Extending Goldberg's Exact Sequence to Braid Groups of Graphs and Simplicial Complexes

Abstract

For a finite connected simplicial complex $X$, the strand map ${\iota }_{\ast }$, from ${\mathbb{P}}_n(X)$ to ${\prod }_{i=1}^n{\pi }_1(X,x_i^0)$, 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 $X$ $\text{weakly Goldberg}$ if some contractible subcomplex $X_0\subseteq X$ realises Goldberg's description, $\ker {\iota }_{\ast }=⟨\mathrm{im}({\mathbb{P}}_n(X_0)\to {\mathbb{P}}_n(X))⟩$, and $\text{Goldberg}$ 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 $X\ncong S^1$; 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 $\text{admissible tree}$ -- 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.