Knot concordance and von Neumann $\rho$-invariants

By Tim D. Cochran, P. Teichner. 2004

47 citations recorded by the source corpus.

We present new results, announced in [T], on the classical knot concordance group C. We establish the nontriviality at all levels of the (n)-solvable filtration · · · ⊆ Fn ⊆ · · · ⊆ F1 ⊆ F0 ⊆ C introduced in [COT1]. Recall that this filtration is significant due to its intimate connection to tower constructions arising in work of A. Casson and M. Freedman on the topological classification problem for 4-manifolds and due to the fact that all previously known concordance invariants are reflected in the first few terms in the filtration. In [COT1], nontriviality at the first new level n = 3 was established. Here, we prove the nontriviality of the filtration for all n, hence giving the ultimate justification to the theory. A broad range of techniques is employed in our proof, including cut-and-paste topology and analytical estimates. We use the Cheeger-Gromov estimate for von Neumann ρ-invariants, a deep analytic result. We also introduce a number of new algebraic arguments involving noncommutative localization and Blanchfield forms. We have attempted to make this article accessible to readers with only passing knowledge of [COT1].

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ee3ed42db89bfea170cac0c661dcabddfd77d721

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Citations in the corpus

Primary decomposition and the fractal nature of knot concordance (2009, methodology): “In [ 20 ] it was shown that each Gn has positive rank and this was extended in [17,36].”
Knot concordance and higher-order Blanchfield duality (2007, methodology): “In fact the techniques of [12], [13], [ 14 ] and [11] were limited to knots of genus at least 2 (note each Jn has genus 1) because of their use of localization techniques.”
A Survey of Classical Knot Concordance (2003, background): “Beginning with [14] and culminating in [ 16 ], there is the following result.”
Topologically slice knots with nontrivial Alexander polynomial (2010, background): “5 for all i was proved in [6] (see also [10] for the existence of elements of infinite order in each of the quotients).”
Homology cobordism invariants and the Cochran-Orr-Teichner filtration of the link concordance group (2006, background): “n G(1), so that G/G(1) r is Z-torsion-free. Proceeding in this way, deﬁning G(n) r to be the radical of [G(n−1) r ,G (n−1) r ], leads to what has been called the rational derived series of G[Ha1] [C] [CT]. This is the most rapidly descending series for which the quotients of successive terms are Z-t”
An infinite-rank summand of topologically slice knots (2013, background): “r nearly a decade from when the homomorphism was deﬁned in 1969 until the advent of Casson-Gordon invariants [CG78, CG86], it was unknown that A was non-trivial. Further work (e.g., [Jia81], [COT04], [CT07]) revealed deeper structure in A. However, these techniques were unable to determine anything ”