Amenable \(L^{2}\)-theoretic methods and knot concordance
type: Item
instance of: scholarly article
zbMATH Open document ID: 1302.57014
author: Jae Choon Cha
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.1058
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57N50
zbMATH DE Number: 6359515
MaRDI profile type: Publication
OpenAlex ID: W2963659869
arXiv ID: 1010.1058
DOI: 10.1093/IMRN/RNT092
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published in: IMRN. International Mathematics Research Notices
title: Amenable \(L^{2}\)-theoretic methods and knot concordance
review text: In recent years, the study of knot concordance has been influenced a lot by the use of \(L^2\)-signatures of spaces associated to knots, starting with the seminal paper ``Knot concordance, Whitney towers and \(L^2\)-signatures'' by \textit{T. D. Cochran} et al. [Ann. Math. (2) 157, No. 2, 433--519 (2003; Zbl 1044.57001)].NEWLINENEWLINEIn particular, there is a geometrically defined filtration \(\dots\subset\mathcal{F}_{n.5}\subset \mathcal{F}_n\subset\dots \subset \mathcal{F_0}\subset\mathcal{C}\) of the topological knot concordance group \(\mathcal{C}\) (defined via the height of Whitney towers and gropes of 4-manifolds bounding the zero surgery of the knot). Refined versions of \(L^2\)-signatures and \(L^2\)-torsion have been used to distinguish different levels of this filtration, using coverings with poly-torsion-free-abelian deck transformation group.NEWLINENEWLINEThe paper at hand reveals new structure in the knot concordance group. The key tool is the extension of the \(L^2\)-methods to more general coverings, namely certain amenable groups (lying in Strebel's class \(D(R)\)), which in particular includes non-solvable groups, and non-torsion-free groups (in contrast to the poly-torsion-free-abelian case). A group \(\Gamma\) lies in \(D(R)\) for a commutative unital ring \(R\), if \(\alpha: P\to Q\) between projective \(R\Gamma\)-modules is injective whenever \(!_R\otimes_{R\Gamma}\alpha: R\otimes_{R\Gamma}P\to R\otimes_{R\Gamma}Q\) is injective.NEWLINENEWLINEThe main results of the paper, making these invariants applicable, are the following vanishing results:NEWLINENEWLINE{ Theorem}. If \(K\) is a slice knot with zero-surgery \(M(K)\) and \(\tilde M\to M(K)\) is a normal covering extending to a covering of a slice disc exterior with deck transformation group \(\Gamma\) amenable and in Strebel's class \(D(R)\) for some \(R\), then the associated \(L^2\)-torsion vanishes: \(\rho^{(2)}(\tilde M;\Gamma)=0\).NEWLINENEWLINEIf \(K\) is an \(n.5\)-solvable knot and …
reviewed by: Thomas Schick
description: scientific article; zbMATH DE number 6359515