A second order algebraic knot concordance group
The classical invariant of knot concordance is the Witt class of the Blanchfield pairing, which is defined in terms of Poincaré duality in the infinite cyclic cover. In higher dimensions this is a complete invariant, but in the classical case there are secondary invariants, such as the Casson-Gordon invariants, which detect non-slice knots with trivial Witt class. Cochran, Orr and Teichner have defined geometric filtrations of the knot concordance group \(\mathcal{C}_1\) whose highest non-trivial quotient is the algebraic concordance group \(\mathcal{AC}_1\). (Here and below ``concordance'' means ``locally flat TOP concordance''.) This paper presents a new algebraic invariant which is a single-stage invariant, is defined in terms of metabelian covers and refines the Witt invariant.NEWLINENEWLINEThe first part of the paper involves defining the invariant. Let \(K\) be a knot in \(S^3\), with exterior \(X\) and group \(\pi=\pi_1(X)\). A pair of parallel meridians in \(\partial{X}\) decomposes \(X\) as a triad \({(X;\partial_1X,\partial_2X;S^1\times{S^0})}\), where \(\partial_1X\cong\partial_2X\cong{S^1\times{D^1}}\). This may be viewed as a relative homology cobordism from \(\partial_1X\) to \(\partial_2X\). A handle decomposition of this cobordism determines an ``algebraic Poincaré triad'' over \(\mathbb{Z}[\pi/\pi'']\). One of the difficulties to be overcome is that the quotient \(\pi/\pi''\) depends on the knot \(K\); the abelianization \(\pi/\pi'\cong\mathbb{Z}\) is the only universal quotient of knot groups. The author successfully defines an equivalence relation on such triads which leads to an algebraically defined group \(\mathcal{AC}_2\). (However this group has not yet been calculated explicitly!)NEWLINENEWLINEIt is then shown that \(1.5\)-solvable knots have trivial image in \(\mathcal{AC}_2\) and that the Witt class map from \(\mathcal{C}_1\) to \(\mathcal{AC}_1\) factors as \(\mathcal{C}_1\to\mathcal{C}_1/\mathcal{F}_{1.5}\to\mathcal{AC}_2\to\mathcal{AC}_1,\) where \(\mathcal{F}_{1.5}\) is the subgroup represented by \(1.5\)-solvable knots. Although the composite map is an epimorphism, it is not clear whether every class in \(\mathcal{AC}_2\) is realized by some knot, nor how close the central map is to being an isomorphism. The obstructions defined by Cochran, Orr and Teichner on \(\mathcal{C}_1/\mathcal{F}_{1.5}\) take values in a pointed set, and factor through \(\mathcal{AC}_2\). In particular, the Cheeger-Gromov von Neumann \(\rho\)-invariant used by Cochran and Teichner are defined and non-trivial on the kernel of the map from \(\mathcal{AC}_2\) to \(\mathcal{AC}_1\). It is shown that this kernel has infinite rank.NEWLINENEWLINEThe author foreshadows forthcoming work on \textit{a priori} invariants which reach deeper into the COT filtrations, but the present work is already a major conceptual advance.
scientific article; zbMATH DE number 6035491
scientific article