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 $f\colon\Gamma\to G$, 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 $p\colon\widetilde G\to G$, it gives a morphism of natural exact sequences. In degree two this yields an epimorphism $ \frac{H_2(G;\mathbb Z)}{f_*H_2(\Gamma;\mathbb Z)} \longrightarrow \ker p\cap[\widetilde G,\widetilde 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.