Year: 2026
Type: preprint
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 $k\le 6$ 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.
Source: arXiv (Cornell University)
Institution: Worcester Polytechnic Institute
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7223821196