Higher discrete homotopy groups of graphs
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 n2. Second we exhibit for each n1 a natural homomorphism ψ:A n (G) n (G), where n (G) is the nth discrete cubical singular homology group, and an infinite family of graphs G for which n (G) is nontrivial and ψ is surjective. It follows that for each n1 there are graphs G for which A n (G) is nontrivial.

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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
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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. http://www.numdam.org/articles/10.5802/alco.151/

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