2020/12/10 by Chih, Tien, Scull, Laura
#05C25 #05C30 #05C38 #05C60 #05E18 #20L05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2012.05378
We develop a theory of ×-homotopy, fundamental groupoids and covering spaces that apply to non-simple graphs, generalizing existing results for simple graphs. We prove that ×-homotopies from finite graphs can be decomposed into moves which adjust at most one vertex at a time, generalizing the spider lemma of \citeCS1. We define a notion of homotopy covering map and develop a theory of universal covers and deck transformations, generalizing \citesTardifWroncha, 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 \citesMatsushita, TardifWroncha.