Models for configuration space in a simplicial complex

Abstract

We produce combinatorial models for configuration space in a simplicial complex, and for configurations near a single point ("local configuration space.") The model for local configuration space is built out of the poset of poset structures on a finite set. The model for global configuration space relies on a combinatorial model for a simplicial complex with a deleted subcomplex. By way of application, we study the nodal curve y 2 z = x 3 + x 2 z, obtaining a presentation for its two-strand braid group, a conjectural presentation for its three-strand braid group, and presentations for its twoand three-strand local braid groups near the singular point.

On the second homology of planar graph braid groups

“…As observed independently in [9] and [15], the space $$B_3({\Theta }_{4})$$ has the homotopy type of a surface of genus 3, whose fundamental class cannot be represented by a map from a torus — here, $${\Theta }_{4}$$ is the suspension of four points. Our main result is that, in the planar case, this theta class is the only ‘exotic’ generator in degree 2.…”

“…In this section, we give an elementary description of the non‐toric class in $$H_2(B_3({\Theta }_{4}))$$ discovered in [9] and [15] in terms of the combinatorics of the Świątkowski complex. We begin with a simple lemma.…”