Nikola Milićević (11/06/2024): Homotopy and Singular Homology Groups of Finite Digraphs

Abstract: Computing algebraic invariants of topological spaces such as singular homology is extremely difficult due to the infinite dimensionality of the associated chain groups, even when the spaces themselves are finite. The same is true for higher homotopy groups. This complexity is a barrier not only in pure mathematics but also in applied settings such as data analysis and network science, where finite structures like graphs and digraphs are common. In this work, we extend McCord's classical theorem, which established a weak homotopy equivalence between finite topological spaces and finite simplicial complexes, to the broader context of finite digraphs. This extension is non-trivial because finite digraphs are not topological spaces, making the classical methods not applicable. This result allows us to compute higher homotopy groups, and singular homology groups, of finite digraphs by replacing them with weakly homotopy equivalent simplicial complexes.