Homotopy commutativity in p-localized gauge groups

Abstract

Let G be a simple, compact Lie group and let $\mathcal{G}_k(G)$ be the gauge group of the principal G-bundle over S4 with second Chern class k. McGibbon classified the groups G that are homotopy commutative when localized at a prime p. We show that in many cases the homotopy commutativity of G, or its failure, determines that of $\mathcal{G}_k(G)$.