On the stable Hopf invariant

Abstract

We provide a simplified approach to the the stable Hopf invariant. We provide short elementary proofs of the Cartan Formula, the Composition Formula, and the Transfer formula. In addition, when $\ddot{I}€$ is a discrete group, we show how to extend these results to the stable category of $\ddot{I}€$-spaces. We also consider the extent to which the stable Hopf invariant is unique.

Added "Theorem E" which shows that an elementary axiomatization of the second Segal-Snaith Hopf invariant is unlikely. To appear in Proc. Roy. Soc. Edinburgh Sec. A