Path homology of circulant digraphs

Abstract

We study GLMY path homology of circulant digraphs through step-word complexes and infinite cyclic lifts. We first prove chain retractions for interval connection sets. For a fixed finite positive connection set $S$ containing $1$, the infinite lift yields a split exact sequence and all-degree stability over every field. We then encode the short additive relations of $S$ in a lattice lift and obtain exterior homology in arbitrarily high degrees for collision-free step sets. Under an exact Freiman admissibility condition, we classify all connection sets with three positive steps. For arbitrarily many steps, we compute rectangular additive configurations and arithmetic progressions over every field, and doubling strings in characteristic zero, including their boundary ranks.

arXiv comment32 pages, 3 figures. Comments are welcome