This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if is a graph containing no 3- or 4-cycles, then the th discrete homotopy group is trivial for all . Second we exhibit for each a natural homomorphism , where is the th discrete cubical singular homology group, and an infinite family of graphs for which is nontrivial and is surjective. It follows that for each there are graphs for which is nontrivial.
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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 = {http://www.numdam.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. http://www.numdam.org/articles/10.5802/alco.151/
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