The hunting of the Hopf ring

Abstract

We provide a new algebraic description of the structure on the set of all unstable cohomology operations for a suitable generalised cohomology theory, $E*(–)$. Our description is as a graded and completed version of a Tall-Wraith monoid. The $E*$-cohomology of a space $X$ is a module for this Tall-Wraith monoid. We also show that the corresponding Hopf ring of unstable co-operations is a module for the Tall-Wraith monoid of unstable operations. Further examples are provided by considering operations from one theory to another.