## 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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### References:

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