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

Theriault, Stephen

Homology Homotopy Appl. 15, No. 2, 83-102 (2013).

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.

Reviewer: Jean Claude Thomas (Angers)

MSC:

55P35	Loop spaces
55T99	Spectral sequences in algebraic topology

Keywords:

Lie group; \(p\)-regular; power map