Homotopy Covers of Graphs
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866914391509696512 |
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| author | Chih, Tien Scull, Laura |
| author_facet | Chih, Tien Scull, Laura |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_05378 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Homotopy Covers of Graphs Chih, Tien Scull, Laura Combinatorics 05C25, 05E18, 05C38, 20L05, 05C30, 05C60 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}. |
| title | Homotopy Covers of Graphs |
| topic | Combinatorics 05C25, 05E18, 05C38, 20L05, 05C30, 05C60 |
| url | https://arxiv.org/abs/2012.05378 |