Unstable atomicity and loop spaces on Lie groups (Q1092449)

Some variants of the definition of atomicity are introduced. The one adopted here is that a space is p-atomic if every self-map is either an equivalence or induces a nilpotent homomorphism in reduced homology with mod p coefficients; this avoids the one-dimensional restriction of the original definition. It is shown that if one excludes the spinors, then the space of loops on a simply connected simple Lie group is 2-atomic. A deeper result is established for G an exceptional Lie group. It is shown that the action of the Steenrod algebra on \(H^ *(\Omega G, {\mathbb{Z}}_ 2)\) and the ring structure of \(H^ *(\Omega G, {\mathbb{Z}}_{(2)})\) are together sufficient to imply that the homomorphism induced on \(H^ *(\Omega G, {\mathbb{Z}}_ 2)\) by any self-map of \(\Omega\) G coincides with that induced by a power map; indeed the near-ring of all such homomorphisms is the ring \({\mathbb{Z}}_ n\) for small n. In particular, any induced algebra homomorphism of \(H^ *(\Omega G, {\mathbb{Z}}_ 2)\) is also a coalgebra homomorphism.