A Kent E. Orr %A Peter Teichner %T Knot concordance, Whitney towers and $L^2$-signatures %J Annals of mathematics %D 2003 %P 433-519 %V 157 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2003.157.433/ %R 10.4007/annals.2003.157.433 %G en %F 10_4007_annals_2003_157_433
Tim D. Cochran; Kent E. Orr; Peter Teichner. Knot concordance, Whitney towers and $L^2$-signatures. Annals of mathematics, Tome 157 (2003) no. 2, pp. 433-519. doi: 10.4007/annals.2003.157.433
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