On the stabilization of the topological complexity of graph braid groups

TL;DR

This paper proves that for most graphs, the topological complexity of their unordered configuration spaces stabilizes at a maximum value, with bounds depending on the graph's structure, extending known stability phenomena.

Contribution

It establishes a geometric lower bound on the topological complexity of graph configuration spaces and demonstrates its stabilization at a maximal value for most graphs.

Findings

Topological complexity stabilizes at the maximum for most graphs.

Stable range depends on the number of trivalent vertices.

Provides a geometric lower bound for the topological complexity.

Abstract

We establish a strong, geometric lower bound on the (sequential) topological complexity of the unordered configuration spaces of a general graph. As an application, we show that, for most graphs, the topological complexity eventually stabilizes at its maximal possible value, a direct analogue of a stability phenomenon in the ordered setting first conjectured by Farber. We estimate the stable range in terms of the number of trivalent vertices.