Lifting Functors and Relative Schur-Baer Theorems

Abstract

We introduce $C$-lifting functors, which axiomatize lifting properties of non-abelian tensor and exterior products with respect to prescribed classes of group extensions. For a homomorphism $f:\Gamma \to G$, we associate to every $C$-lifting functor $F$ a relative quotient $F_f(G)$. This quotient maps epimorphically onto the subgroup determined by $F$ in every $f$-extension belonging to $C$. We show that the construction is functorial in $f$ and that, for an $f$-extension $p:\tilde{G}\to G$, it gives a morphism of natural exact sequences. In degree two this yields an epimorphism $\frac{H_2(G;\mathbb{Z})}{f_{\ast }{H}_2(\Gamma ;\mathbb{Z})}⟶\ker p\cap [\tilde{G},\tilde{G}].$ Applying the construction to iterated tensor and exterior powers, we obtain relative Schur-Baer theorems for the lower central and derived series. We also compare the relative tensor and exterior squares, relate the exterior construction to the relative Schur multiplier $H_2(G,\Gamma ;\mathbb{Z})$, and derive applications to orderability. As a consequence, we prove that a virtual knot group is left-orderable if and only if it is circularly orderable.