Estimates for homological dimension of configuration spaces of graphs (Q2717742)

For a finite graph \(\Gamma\) and a natural number \(n \geq 2\), the marked \(n\)-point configuration space of \(\Gamma\) is the subspace \(\widetilde{C_n\Gamma} = \{ (x_1, \cdots ,x_n) \in \Gamma\mid x_i \neq x_j\) for \(i \neq j \}\) of the product space \(\Gamma^n\), and the \(n\)-point configuration space \(C_n\Gamma\) is given by the orbit space of \(\widetilde{C_n\Gamma}\) under the natural free action of the symmetric group \(S_n\) defined by \(\sigma(x_1,\cdots,x_n) = (x_{\sigma(1)},\cdots,x_{\sigma(n)})\). \(C_n\Gamma\) is a \(k(\pi,1)\)-space (see \textit{M. Davis, T. Januszkiewicz} and \textit{R. Scott} [Sel. Math., New Ser. 4, No. 4, 491-547 (1998; Zbl 0924.53033)] or this article). The author is interested in estimating the homological dimension of \(C_n\Gamma\). This is achieved by construction of a cube complex \(K_n\Gamma\) and its embedding into the configuration space \(C_n\Gamma\) as a deformation retract. By construction \(\dim K_n\Gamma = \min \{ b(\Gamma),n \}\), where \(b(\Gamma)\) is the number of branched vertices of \(\Gamma\). The author constructs locally isometric mappings of \(k\)-dimensional tori into \(K_n\Gamma\) for \(k \leq \min \{ b(\Gamma),[n/2] \}\), where \([n/2]\) is the integer part of \(n/2\). Hence \(\pi_1(K_n\Gamma)\) contains a subgroup isomorphic to \(\mathbb{Z}^k\). Since \(K_n\Gamma\) is a deformation retract of \(C_n\Gamma\), we can obtain interesting results for homological dimension of configuration spaces of finite graphs.