We show that the homology of ordered configuration spaces of finite trees with loops is torsion free. We introduce configuration spaces with sinks, which allow for taking quotients of the base space. Furthermore, we give a concrete generating set for all homology groups of configuration spaces of trees with loops and the first homology group of configuration spaces of general finite graphs. An important technique in the paper is the identification of the E 1 -page and differentials of Mayer-Vietoris spectral sequences for configuration spaces.
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 present a new computing package Flagser, designed to construct the directed flag complex of a finite directed graph, and compute persistent homology for flexibly defined filtrations on the graph and the resulting complex. The persistent homology computation part of Flagser is based on the program Ripser by U. Bauer, but is optimised specifically for large computations. The construction of the directed flag complex is done in a way that allows easy parallelisation by arbitrarily many cores. Flagser also has the option of working with undirected graphs. For homology computations Flagser has an approximate option, which shortens compute time with remarkable accuracy. We demonstrate the power of Flagser by applying it to the construction of the directed flag complex of digital reconstructions of brain microcircuitry by the Blue Brain Project and several other examples. In some instances we perform computation of homology. For a more complete performance analysis, we also apply Flagser to some other data collections. In all cases the hardware used in the computation, the use of memory and the compute time are recorded.
We consider for two based graphs G and K the sequence of graphs G k given by the wedge sum of G and k copies of K. These graphs have an action of the symmetric group Σ k by permuting the K-summands. We show that the sequence of representations of the symmetric group Hq(Confn(G•); Q), the homology of the ordered configuration space of these spaces, is representation stable in the sense of Church and Farb. In the case where G and K are trees (with loops), we provide a similar result for glueing along arbitrary subtrees instead of the base point. Furthermore, we show that stabilization alway holds for q = 1.
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