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The homology of configuration spaces of trees with loops. (English) Zbl 1396.55012

The \(n^{\text{th}}\)-ordered configuration space of a topological space \(X\) is defined to be \[ \text{Conf}_n(X):=\{(x_1,\ldots,x_n) \in X^n\mid x_i\neq x_j \text{~for~ } i\neq j\}. \] The ordered configuration spaces are of particular importance in mathematics. For example, the ordered configuration space \(\text{Conf}_n(\mathbb{C})\) is a classifying space for the pure braid group \(P_n\), while the configuration space \(\text{Conf}_n(\Sigma_g)\) is a classifying space for \(P_{g,n}\), the pure braid group on a compact Riemann surface \(\Sigma_g\) of genus \(g\). Computing the (co)homology of configuration spaces is a popular question in algebraic topology. The paper under review considers the homology of the \(n^{\text{th}}\)-ordered configuration space of a finite connected graph.
Let \(G\) be a tree with loops, i.e., \(G\) can be constructed as an iterated wedge of star graphs and copies of circles. Then the main result of the paper shows that the integral homology \(H_q(\text{Conf}_n(G);\mathbb{Z})\) is torsion-free and generated by ‘products of basic classes’ for each \(q\geq 0\). The proof of the main theorem relies on constructing combinatorial models for configuration spaces with ‘sinks’, and the identification of the \(E_1\)-page and differentials of the Mayer-Vietoris spectral sequence for configuration spaces. Let \(G\) be a finite graph. The authors show that the first homology group \(H_1(\text{Conf}_n(G);\mathbb{Z})\) is generated by basic classes. For each \(i\geq 2\), the authors provide explicit examples to show that there exists a finite graph \(G\) and a number \(n\) such that \(H_i(\text{Conf}_n(G);\mathbb{Z})\) is not generated by products of \(1\)-classes. The authors conjecture that the integral homology \(H_q(\text{Conf}_n(G);\mathbb{Z})\) is torsion-free for each \(q\geq 0\).
Reviewer: He Wang (Reno)

MSC:

55R80 Discriminantal varieties and configuration spaces in algebraic topology
57M15 Relations of low-dimensional topology with graph theory
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