Twisted homology cobordism invariants of knots in aspherical manifolds (Q2909341)

After fundamental work on the classical concordance group involving \(L^2\)-signatures during the past decade, by J. C. Cha, T. Cochran, C. Livingston, K. Orr and P. Teichner, this work embarks on applying similar methods to study concordance of knots in more general \(3\)-manifolds. The outcome is, given a suitable knot \(J\) in \(M\), an infinite set of nonconcordant knots in \(M\), all homotopic to \(J\). The knots considered are closed curves in closed, oriented, irreducible, aspherical \(3\)-manifolds, nullhomologous and representing a homotopy class of infinite order. The concordance invariants distinguishing the knots are Cheeger-Gromov von Neumann \(\rho\)-invariants, i.e., the difference of an \(L^2\)-signature and the classical signature, involving adequate \(4\)-manifolds and coefficient systems. The manifolds occurring come up rather geometrically by surgery methods. The crux lies in the construction of the coefficient systems. This imposes an additional technical condition on \(M\), namely that \(M\) is PTFA (poly-torsion-free abelian), and requires the extensive construction of the rational \(\pi_1(M)\)-local derived series, whose presentation along the lines of work done by J. Levine and more recently by S. L. Harvey takes up the final third of the paper.