AI 中文总结
本文引入$C$-提升函子，构造相对商并证明其函子性，导出二阶满射，得到相对舒尔-鲍尔定理，关联外构造与相对舒尔乘子，证明虚拟纽结群左可序当且仅当圆可序。
AI 中文摘要
我们引入了$C$-提升函子，该函子公理化了非阿贝尔张量积与外积关于指定群扩张类的提升性质。对于同态$f\colon\Gamma\to G$，我们将每个$C$-提升函子$F$关联到一个相对商$F_f(G)$，该商可满射到$C$中每个$f$-扩张所确定的$F$对应的子群。我们证明该构造关于$f$是函子性的，且对于$f$-扩张$p\colon\widetilde G\to G$，它给出自然正合序列的态射。在二阶情形，这导出一个满射：$\frac{H_2(G;\mathbb Z)}{f_*H_2(\Gamma;\mathbb Z)} \longrightarrow \ker p\cap[\widetilde G,\widetilde G]$。将该构造应用于迭代张量积与外幂，我们得到关于下中心列与导出列的相对舒尔-鲍尔定理。我们还比较了相对张量平方与外平方，将外构造与相对舒尔乘子$H_2(G,\Gamma;\mathbb Z)$关联，并导出了可序性的应用结果。作为推论，我们证明虚拟纽结群是左可序的当且仅当它是圆可序的。
英文摘要
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 $f\colonΓ\to G$, 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(Γ;\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 $H_2(G,Γ;\mathbb Z)$, 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.
