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The real Brown-Peterson homology of $Ω^ρS^{ρ+ 1}$

Christian Carrick, Bertrand Guillou, Sarah Petersen

TL;DR

We compute the $RO(C_2)$-graded real Brown–Peterson homology of the representation-loop space $\, ext{Ω}^ ho S^{ ho+1}$, providing a $C_2$-equivariant analogue of Ravenel’s classical calculation for $ ext{Ω}^2 S^3$. The main input is the $A^{C_2}_igstar$-coaction on $ extbf{H}_igstar ext{Ω}^ ho S^{ ho+1}$, derived from Behrens–Wilson’s computation and organized via comodule Nishida relations for $ ho$-loop spaces. The calculation proceeds through a Snaith splitting and a Borel Adams spectral sequence (via an $a_\sigma$-Bockstein filtration) to produce an $E_2$-term description and show collapse, yielding explicit information about $ extbf{BP}_{R,*} ext{Ω}^ ho S^{ ho+1}$ and its restriction to the nonequivariant case. The results illuminate how equivariant Dieudonné-type phenomena might arise and provide a tractable route toward a larger equivariant Dieudonné theory in the $C_2$-setting.

Abstract

We compute the $RO(C_2)$-graded real Brown--Peterson homology of the representation-loop space $Ω^ρS^{ρ+ 1}$, where $ρ$ is the regular representation of the cyclic group of order two. This calculation gives a $C_2$-equivariant analogue of the classical computation of Brown--Peterson homology of the double loop space $Ω^2 S^3$ due to Ravenel. Along the way, we develop comodule Nishida relations for $ρ$-loop spaces.

The real Brown-Peterson homology of $Ω^ρS^{ρ+ 1}$

TL;DR

We compute the -graded real Brown–Peterson homology of the representation-loop space , providing a -equivariant analogue of Ravenel’s classical calculation for . The main input is the -coaction on , derived from Behrens–Wilson’s computation and organized via comodule Nishida relations for -loop spaces. The calculation proceeds through a Snaith splitting and a Borel Adams spectral sequence (via an -Bockstein filtration) to produce an -term description and show collapse, yielding explicit information about and its restriction to the nonequivariant case. The results illuminate how equivariant Dieudonné-type phenomena might arise and provide a tractable route toward a larger equivariant Dieudonné theory in the -setting.

Abstract

We compute the -graded real Brown--Peterson homology of the representation-loop space , where is the regular representation of the cyclic group of order two. This calculation gives a -equivariant analogue of the classical computation of Brown--Peterson homology of the double loop space due to Ravenel. Along the way, we develop comodule Nishida relations for -loop spaces.

Paper Structure

This paper contains 23 sections, 40 theorems, 191 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Notation
  4. Background
  5. The dual Steenrod Algebra
  6. Dyer-Lashof operations
  7. The $\mathcal{A}_*$-comodule ${\mathrm{H}_* \Omega^2 S^3}$
  8. The $C_2$-equivariant Steenrod algebra
  9. Equivariant Dyer-Lashof operations
  10. Comparison to Wilson's stable operations
  11. The Cartan formula
  12. The equivariant Nishida relations
  13. The $\mathcal{A}^{{\mathrm{C}_2}}_\bigstar$-comodule $\mathbf{H}_\bigstar \Omega^\rho S{^{\rho+1}}$
  14. The $\mathcal{A}^h_\bigstar$-comodule $\mathbf{H}^h_\bigstar \Omega^\rho S{^{\rho+1}}$
  15. The Snaith Splitting
  16. ...and 8 more sections

Key Result

Theorem A

Let $X$ be an $E_\rho$-algebra. Then for $x\in \mathrm{H}_{k\rho}X$ and $y\in \mathrm{H}_{k\rho+1}X$, the (right) coaction satisfies where ${\color{purple!50!blue}{\boldsymbol{\tau}}}_0\in \mathcal{A}^{{\mathrm{C}_2}}_\bigstar$ is in degree 1.

Figures (5)

  • Figure 4.1: The $RO(C_2)$-graded coefficients $\mathbf{H}_\bigstar$ and $\mathbf{H}_\bigstar^h$, as described in \ref{['eq:Hcoeffs']}. The class $a_\sigma$ is in degrees $n+k=-1$ and $k=-1$ while $u_\sigma$ is in degrees $n+k=0$ and $k=-1$.
  • Figure 4.2: The $C_2$-equivariant dual Steenrod algebra $\mathcal{A}^{{\mathrm{C}_2}}_{n+k\sigma}$, using the "motivic" grading in which the vertical direction indicates multiples of $\sigma$ and the horizontal is the underlying topological dimension. Here $\mathcal{E}^{{\mathrm{C}_2}}_\bigstar$ is indicated in blue. Each copy of $\mathbf{H}_\bigstar$ contributes a "positive" cone (pointing down) and a "negative" cone (point up).
  • Figure 4.3: The Borel $C_2$-equivariant dual Steenrod algebra $\mathcal{A}^{h{\mathrm{C}_2}}_{n+k\sigma}$, using the "motivic" grading in which the vertical direction indicates multiples of $\sigma$ and the horizontal is the underlying topological dimension. Here $\mathcal{E}^{h{\mathrm{C}_2}}_\bigstar$ is indicated in blue. Each copy of $\mathbf{H}_\bigstar^h$ contributes a left half-plane, the $u_\sigma$-localization of the positive cone in $\mathbf{H}_\bigstar$.
  • Figure 12.1: The $E_2$-page for summands of $BP\mathbb{R}_\bigstar\Omega^\rho S^{\rho+1}$
  • Figure 12.2:

Theorems & Definitions (80)

  • Theorem A
  • Theorem B: \ref{['cor:EinftyBSS']}
  • Theorem C: \ref{['cor:adamscollapse']}
  • Remark 1.1
  • Theorem D: \ref{['prop:evenoddsplit', 'prop:resinjective']}
  • Remark 1.2
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Proposition 4.1: HK,Voev
  • ...and 70 more