编织曲面及其特征映射
Braided surfaces and their characteristic maps
AI总结：
本文研究曲面分支覆盖何时成为编织曲面的特征映射，利用经自由群与辫群的单值分解给出构造，并以辫群 Hermitian 表示刻画其强等价类的有限完备可区分性。
摘要：
我们证明，当曲面的亏格足够大时，该曲面的分支覆盖可作为编织曲面的特征映射出现；也就是说，它们可以提升为该曲面与 $\mathbb R^2$ 的乘积中的嵌入。即使对无分歧覆盖而言，这一结果也并非平凡：在这种情形下，提升问题众所周知地归结为一个纯代数问题，即把到对称群 $S_n$ 的单值映射通过辫群 $B_n$ 进行分解。在我们的方法中，这种分解通常是由一个更强性质得到的：通过自由群进行分解。在反方向上，我们证明任意非 Abel 曲面群都有无穷多个有限单非 Abel 群商，这些商具有特征核，而这些核不包含任何简单闭曲线，因此相应的商映射不能通过自由群分解。通过拉回构造，辫群的有限维 Hermitian 表示为编织曲面提供不变量。我们证明，这些不变量能够区分编织曲面的强等价类，当且仅当这些强等价类在有限完备意义下可被区分。
英文摘要：
We show that branched coverings of surfaces of large enough genus arise as characteristic maps of braided surfaces that is, lift to embeddings in the product of the surface with $\mathbb R^2$. This result is nontrivial already for unramified coverings, in which case the lifting problem is well-known to reduce to the purely algebraic problem of factoring the monodromy map to the symmetric group $S_n$ through the braid group $B_n$. In our approach, this factorization is often achieved as a consequence of a stronger property: a factorization through a free group. In the reverse direction we show that any non-abelian surface group has infinitely many finite simple non-abelian groups quotients with characteristic kernels which do not contain any simple loop and hence the quotient maps do not factor through free groups. By a pullback construction, finite dimensional Hermitian representations of braid groups provide invariants for the braided surfaces. We show that the strong equivalence classes of braided surfaces are separated by such invariants if and only if they are profinitely separated.
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