Twisted Homology Cobordism Invariants of Knots in Aspherical Manifolds

Abstract

We study concordance of homotopically essential, null-homologous knots in three-manifolds M with poly-torsion-free-abelian fundamental group. We fix a knot J and, using L2-signature techniques, construct a family of concordance invariants of knots homotopic to J. We then construct an infinite family of non-concordant knots that are characteristic to J. Our invariants are ρ-invariants of certain three-manifolds associated to these knots, where the three-manifolds depend on the fixed knot J. In order to obtain concordance invariants, we define a series that admits an injectivity theorem (that is, a theorem reminiscent of Stallings’ theorem). We define this series by constructing a localization on the category of groups over π1(M), following Levine’s construction of the algebraic closure for groups [10].