Asphericity and Stable Complexity of Graph Configuration Spaces with Sinks

Let \(G\) be a finite graph and let \(S\subseteq V(G)\) be a set of vertices at which collisions are permitted. We prove that every path component of the ordered sink configuration space \(\Conf_n(G,S)\) is aspherical by applying the cubical link criterion to the Chettih--Lütgehetmann model. The unordered analogue fails. For the \(m\)-petal rose \(R_m\) with its unique vertex \(v\) declared a sink, we construct a deformation retraction \(\BConf_n(R_m,\{v\})\simeq \sk_{\min\{n,m\}}(\T^m)\). Consequently this unordered space is aspherical exactly for \(n\leq1\) or \(n\geq m\). If \(2\leq d=\min\{n,m\}<m\), its first nonzero higher homotopy group is the image of the \((d+1)\)-st differential in the Laurent--Koszul resolution of \(\Z\) over \(\Z[\Z^m]\), and has module rank \(\binom{m-1}{d}\). In particular, \(\pi_2(\BConf_2(R_3,\{v\}))\cong\Z[\Z^3]\). Finally, for reduced sequential topological complexity and every \(r\geq2\), \(\TC_r\bigl(\BConf_n(R_m,\{v\})\bigr)=\min\{r\min(n,m),(r-1)m\}\). The sharp numerical stabilization threshold is \(\lceil(r-1)m/r\rceil\), while the homotopy type stabilizes at \(n=m\). Any finite torsion-free cover used to remove the residual orbifold isotropy is shown to have degree divisible by the least common multiple of the local stabilizer orders, giving a sharp arithmetic obstruction.