In this paper we determine the topological complexity of configuration spaces of graphs which are not necessarily trees, which is a crucial assumption in previous results. We do this for two very different classes of graphs: fully articulated graphs and banana graphs.We also complete the computation in the case of trees to include configuration spaces with any number of points, extending a proof of Farber.At the end we show that an unordered configuration space on a graph does not always have the same topological complexity as the corresponding ordered configuration space (not even when they are both connected). Surprisingly, in our counterexamples the topological complexity of the unordered configuration space is in fact smaller than for the ordered one.Definition 1.1. The topological complexity of X, denoted TC(X), is defined to be the minimal k such that X × X admits a cover by k + 1 open sets U 0 , U 1 , . . . , U k , on each of which there exists a local section of p X (that is, a continuous mapNote that here we use the reduced version of TC(X), which is one less than the original definition by Farber.
We determine the topological complexity of unordered configuration spaces on almost all punctured surfaces (both orientable and nonorientable). We also give improved bounds for the topological complexity of unordered configuration spaces on all aspherical closed surfaces, reducing it to three possible values. The main methods used in the proofs were developed in 2015 by Grant, Lupton and Oprea to give bounds for the topological complexity of aspherical spaces. As such this paper is also part of the current effort to study the topological complexity of aspherical spaces and it presents many further examples where these methods strongly improve upon the lower bounds given by zero-divisor cup-length. arXiv:1712.07068v3 [math.AT]
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