Abrams's stable equivalence for graph braid groups

TL;DR

This paper extends Abrams's results on the deformation retraction of graph braid configuration spaces by relaxing the length conditions on paths between essential vertices, using discrete Morse theory.

Contribution

It generalizes Abrams's conditions for deformation retraction of graph braid spaces by applying Forman's discrete Morse theory to relax path length requirements.

Findings

The first condition on path length can be reduced from n+1 to n-1 edges.

The deformation retraction still holds under the relaxed conditions.

Discrete Morse theory provides the key tool for this generalization.

Abstract

In his PhD thesis, Abrams proved that, for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G deformation retracts to a compact subspace, the discretized n-strand configuration space, provided G satisfies two conditions: each path between distinct essential vertices (vertices of degree not equal to 2) is of length at least n+1 edges, and each path from a vertex to itself which is not nullhomotopic is of length at least n+1 edges. Using Forman's discrete Morse theory for CW-complexes, we show the first condition can be relaxed to require only that each path between distinct essential vertices is of length at least n-1.