将Goldberg正合序列推广到图与单纯复形的辫群
Extending Goldberg's Exact Sequence to Braid Groups of Graphs and Simplicial Complexes
AI总结：
该研究将Goldberg关于闭曲面纯辫群链映射核的定理推广到任意有限连通单纯复形，定义了弱Goldberg与Goldberg复形的概念，刻画了其等价条件，还分类了核平凡的复形并解决了例外曲面等情形。
AI中文摘要：
对于有限连通单纯复形$X$，从纯辫群$\n\b_p_n(X)$到乘积群$\n\b_p_{i=1}^n\ng_1(X,x_i^0)$的链映射$ι_\n\u2217$，将一个纯辫映射为其各条链的同伦类。Goldberg于1973年证明的定理给出了当$X$为除$S^2$与$\n\b_RP^2$之外的闭曲面时该映射的核：核是支撑在一个嵌入圆盘内的纯辫的正规闭包。我们将这一结论推广到任意有限连通单纯复形。若存在可缩子复形$X_0\nsubseteq X$使得Goldberg的描述成立，即$\n\bkerι_\n\u2217=\n\left\ngle \n\b_\operatorname{im}(\n\b_p_n(X_0)\n\to\n\b_p_n(X)) \n\right\ngle$，则称$X$为“弱Goldberg复形”；若还能选取这样的$X_0$使得映射$\n\b_p_n(X_0)\n\to\n\b_p_n(X)$是单射，则称$X$为“Goldberg复形”。我们证明：链映射是满射当且仅当$X\n\not\ngcong S^1$；$X$是弱Goldberg复形当且仅当其自由部分是森林；$X$是Goldberg复形当且仅当它存在“容许树”——即脚手架的极大树，且与自由部分附着到厚分量的边界和内部类型相容。我们还分类了核平凡的复形，解决了例外曲面$S^2$和$\n\b_RP^2$的情况，并得到了流形与图情形下的完整结论。主要工具是$X$上一点处配置空间的空间图分解，以及将任意复形约化为简单模型的消解程序。
英文摘要：
For a finite connected simplicial complex $X$, the strand map $ι_\ast$, from $\mathbb{P}_n(X)$ to $\prod_{i=1}^nπ_1(X,x_i^0)$, sends a pure braid to the homotopy classes of its strands. A theorem of Goldberg (1973) computes its kernel when $X$ is a closed surface other than $S^2$ and $\mathbb{RP}^2$: the kernel is the normal closure of the pure braids supported in an embedded disc. We extend this picture to arbitrary finite connected simplicial complexes. Call $X$ $\textit{weakly Goldberg}$ if some contractible subcomplex $X_0\subseteq X$ realises Goldberg's description, $\kerι_\ast=\left\langle \operatorname{im}(\mathbb{P}_n(X_0)\to\mathbb{P}_n(X)) \right\rangle$, and $\textit{Goldberg}$ if $X_0$ can moreover be chosen so that $\mathbb{P}_n(X_0)\to\mathbb{P}_n(X)$ is injective. We prove that the strand map is surjective if and only if $X\not\cong S^1$; that $X$ is weakly Goldberg if and only if its free part is a forest; and that $X$ is Goldberg if and only if it admits an $\textit{admissible tree}$ -- a maximal tree of a scaffold, compatible with the boundary and interior types of the attachments of the free part to the thick components. We also classify the complexes for which the kernel is trivial, settle the exceptional surfaces $S^2$ and $\mathbb{RP}^2$, and obtain complete answers for manifolds and for graphs. The main tools are a graph-of-spaces decomposition of the configuration space at a point of $X$ and a resolution procedure reducing an arbitrary complex to a simple model.
