Homotopy Groups and Puppe Sequence of Digraphs
Jingyan Li, Jie Wu, Shing-Tung Yau, Mengmeng Zhang
TL;DR
The paper develops a grid-oriented digraph homotopy theory by introducing $n$-grid maps and the reduced homotopy groups $\overline{\pi}_n(G)$, providing a grid-compatible extension of GLMY theory. It defines $\overline{\pi}_n(G)$ as a direct limit over subdivisions of maps from the standard $n$-grid into $(G,*)$, endows these groups with a concatenation-based structure (abelian for $n\ge2$), and ties them to the reduced loop-digraph via $\overline{\pi}_n(G) \cong \overline{\pi}_{n-1}(\overline{L}G)$. A central contribution is the construction of a Puppe sequence for based digraph maps using the mapping path digraph $P_f$, with $\overline{P}G$ shown to be weakly contractible and a long exact sequence of $\overline{\pi}_n$ established (groups for $n\ge1$). These results yield a concrete, grid-based framework for analyzing higher-order structure in directed networks and set the stage for fibration- and fibre-bundle-inspired approaches in digraphs.
Abstract
We introduce homotopy groups of digraphs that admit an intuitive description of grid structures, which is a variation of the GLMY homotopy groups introduced by Grigor'yan, Lin, Muranov and Yau in 2014. This direct approach enables a descriptive interpretation of GLMY theory in applications such as network science. Furthermore, we prove that there exists a long exact sequence of homotopy groups of digraphs associated to any based digraph map, that is, there exists a digraph version of the Puppe sequence.