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arxiv: 2503.17149 · v3 · submitted 2025-03-21 · 🧮 math.AT

A spectrum-level splitting of the ku_mathbb{R}-cooperations algebra

Guchuan Li , Sarah Petersen , Elizabeth Tatum This is my paper

Pith reviewed 2026-05-22 22:54 UTC · model grok-4.3

classification 🧮 math.AT
keywords C2-equivariant spectraku-cooperations algebraBrown-Gitler spectraMahowald-Kane splittingequivariant Adams coversoperations algebrasstable homotopy theory
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The pith

The classical Mahowald-Kane splitting of ku ∧ ku lifts to a C2-equivariant version.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a C2-equivariant lift of the Mahowald-Kane splitting of ku ∧ ku into a sum of finitely generated ku-module spectra, originally obtained via integral Brown-Gitler spectra. This construction also yields a description in terms of C2-equivariant Adams covers and produces an analogous splitting for the smash product of two copies of the equivariant Eilenberg-MacLane spectrum H underline Z. Along the way the authors compute the full ku_R and H Z operations and cooperations algebras. A sympathetic reader cares because the original non-equivariant splitting drove progress on v1-periodicity and sphere homotopy groups, so the lift supplies the same tool in the C2-equivariant setting.

Core claim

We construct a C2-equivariant lift of Mahowald and Kane's splitting of ku ∧ ku. The lift is obtained by producing compatible C2-equivariant lifts of the integral Brown-Gitler spectra that preserve the ku-module structures, so the decomposition descends to the equivariant cooperations algebra; the resulting splitting is expressed via C2-equivariant Adams covers, and a parallel splitting is recorded for H underline Z ∧ H underline Z.

What carries the argument

C2-equivariant lifts of integral Brown-Gitler spectra that preserve ku-module structures and allow the classical decomposition to descend.

If this is right

  • The splitting descends to the equivariant setting and therefore supplies summands for computing ku_R-cooperations.
  • The decomposition can be rewritten in terms of C2-equivariant Adams covers.
  • An analogous direct-sum decomposition holds for H underline Z ∧ H underline Z.
  • Complete calculations of the ku_R and H Z operations and cooperations algebras become available as a byproduct.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same lifting technique may apply to other classical splittings that rely on Brown-Gitler spectra.
  • Equivariant v1-periodicity statements could now be attacked with the same module-summand approach used in the non-equivariant case.
  • The cooperations algebra computations provide raw input for spectral sequence calculations in real K-theory.

Load-bearing premise

The integral Brown-Gitler spectra used in the classical splitting admit compatible C2-equivariant lifts that preserve the ku-module structures.

What would settle it

Explicit failure to produce C2-equivariant Brown-Gitler spectra whose module actions commute with the ku_R action and whose summands reproduce the non-equivariant splitting after forgetting the C2-action.

read the original 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 describe the resulting $C_2$-equivariant splitting in terms of $C_2$-equivariant Adams covers and record an analogous splitting for $H\underline{\mathbb{Z}} \wedge H \underline{\mathbb{Z}}$. Along the way, we give complete computations of the $ku_{\mathbb{R}}$ and $H \mathbb{Z}$ operations and cooperations algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the claim rests on standard properties of equivariant spectra and the existence of lifts of Brown-Gitler spectra; no free parameters or invented entities are indicated.

axioms (1)
  • standard math Standard properties of C2-equivariant spectra, smash products, and module structures hold and are compatible with the classical constructions.
    Implicitly required for the lift to exist and be described in terms of Adams covers.

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