## Power maps on $$p$$-regular Lie groups.(English)Zbl 1280.55006

Author’s 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.

### MSC:

 55P35 Loop spaces 55T99 Spectral sequences in algebraic topology

### Keywords:

Lie group; $$p$$-regular; power map
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