The directed Vietoris-Rips complex and homotopy and singular homology groups of finite digraphs

TL;DR

This paper develops a framework connecting algebraic topology with directed graphs by constructing homotopy equivalences between directed Vietoris-Rips complexes and graphs, enabling efficient computation of homology groups.

Contribution

It introduces a novel approach to analyze homotopy and homology groups of directed graphs using pseudotopological spaces and directed Vietoris-Rips complexes.

Findings

Existence of long exact sequences for homotopy groups of pseudotopological spaces

Weak homotopy equivalences induce isomorphisms in homology groups

Efficient computation of singular homology for finite directed graphs

Abstract

We prove analogues of classical results for higher homotopy groups and singular homology groups of pseudotopological spaces. Pseudotopological spaces are a generalization of (\v{C}ech) closure spaces which are in turn a generalization of topological spaces. Pseudotopological spaces also include graphs and directed graphs as full subcategories. Thus they are a bridge that connects classical algebraic topology with the more applied side of topology. More specifically, we show 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. Our main result is the construction of weak homotopy equivalences between the geometric realizations of directed Vietoris-Rips complexes and their underlying directed graphs. This 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. 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.