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Additive power operations in equivariant cohomology

Peter J. Bonventre, Bertrand J. Guillou, Nathaniel J. Stapleton

TL;DR

This work identifies the minimal Mackey-ideal $\\\underline{J}$ in $\\underline{E}^0(X \\times B\\Sigma_m)$ so that the reduced $m$th power operation $P_m$ becomes a map of Green functors in equivariant cohomology theories represented by $H_ obreak ext{-} obreak$r ing $G$-spectra (and its global analogue). It develops a general framework for additive power operations via $G \\times \\Sigma_m$-Green functors, introduces explicit transfer-based ideals $J^G$ and $J_H^G$ (and the induced Mackey ideal $\\underline{J}$), and proves their compatibility with transfers across all subgroups. The theory specializes to Borel, genuine, and global settings, with concrete computations in ordinary cohomology, the sphere spectrum, global $KU$, class functions, and height $2$ Morava $E$-theories, revealing how additivity and transfer-compatibility are governed by the described transfer ideals. Collectively, these results provide a robust algebraic-structure toolkit to understand and compute additive power operations in a broad spectrum of equivariant and global cohomology theories, including explicit phenomena at primes and in chromatic settings.

Abstract

Let $G$ be a finite group and $E$ be an $H_\infty$-ring $G$-spectrum. For any $G$-space $X$ and positive integer $m$, we give an explicit description of the smallest Mackey ideal $\underline{J}$ in $\underline{E}^0(X\times BΣ_m)$ for which the reduced $m$th power operation $\underline{E}^0(X) \to \underline{E}^0(X \times BΣ_m )/\underline{J}$ 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 $G\timesΣ_m$-Green functors. This theorem also specializes to characterize the appropriate ideal $\underline{J}$ when $E$ is a $G_\infty$-ring in global spectra. We give example computations for the sphere spectrum, complex $K$-theory, and Morava $E$-theory.

Additive power operations in equivariant cohomology

TL;DR

This work identifies the minimal Mackey-ideal in so that the reduced th power operation becomes a map of Green functors in equivariant cohomology theories represented by r ing -spectra (and its global analogue). It develops a general framework for additive power operations via -Green functors, introduces explicit transfer-based ideals and (and the induced Mackey ideal ), and proves their compatibility with transfers across all subgroups. The theory specializes to Borel, genuine, and global settings, with concrete computations in ordinary cohomology, the sphere spectrum, global , class functions, and height Morava -theories, revealing how additivity and transfer-compatibility are governed by the described transfer ideals. Collectively, these results provide a robust algebraic-structure toolkit to understand and compute additive power operations in a broad spectrum of equivariant and global cohomology theories, including explicit phenomena at primes and in chromatic settings.

Abstract

Let be a finite group and be an -ring -spectrum. For any -space and positive integer , we give an explicit description of the smallest Mackey ideal in for which the reduced th power operation 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 -Green functors. This theorem also specializes to characterize the appropriate ideal when is a -ring in global spectra. We give example computations for the sphere spectrum, complex -theory, and Morava -theory.

Paper Structure

This paper contains 24 sections, 47 theorems, 159 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conventions
  3. Organization
  4. Acknowledgments
  5. The Borel equivariant case
  6. Overview
  7. Stabilizers of elements in $(G/H)^{\times m}$
  8. The absolute transfer ideal $J^G$
  9. The relative ideal $J_H^G$ for normal subgroups $H \trianglelefteq G$
  10. Specializing to a prime
  11. The relatively prime case
  12. Additive power operations and Green functors
  13. Reminder on Mackey functors and Green functors
  14. tocCertain ideals in the induced $G$-Green functortocCertain ideals in $\hbox{${\underline{R}} \space \uparrow_{G\times \Sigma_m}^{G}$}$
  15. Power operations on Green functors
  16. ...and 9 more sections

Key Result

Proposition 1

Assume that $E$ is an $H_{\infty}$-ring in genuine $G$ spectra. The ideals $\ul{J}(G/H)$, defined above, assemble to a Mackey ideal $\ul{J} \subseteq \ul{E}^0(B\Sigma_p)$, minimal with the property that the composite is a map of $G$-Green functors.

Figures (1)

  • Figure 1: The $(C_2\times \Sigma_2)$-set $(nC_2+k)\times (nC_2+k)$

Theorems & Definitions (106)

  • Proposition
  • Theorem
  • Theorem
  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • ...and 96 more