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