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On the stabilization of the topological complexity of graph braid groups

Ben Knudsen

TL;DR

The paper addresses the stabilization of topological complexity for unordered graph configuration spaces by deriving a strong geometric lower bound via disjoint-conjugate obstructions and local-graph analysis. It leverages configuration spaces with sinks, edge/sink stabilization, and toric/detection homomorphisms to reduce the problem to graph braid groups and aspherical spaces, yielding a universal lower bound and a stability result under a mild vertex-condition: for connected graphs with $m(\mathsf{\Gamma})\ge 2$ and no non-separating trivalent vertices, $(1/r)TC_r(B_k(\mathsf{\Gamma}))=m(\mathsf{\Gamma})$ for large $k$, with an explicit stable range $k_0=2m(\mathsf{\Gamma})$ plus the count of trivalent vertices. The methodology combines a geometric framework with group-theoretic TC_r results, providing a robust unordered-graph analogue to Farber's stability conjecture and expanding understanding of graph braid groups. This work also clarifies the role of local structures (local graphs) and stabilization operations in shaping the global TC_r$-invariants for graphs.

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.

On the stabilization of the topological complexity of graph braid groups

TL;DR

The paper addresses the stabilization of topological complexity for unordered graph configuration spaces by deriving a strong geometric lower bound via disjoint-conjugate obstructions and local-graph analysis. It leverages configuration spaces with sinks, edge/sink stabilization, and toric/detection homomorphisms to reduce the problem to graph braid groups and aspherical spaces, yielding a universal lower bound and a stability result under a mild vertex-condition: for connected graphs with and no non-separating trivalent vertices, for large , with an explicit stable range plus the count of trivalent vertices. The methodology combines a geometric framework with group-theoretic TC_r results, providing a robust unordered-graph analogue to Farber's stability conjecture and expanding understanding of graph braid groups. This work also clarifies the role of local structures (local graphs) and stabilization operations in shaping the global TC_r$-invariants for graphs.

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.
Paper Structure (11 sections, 18 theorems, 16 equations)

This paper contains 11 sections, 18 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Relation to other work
  3. Future directions
  4. Conventions and notation
  5. Configurations with controlled collisions
  6. Edge and sink stabilization
  7. Local graphs
  8. Motion planning in aspherical spaces
  9. Local calculation
  10. Toric and detection homomorphisms
  11. The main result

Key Result

Theorem 1.1

Let $\mathsf{\Gamma}$ be a connected graph with $m(\mathsf{\Gamma})\geq2$, and fix $r>0$. If $\mathsf{\Gamma}$ has no non-separating trivalent vertices, then there exists $k_0$ such that, for all $k\geq k_0$, we have the equality We may take $k_0$ to be $2m(\mathsf{\Gamma})$ plus the number of trivalent vertices of $\mathsf{\Gamma}$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Proposition 2.4
  • Example 2.5
  • Definition 2.6
  • Theorem 2.7: Abrams:CSBGGSwiatkowski:EHDCSGGhrist:CSBGGR
  • Remark 2.8
  • ...and 39 more