Calculating Higher Digraph Homotopy Groups
Stephen Theriault, Jie Wu, Shing-Tung Yau, Mengmeng Zhang
TL;DR
The paper develops a systematic framework for higher digraph homotopy groups by introducing relative digraph homotopy groups, digraph suspension, and a digraph Hurewicz map. It proves a long exact sequence for based digraph pairs, constructs a compatible suspension that interplays with cubical and GLMY homology, and establishes that the Hurewicz map commutes with suspension. This machinery reduces the problem of computing higher digraph homotopy groups to degree-1 path homology and enables explicit existence results: for every $n\ge1$ there are digraphs with nontrivial $\overline{\pi}_n$ including $\mathbb{Z}$-summands, higher-rank free parts, and torsion. Consequently, nontrivial higher digraph homotopy groups exist in great variety, enriching GLMY theory and bridging digraph topology with combinatorial homology. The results also suggest a path toward broader computations via the abelianization of $\pi_1$ and suspensions, opening avenues for further torsion phenomena in higher degrees.
Abstract
We give the first tractable and systematic examples of nontrivial higher digraph homotopy groups. To do this we define relative digraph homotopy groups and show these satisfy a long exact sequence analogous to the relative homotopy groups of spaces. We then define digraph suspension and Hurewicz homomorphisms and show they commute with each other. The existence of nontrivial digraph homotopy groups then reduces to the existence of corresponding groups in the degree 1 path homology of digraphs.