Amenable \(L^{2}\)-theoretic methods and knot concordance
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 \(\Gamma\) is \(n+1\)-step solvable (i.e.~\(\Gamma^{(n+1)}=\{1\}\)) and lies in \(D(R)\) for \(R=\mathbb{Q}\) or \(R=\mathbb{Z}_p\), and if \(\tilde M\to M(K)\) is a normal \(\Gamma\)-covering extending to an \(n.5\)-solution, and such that the meridian is covered by a union of lines, then \(\rho^{(2)}(\tilde M;\Gamma)=0\).NEWLINENEWLINE\smallskip As an application, the author produces for each \(n\) a large family of \(n\)-solvable knots which are not \(n.5\)-solvable, but not detected by previously know \(L^2\)-signature methods.NEWLINENEWLINEThe new algebraic ingredients underlying these applications (to obtain the required deck transformation groups) use mixed-coefficient commutator series, and mod \(p\) versions of non-commutative higher Blanchfield pairings.
scientific article; zbMATH DE number 6359515