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The Hunting of the Hopf Ring

2007/11/23 by Andrew Stacey, Stacey, Andrew, Sarah Whitehouse +1
Mathematics · #16W99 #55N20 #55S25 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.AT #math.KT #math.RA #msc:16W99 #msc:55N20 #msc:55S25

paper · pdf · doi:10.48550/arxiv.0711.3722

61 pages, no figures; uses xy, pxfonts

arxiv created 2007/11/23 · openalex publication_date 2007/11/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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