Additive power operations in equivariant cohomology

2020/01/29 by Bonventre, Peter J., Guillou, Bertrand J., Stapleton, Nathaniel J.
#55N91 #55S25 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2001.11078

Abstract

Let G be a finite group and E be an H_∞-ring G-spectrum. For any G-space X and positive integer m, we give an explicit description of the smallest Mackey ideal \underlineJ in \underlineE⁰(X× BΣ_m) for which the reduced mth power operation \underlineE⁰(X) → \underlineE⁰(X × BΣ_m )/\underlineJ is a map of Green functors. We obtain this result as a special case of a general theorem that we establish in the context of G×Σ_m-Green functors. This theorem also specializes to characterize the appropriate ideal \underlineJ when E is an ultra-commutative global ring spectrum. We give example computations for the sphere spectrum, complex K-theory, and Morava E-theory.

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