Higher discrete homotopy groups of graphs

Lutz, Bob

doi:10.5802/alco.151

Geodesic

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Higher discrete homotopy groups of graphs

Lutz, Bob ¹

¹ Life Cycle Engineering, Inc. 4900 S. Broad St. Philadelphia, PA 19112, USA

Algebraic Combinatorics, Tome 4 (2021) no. 1, pp. 69-88.

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Résumé

This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if $G$ is a graph containing no 3- or 4-cycles, then the $n$ th discrete homotopy group $A_{n} (G)$ is trivial for all $n \geq 2$ . Second we exhibit for each $n \geq 1$ a natural homomorphism $ψ : A_{n} (G) \to ℋ_{n} (G)$ , where $ℋ_{n} (G)$ is the $n$ th discrete cubical singular homology group, and an infinite family of graphs $G$ for which $ℋ_{n} (G)$ is nontrivial and $ψ$ is surjective. It follows that for each $n \geq 1$ there are graphs $G$ for which $A_{n} (G)$ is nontrivial.

Reçu le : 2020-03-30
Accepté le : 2020-09-12
Publié le : 2021-02-16

DOI : 10.5802/alco.151

Classification : 05C99, 55Q99
Mots-clés : Discrete homotopy, discrete singular cubical homology,

A

-theory, Hurewicz theorem

Affiliations des auteurs :

Lutz, Bob ¹

¹ Life Cycle Engineering, Inc. 4900 S. Broad St. Philadelphia, PA 19112, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Lutz, Bob. Higher discrete homotopy groups of graphs. Algebraic Combinatorics, Tome 4 (2021) no. 1, pp. 69-88. doi : 10.5802/alco.151. https://geodesic-test.mathdoc.fr/articles/10.5802/alco.151/

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Cité par

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