The topological complexity of pure graph braid groups is stably maximal

Abstract

We prove Farber’s conjecture on the stable topological complexity of configuration spaces of graphs. The conjecture follows from a general lower bound derived from recent insights into the topological complexity of aspherical spaces. Our arguments apply equally to higher topological complexity.

On the stabilization of the topological complexity of graph braid groups

“…By construction, if a = 1 and v ∈ W 2 , then the restriction of τ a W to the factor in question factors through the homomorphism induced by ϕ 1 v , which is a constant map. In the case v = w, the argument is essentially identical to that of [Knu22,Lem. 6.1], the essential point being that the image of the embedding ϕ a v : S d(v) → Γ is contained in the closed star of v, hence disjoint from the open star of w by our assumption on the subdivision of Γ.…”

“…We may take k 0 = 2m(Γ). What Farber conjectured, and what the author later proved [Knu22] and generalized to the "higher" topological complexity TC r [Rud10], is that these three statements hold provided Γ is connected with m(Γ) ≥ 2.…”

“…It is a standard fact that ǫ : S 1 → B 2 (S 3 ) is a homotopy equivalence. More precisely, we have the following (see [Knu22,Lem. 4.2], for example).…”

“…By arguing geometrically, we further circumvent the fearsome combinatorial challenges posed by the standard Morse theoretic presentations of these groups [FS05,FS12]. This geometric flavor is maintained through the use of configuration spaces with sinks, an approach inspired by [LRM19] and already present implicitly in [Knu22].…”