ALGEBRAIC COMBINATORICS

no. 1
Higher discrete homotopy groups of graphs
Lutz, Bob1
1 Life Cycle Engineering, Inc. 4900 S. Broad St. Philadelphia, PA 19112, USA
Algebraic Combinatorics, Volume 4 (2021) no. 1, pp. 69-88.

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 nth discrete homotopy group A n (G) is trivial for all n≥2. Second we exhibit for each n≥1 a natural homomorphism ψ:A n (G)→ℋ n (G), where ℋ n (G) is the nth 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≥1 there are graphs G for which A n (G) is nontrivial.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.151
Classification: 05C99, 55Q99
Keywords: Discrete homotopy, discrete singular cubical homology, $A$-theory, Hurewicz theorem
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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