On Lewis's homotopy in distance-regular graphs

Source 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.