Higher discrete homotopy groups of graphs

Lutz, Bob

doi:10.5802/alco.151

Lutz, Bob ¹

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

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

Abstract

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.

Received: 2020-03-30
Accepted: 2020-09-12
Published online: 2021-02-16

DOI: 10.5802/alco.151

Classification: 05C99, 55Q99
Keywords: Discrete homotopy, discrete singular cubical homology, $A$-theory, Hurewicz theorem

Author's affiliations:

Lutz, Bob ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Lutz, Bob. Higher discrete homotopy groups of graphs. Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 69-88. doi : 10.5802/alco.151. https://alco.centre-mersenne.org/articles/10.5802/alco.151/

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