ALGEBRAIC COMBINATORICS

no. 1

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 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: https://doi.org/10.5802/alco.151
Classification: 05C99,  55Q99
Keywords: Discrete homotopy, discrete singular cubical homology, A-theory, Hurewicz theorem 
@article{ALCO_2021__4_1_69_0,
     author = {Lutz, Bob},
     title = {Higher discrete homotopy groups of graphs},
     journal = {Algebraic Combinatorics},
     pages = {69--88},
     publisher = {MathOA foundation},
     volume = {4},
     number = {1},
     year = {2021},
     doi = {10.5802/alco.151},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.151/}
}
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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