A McCord-type theorem for pseudotopological spaces and directed graphs

Abstract We construct weak homotopy equivalences between the geometric realizations of directed Vietoris-Rips complexes and their underlying directed graphs, seen as pseudotopological spaces. Pseudotopological spaces are a generalization of (Čech) closure spaces which in turn generalize topological spaces, but they also include graphs and directed graphs as full subcategories, making them a natural bridge that connects classical algebraic topology with the more applied side of topology. This weak homotopy equivalence implies that singular homology groups of finite directed graphs can be efficiently calculated from finite combinatorial structures, despite their associated chain groups being infinite dimensional. Along the way, we establish analogues of classical results such as the existence of a long exact sequence for homotopy groups of pairs of pseudotopological spaces and that a weak homotopy equivalence induces isomorphisms for homology groups. This work is similar to the work of McCord for finite topological spaces but in the context of pseudotopological spaces. Our results also give a novel approach for studying (higher) homotopy groups of discrete mathematical structures such as (directed) graphs or digital images.