The odd primary order of the commutator on low rank Lie groups

Abstract

Let $G$ be a simply-connected, compact, simple Lie group of low rank relative to a fixed prime $p$. After localization at $p$, there is a space $A$ which "generates" $G$ in a certain sense. Assuming $G$ satisfies a homotopy nilpotency condition relative to $p$, we show that the Samelson product $⟨I{d}_G,I{d}_G⟩$ of the identity of $G$ equals the order of the Samelson product $⟨ı,ı⟩$ of the inclusion $ı:A\to G$. Applying this result, we calculate the orders of $⟨I{d}_G,I{d}_G⟩$ for all $p$-regular Lie groups and give bounds on the orders of $⟨I{d}_G,I{d}_G⟩$ for certain quasi-$p$-regular Lie groups.

arXiv comment18 pages; Accepted by Topology and its Applications