Homotopy Covers of Graphs

TL;DR

This paper extends the theory of homotopy, fundamental groupoids, and covering spaces to non-simple graphs, providing new decomposition methods and a universal cover theory applicable to a broader class of graphs.

Contribution

It generalizes existing homotopy and covering space theories from simple to non-simple graphs, including universal covers and homotopy lifting properties.

Findings

Decomposition of homotopies into single-vertex moves

Definition of homotopy covering maps and universal covers

Homotopy lifting property for non-simple graphs

Abstract

We develop a theory of $\times$-homotopy, fundamental groupoids and covering spaces that apply to non-simple graphs, generalizing existing results for simple graphs. We prove that $\times$-homotopies from finite graphs can be decomposed into moves which adjust at most one vertex at a time, generalizing the spider lemma of \cite{CS1}. We define a notion of homotopy covering map and develop a theory of universal covers and deck transformations, generalizing \cites{TardifWroncha, Matsushita} to non-simple graphs. We examine the case of reflexive graphs, where each vertex has at least one loop. We also prove that these homotopy covering maps satisfy a homotopy lifting property for arbitrary graph homomorphisms, generalizing path lifting results of \cites{Matsushita, TardifWroncha}.