Extending Goldberg's Exact Sequence to Braid Groups of Graphs and Simplicial Complexes
Abstract
For a finite connected simplicial complex X, the strand map ι∗, from ℙn(X) to ∏i=1nπ1(X,xi0), 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 S2 and ℝℙ2: 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 weakly Goldberg if some contractible subcomplex X0⊆ X realises Goldberg's description, kerι∗=⟨ im(ℙn(X0)→ℙn(X)) ⟩, and Goldberg if X0 can moreover be chosen so that ℙn(X0)→ℙn(X) is injective. We prove that the strand map is surjective if and only if Xnot≅ S1; 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 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 S2 and ℝℙ2, 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.
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BibTeX
@misc{an2026extending,
title = {{Extending Goldberg's Exact Sequence to Braid Groups of Graphs and Simplicial Complexes}},
author = {Byung Hee An},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.14350},
eprint = {2608.14350},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14350},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extending Goldberg's Exact Sequence to Braid Groups of Graphs and Simplicial Complexes
AU - Byung Hee An
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2608.14350
UR - https://arxiv.org/abs/2608.14350
AB - For a finite connected simplicial complex X, the strand map ι_∗, from ℙ_n(X) to ∏_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 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 weakly Goldberg if some contractible subcomplex X_0⊆ X realises Goldberg's description, kerι_∗=⟨ im(ℙ_n(X_0)→ℙ_n(X)) ⟩, and Goldberg if X_0 can moreover be chosen so that ℙ_n(X_0)→ℙ_n(X) is injective. We prove that the strand map is surjective if and only if Xnot≅ 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 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 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.
ER -