On the path homology of Cayley digraphs and covering digraphs

TL;DR

This paper introduces a theory of covering digraphs and applies it to Cayley digraphs, establishing a connection between path homology and group homology, enabling simplified calculations and explicit examples.

Contribution

It develops a novel theory of covering digraphs, linking path homology to group homology, and demonstrates this with explicit computations for specific Cayley digraphs.

Findings

Path homology can be expressed in terms of group homology in certain cases.

A new filtered nerve construction for digraphs is introduced.

Explicit path homology computation for Cayley digraph of rationals with factorial inverses.

Abstract

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a "bridge" between GLMY-theory and group homology theory, which helps to reduce path homology calculations to group homology computations. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers with a generating set consisting of inverses to factorials. The main tool in our work is a filtered simplicial set associated with a digraph, which we call the filtered nerve of a digraph.