Homology, Homotopy and Applications

Volume 11 (2009)

Number 2

The hunting of the Hopf ring

Pages: 75 – 132

DOI: https://dx.doi.org/10.4310/HHA.2009.v11.n2.a6


Andrew Stacey (Institutt for Matematiske Fag, NTNU, Trondheim, Norway)

Sarah Whitehouse (Department of Pure Mathematics, University of Sheffield, United Kingdom)


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.


unstable cohomology operations; Hopf ring

2010 Mathematics Subject Classification

Primary 55S25. Secondary 16Wxx, 55N20.

Published 1 January 2009