Combinatorial Conditions for Directed Collapsing

While collapsibility of CW complexes dates back to the 1930s, collapsibility of  directed  Euclidean cubical complexes has not been well studied to date. The classical definition of collapsibility involves certain conditions on pairs of cells of the complex. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We call this type of collapse a  link-preserving directed collapse . In the undirected setting, pairs of cells are removed that create a deformation retract. In the directed setting, topological properties—in particular, properties of spaces of directed paths—are not always preserved. In this paper, we give computationally simple conditions for preserving the topology of past links. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of spaces of directed paths. Throughout, we provide illustrative examples.