On Lewis's homotopy in distance-regular graphs
Abstract
In 2000, Lewis introduced a nested sequence of normal subgroups of the fundamental group of a distance-regular graph Γ. While the full homotopy group itself is simply a free group, it is interesting to determine the smallest integer k such that every closed walk in Γ can be built, up to homotopy, from closed walks of "essential length" k or less. Lewis proved that k≤ 6 for any Q-polynomial distance-regular graph Γ that is not a pseudoquotient. In this paper, we prove that k=3 for Johnson graphs and that k=4 in the case of Hamming graphs. We also compute the homotopy groups of natural cell complexes associated to the folded Johnson graphs and folded cubes. The paper finishes with some open questions and suggestions for further lines of investigation.
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BibTeX
@misc{burkey2026lewiss,
title = {{On Lewis's homotopy in distance-regular graphs}},
author = {MollyAnn Burkey and William J. Martin and Freud Oulon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33869},
eprint = {2610.33869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33869},
}
RIS
TY - UNPB
TI - On Lewis's homotopy in distance-regular graphs
AU - MollyAnn Burkey
AU - William J. Martin
AU - Freud Oulon
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33869
UR - https://arxiv.org/abs/2610.33869
AB - In 2000, Lewis introduced a nested sequence of normal subgroups of the fundamental group of a distance-regular graph Γ. While the full homotopy group itself is simply a free group, it is interesting to determine the smallest integer k such that every closed walk in Γ can be built, up to homotopy, from closed walks of "essential length" k or less. Lewis proved that k≤ 6 for any Q-polynomial distance-regular graph Γ that is not a pseudoquotient. In this paper, we prove that k=3 for Johnson graphs and that k=4 in the case of Hamming graphs. We also compute the homotopy groups of natural cell complexes associated to the folded Johnson graphs and folded cubes. The paper finishes with some open questions and suggestions for further lines of investigation.
ER -