Knot concordance, Whitney towers and L2-signatures

• 成果类型: Article
• 署名作者: Cochran, TD; Orr, KE; Teichner, P
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2003.157.433
• 发表日期: 2003
• 页码: 433-519
• 关键词: invariants; MANIFOLDS; dimension; signature; cobordism; TOPOLOGY

摘要

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.