# ALGEBRAIC COMBINATORICS

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 $n$th discrete homotopy group ${A}_{n}\left(G\right)$ is trivial for all $n\ge 2$. Second we exhibit for each $n\ge 1$ a natural homomorphism $\psi :{A}_{n}\left(G\right)\to {ℋ}_{n}\left(G\right)$, where ${ℋ}_{n}\left(G\right)$ is the $n$th discrete cubical singular homology group, and an infinite family of graphs $G$ for which ${ℋ}_{n}\left(G\right)$ is nontrivial and $\psi$ is surjective. It follows that for each $n\ge 1$ there are graphs $G$ for which ${A}_{n}\left(G\right)$ is nontrivial.

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/

[1] Babson, Eric; Barcelo, Hélène; de Longueville, Mark; Laubenbacher, Reinhard Homotopy theory of graphs, J. Algebraic Combin., Volume 24 (2006) no. 1, pp. 31-44 | Article | MR 2245779 | Zbl 1108.05030

[2] Baillif, Mathieu; Gabard, Alexandre Manifolds: Hausdorffness versus homogeneity, Proc. Amer. Math. Soc., Volume 136 (2008) no. 3, pp. 1105-1111 | Article | MR 2361887 | Zbl 1140.57014

[3] Barcelo, Hélène; Capraro, Valerio; White, Jacob A. Discrete homology theory for metric spaces, Bull. Lond. Math. Soc., Volume 46 (2014) no. 5, pp. 889-905 | Article | MR 3262192 | Zbl 1308.55004

[4] Barcelo, Hélène; Greene, Curtis; Jarrah, Abdul Salam; Welker, Volkmar Discrete cubical and path homologies of graphs, Algebr. Comb., Volume 2 (2019) no. 3, pp. 417-437 | Article | MR 3968745 | Zbl 1414.05280

[5] Barcelo, Hélène; Greene, Curtis; Jarrah, Abdul Salam; Welker, Volkmar On the vanishing of discrete singular cubical homology for graphs (2019) (https://arxiv.org/abs/1909.02901)

[6] Barcelo, Hélène; Kramer, Xenia; Laubenbacher, Reinhard; Weaver, Christopher Foundations of a connectivity theory for simplicial complexes, Adv. in Appl. Math., Volume 26 (2001) no. 2, pp. 97-128 | Article | MR 1808443 | Zbl 0984.57014

[7] Barcelo, Hélène; Laubenbacher, Reinhard Perspectives on $A$-homotopy theory and its applications, Discrete Math., Volume 298 (2005) no. 1-3, pp. 39-61 | Article | MR 2163440 | Zbl 1082.37050

[8] Barcelo, Hélène; Severs, Christopher; White, Jacob A. $k$-parabolic subspace arrangements, Trans. Amer. Math. Soc., Volume 363 (2011) no. 11, pp. 6063-6083 | Article | MR 2817419 | Zbl 1234.52016

[9] Barcelo, Hélène; Smith, Shelly The discrete fundamental group of the order complex of ${B}_{n}$, J. Algebraic Combin., Volume 27 (2008) no. 4, pp. 399-421 | Article | MR 2393249 | Zbl 1200.05257

[10] Blakers, Albert L.; Massey, William S. The homotopy groups of a triad. II, Ann. of Math. (2), Volume 55 (1952), pp. 192-201 | Article | MR 44836 | Zbl 0046.40604

[11] Hatcher, Allen Algebraic topology, Cambridge University Press, Cambridge, 2002, xii+544 pages | MR 1867354 | Zbl 1044.55001

[12] Hurewicz, Witold Homotopie und Homologiegruppen, Proc. Akad. Wetensch. Amsterdam, Volume 38 (1935), pp. 521-528

[13] tom Dieck, Tammo Algebraic topology, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2008, xii+567 pages | Article | MR 2456045 | Zbl 1156.55001