The homology of configuration spaces of trees with loops

Chettih, Safia; Lütgehetmann, Daniel

doi:10.2140/agt.2018.18.2443

Cited by 17 publications

(18 citation statements)

References 12 publications

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“…The uncompactified configuration spaces of graphs are objects of intense study, see e.g. [Ghr01], [Abr00], [BF09], [KP12], [CL18], but they are quite far from the objects of study in this paper, and our techniques do not shed light on these spaces at this time.…”

Section: Introductionmentioning

confidence: 99%

Homology representations of compactified configurations on graphs applied to $\mathcal{M}_{2,n}$

Bibby¹,

Chan²,

Gadish³

et al. 2021

Preprint

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The homology of a compactified configuration space of a graph is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We construct an efficient free resolution for these homology representations. Using the Peter-Weyl Theorem for symmetric groups, we consider irreducible representations individually, vastly simplifying the calculation of these homology representations from the free resolution.As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces M 2,n , equivalently the rational homology of the tropical moduli spaces ∆ 2,n , as a representation of S n acting by permuting point labels for all n ≤ 10. We further give new multiplicity calculations for specific irreducible representations of S n appearing in cohomology for n ≤ 17. Our approach produces information about these homology groups in a range well beyond what was feasible with previous techniques.

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Section: Introductionmentioning

confidence: 99%

Homology representations of compactified configurations on graphs applied to $\mathcal{M}_{2,n}$

Bibby¹,

Chan²,

Gadish³

et al. 2021

Preprint

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“…It is unclear at this time whether or not H i (Conf sink n (G, V )) will have torsion if (G, V, E) is not a tree. The above corollary tells us that any torsion would have uniformly bounded exponent, and would necessary appear before some finite n. Note that the homologies of the usual configuration spaces of graphs are always torsion free [CL,Theorem 1], while the so-called unordered configuration spaces of graphs can have torsion in their homology [KP,Theorem 3.6]. We saw previously that H i (Conf sink n (G, V )) is always torsion free if (G, V, E) is a tree (Corollary 3.25) Our next goal will be to develop a better understanding of the polynomials p i j of Corollary 4.3.…”

Section: Proofmentioning

confidence: 89%

“…We note that DConf sink n (e 0 , {v 0 , v 1 }) is a 1-dimensional CW-complex, and so its fundamental group is free. The rank of π 1 (DConf sink n (e 0 , {v 0 , v 1 }) was computed in [CL,Proposition 2.2] to be (n − 2)2 n−1 + 1. Our result now follows by induction.…”

Section: Proofmentioning

confidence: 99%

Configuration Spaces of Graphs with Certain Permitted Collisions

Ramos

2018

Discrete Comput Geom

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If G is a graph with vertex set V , let Conf sink n (G, V ) be the space of n-tuples of points on G, which are only allowed to overlap on elements of V . We think of Conf sink n (G, V ) as a configuration space of points on G, where points are allowed to collide on vertices. In this paper, we attempt to understand these spaces from two separate, but closely related, perspectives. Using techniques of combinatorial topology we compute the fundamental groups and homology groups of Conf sink n (G, V ) in the case where G is a tree. Next, we use techniques of asymptotic algebra to prove statements about Conf sink n (G, V ), for general graphs G, whenever n is sufficiently large. It is proven that, for general graphs, the homology groups exhibit generalized representation stability in the sense of [R2].

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“…For more details on those classes, see for example [Lü17]. We now recall the construction of the Mayer-Vietoris spectral sequence for configuration spaces discussed in [CL18] .…”

Section: A Mayer-vietoris Spectral Sequence For Configuration Spacesmentioning

confidence: 99%

Topological Complexity of Configuration Spaces of Fully Articulated Graphs and Banana Graphs

Lütgehetmann

Recio-Mitter

2019

Discrete Comput Geom

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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.

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The homology of configuration spaces of trees with loops

Cited by 17 publications

References 12 publications

Homology representations of compactified configurations on graphs applied to $\mathcal{M}_{2,n}$

Homology representations of compactified configurations on graphs applied to $\mathcal{M}_{2,n}$

Configuration Spaces of Graphs with Certain Permitted Collisions

Topological Complexity of Configuration Spaces of Fully Articulated Graphs and Banana Graphs

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