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A spectrum-level splitting of the $ku_\mathbb{R}$-cooperations algebra

Guchuan Li, Sarah Petersen, Elizabeth Tatum

TL;DR

This work extends the classical nonequivariant splitting of the cooperations algebras for $ku$ and $ko$ to the $C_2$-equivariant setting by constructing a spectrum-level lift of Mahowald–Kane’s splitting. The authors prove a $C_2$-equivariant, $2$-completed decomposition $ku_{\mathbb{R}}\wedge ku_{\mathbb{R}} \simeq \bigvee_{k=0}^{\infty} ku_{\mathbb{R}}\wedge \Sigma^{\rho k}{\mathcal B}_0(k)$, and describe it via $C_2$-equivariant Adams covers, along with a parallel splitting for $H\underline{\mathbb{Z}}\wedge H\underline{\mathbb{Z}}$. The paper develops extensive $C_2$-equivariant homology technology, including a Whitehead theorem for Margolis homology, a computation of $H_{\starBP_{\mathbb{R}}\langle n\rangle}$, and the structure of $ku_{\mathbb{R}}$-cooperations and $[ku_{\mathbb{R}},ku_{\mathbb{R}}]$-operations. These results yield explicit $E_1$-pages for the $ku_{\mathbb{R}}$-based Adams spectral sequence and lay groundwork for higher-height $C_2$-equivariant Brown–Gitler theory, offering powerful computational tools for $C_2$-equivariant stable stems and $v_1$-periodicity problems. The techniques bridge nonequivariant decompositions with equivariant Adams spectral sequence analysis, enabling new calculations in the $C_2$-equivariant realm and setting the stage for future explorations of $ko_{C_2}$-based methods and higher-height analogues.

Abstract

In the 1980's, Mahowald and Kane used integral Brown--Gitler spectra to decompose $ku \wedge ku$ as a sum of finitely generated $ku$-module spectra. This splitting, along with an analogous decomposition of $ko \wedge ko$ led to a great deal of progress in stable homotopy computations and understanding of $v_1$-periodicity in the stable homotopy groups of spheres. In this paper, we construct a $C_2$-equivariant lift of Mahowald and Kane's splitting of $ku \wedge ku$. We also give a description of the resulting $C_2$-equivariant splitting in terms of $C_2$-equivariant Adams covers and record an analogous splitting for $H\underline{\mathbb{Z}} \wedge H \underline{\mathbb{Z}}$. Similarly to the nonequivariant story, we expect the techniques of this paper to facilitate further $C_2$-equivariant stable homotopy computations and understanding of $v_1$-periodicity in $C_2$-equivariant stable stems.

A spectrum-level splitting of the $ku_\mathbb{R}$-cooperations algebra

TL;DR

This work extends the classical nonequivariant splitting of the cooperations algebras for and to the -equivariant setting by constructing a spectrum-level lift of Mahowald–Kane’s splitting. The authors prove a -equivariant, -completed decomposition , and describe it via -equivariant Adams covers, along with a parallel splitting for . The paper develops extensive -equivariant homology technology, including a Whitehead theorem for Margolis homology, a computation of , and the structure of -cooperations and -operations. These results yield explicit -pages for the -based Adams spectral sequence and lay groundwork for higher-height -equivariant Brown–Gitler theory, offering powerful computational tools for -equivariant stable stems and -periodicity problems. The techniques bridge nonequivariant decompositions with equivariant Adams spectral sequence analysis, enabling new calculations in the -equivariant realm and setting the stage for future explorations of -based methods and higher-height analogues.

Abstract

In the 1980's, Mahowald and Kane used integral Brown--Gitler spectra to decompose as a sum of finitely generated -module spectra. This splitting, along with an analogous decomposition of led to a great deal of progress in stable homotopy computations and understanding of -periodicity in the stable homotopy groups of spheres. In this paper, we construct a -equivariant lift of Mahowald and Kane's splitting of . We also give a description of the resulting -equivariant splitting in terms of -equivariant Adams covers and record an analogous splitting for . Similarly to the nonequivariant story, we expect the techniques of this paper to facilitate further -equivariant stable homotopy computations and understanding of -periodicity in -equivariant stable stems.

Paper Structure

This paper contains 44 sections, 65 theorems, 241 equations, 14 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. New directions
  5. Outline of the Paper
  6. Notation
  7. Underlying nonequivariant splittings
  8. Nonequivariant Brown--Gitler spectra
  9. Nonequivariant Splittings
  10. Literature on nonequivariant splittings
  11. Strategies for the nonequivariant splitting
  12. Constructing the nonequivariant splitting
  13. Equivariant preliminaries
  14. The $C_2$-equivariant homology of a point
  15. Comparison with nonequivariant homology
  16. ...and 29 more sections

Key Result

Theorem 1

Up to $2$-completion, there is a splitting of $ku_{\mathbb R}$-modules

Figures (14)

  • Figure 2.1: $\mathop{\mathrm{Ext}}\nolimits_{E(1)_*} ({\mathbb F}_p, L(3))$
  • Figure 2.2: $\mathop{\mathrm{Ext}}\nolimits_{E(1)_*} (L(3), {\mathbb F}_p)$
  • Figure 3.1: ${\mathbb M}_2$
  • Figure 4.1: Homological lightning flash module: $L(4)$
  • Figure 5.1: $\mathop{\mathrm{Ext}}\nolimits_{{\mathcal{E}}^{-}_{\star}(0)}({\mathbb M}_2^{\mathbb R}, {\mathbb M}_{2}^{{\mathbb R}})$
  • ...and 9 more figures

Theorems & Definitions (98)

  • Theorem : \ref{['thm:mainTheorem']}
  • Corollary : \ref{['thm:C2AdamsCovers']}
  • Theorem : \ref{['thm:cooperationsAlg']}
  • Corollary : \ref{['cor:kuglcooperations']}
  • Theorem : \ref{['thm:operationsAlgebra']}
  • Theorem : \ref{['thm: height zero splitting']}
  • Corollary : \ref{['cor: height zero simples']}
  • Theorem : \ref{['thm:homology:decomp']}
  • Proposition 2.1
  • Proposition 2.2
  • ...and 88 more