## Homology, Homotopy and Applications

### Power maps on $p$-regular Lie groups

Stephen Theriault

#### 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.

#### Article information

Source
Homology Homotopy Appl., Volume 15, Number 2 (2013), 83-102.

Dates
First available in Project Euclid: 8 November 2013

https://projecteuclid.org/euclid.hha/1383945277

Mathematical Reviews number (MathSciNet)
MR3117388

Zentralblatt MATH identifier
1280.55006

Subjects
Primary: 55P35: Loop spaces
Secondary: 55T99: None of the above, but in this section

Keywords
Lie group $p$-regular power map

#### Citation

Theriault, Stephen. Power maps on $p$-regular Lie groups. Homology Homotopy Appl. 15 (2013), no. 2, 83--102. https://projecteuclid.org/euclid.hha/1383945277