A discrete model for surface configuration spaces

One of the primary methods of studying the topology of configurations of points in a graph and configurations of disks in a planar region has been to examine discrete combinatorial models arising from the underlying spaces. Despite the success of these models in the graph and disk settings, they have not been constructed for the vast majority of surface configuration spaces. In this paper, we construct such a model for the ordered configuration space of $m$ points in an oriented surface $\Sigma$. More specifically, we prove that if we give $\Sigma$ a certain cube complex structure $K$, then the ordered configuration space of $m$ points in $\Sigma$ is homotopy equivalent to a subcomplex of $K^{m}$