Higher discrete homotopy groups of graphs

Lutz, Bob¹

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

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

Résumé

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 $\psi :A_n\left(G\right)\to {\mathscr{H}}_n\left(G\right)$, where ${\mathscr{H}}_n\left(G\right)$ is the $n$th discrete cubical singular homology group, and an infinite family of graphs $G$ for which ${\mathscr{H}}_n\left(G\right)$ 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.

Détail

Reçu le : 2020-03-30
Accepté le : 2020-09-12
Publié le : 2021-02-16

DOI : 10.5802/alco.151

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

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits