Power maps on $p$-regular Lie groups

Abstract

A simple, simply-connected, compact Lie group $G$ is $p$-regular if it is homotopy equivalent to a product of spheres when localized at $p$. If $A$ is the corresponding wedge of spheres, then it is well known that there is a $p$-local retraction of $G$ off $\Omega\Sigma A$. We show that that complementary factor is very well behaved, and this allows us to deduce properties of $G$ from those of $\Omega\Sigma A$. We apply this to show that, localized at $p$, the $p$^th-power map on $G$ is an $H$-map. This is a significant step forward in Arkowitz-Curjel and McGibbon's programme for identifying which power maps between finite $H$-spaces are $H$-maps.