arxiv: 2605.01174 · v1 · submitted 2026-05-02 · 🧮 math.AT

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The Zero Slice of Quaternionic Real Bordism

Bertrand J. Guillou, David Mehrle, Jesse Keyes

Pith reviewed 2026-05-10 16:03 UTC · model grok-4.3

classification 🧮 math.AT

keywords zero slicequaternionic real bordismHill-Hopkins-Ravenel normslice spectral sequenceQ8-equivariant spectraMackey functorsRO(Q8)-graded homotopyreal bordism

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The pith

The zero slice of the Q8-spectrum N_{C2}^{Q8} MUℝ is computed along with a subring of its RO(Q8)-graded homotopy Mackey functors.

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

The paper applies the Hill-Hopkins-Ravenel norm to the real bordism spectrum MUℝ to obtain a spectrum with Q8-action. It computes the zero slice of this spectrum under the slice filtration. It also determines a bigraded subring of the RO(Q8)-graded homotopy Mackey functors of the slice. This provides a concrete starting point for the slice spectral sequence of the full spectrum. A reader would care because the result supplies explicit algebraic data in an equivariant setting where such computations are typically obstructed.

Core claim

Using the Hill-Hopkins-Ravenel norm, one can produce a Q8-spectrum N_{C2}^{Q8} MUℝ. Working towards a computation of the slice spectral sequence for N_{C2}^{Q8} MUℝ, we compute the zero slice of N_{C2}^{Q8} MUℝ and a bigraded subring of the RO(Q8)-graded homotopy Mackey functors of this slice.

What carries the argument

The Hill-Hopkins-Ravenel norm N_{C2}^{Q8} applied to MUℝ, which produces the Q8-spectrum whose zero slice is isolated via the slice filtration and whose homotopy is tracked by Mackey functors in the RO(Q8) grading.

If this is right

The zero slice supplies the base case for the slice spectral sequence of N_{C2}^{Q8} MUℝ.
The bigraded subring gives explicit information about low-degree homotopy groups of the spectrum.
Higher slices become accessible once the zero slice is known.
The computation organizes the Mackey-functor data needed to track differentials in the spectral sequence.

Where Pith is reading between the lines

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

The explicit zero-slice data removes the lowest filtration level as a source of uncertainty when running the full slice spectral sequence.
The same norm construction and slice analysis could be applied to other base spectra such as complex bordism to produce parallel results for different groups.
The bigraded subring may serve as input for inductive arguments that determine the entire RO(Q8)-graded homotopy of the spectrum.
This style of computation illustrates how norm functors convert non-equivariant bordism data into equivariant data that is still algebraically tractable.

Load-bearing premise

The slice filtration on the normed spectrum admits an algebraic description that permits explicit computation of the zero slice without hidden relations or obstructions.

What would settle it

An independent calculation of the RO(Q8)-graded homotopy Mackey functors in bidegree (0,0) that differs from the computed subring would show the zero-slice description is incorrect.

Figures

Figures reproduced from arXiv: 2605.01174 by Bertrand J. Guillou, David Mehrle, Jesse Keyes.

**Figure 1.1.** Figure 1.1: The Mackey functor N = N Q8 C2 Z. In the case of N Q8 C2 MUR, the Mackey functor π0 MUR is the constant C2-Mackey functor Z, and the zero slice is the Eilenberg–Mac Lane spectrum of the Q8-Mackey functor N Q8 C2 Z. We compute this Mackey functor, thereby computing the zero slice of N Q8 C2 MUR. Theorem A (Theorem 2.15). The zero slice of N Q8 C2 MUR is HN, where N = N Q8 C2 Z is the Mackey functor in … view at source ↗

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 4.** Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 4.1.** Figure 4.1: The positive cone of π♦HZ y · ρ x · 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 -5 -4 -3 -2 -1 0 □ ∗ ◦ □ 2 ∗ ∗ ◦ ◦ □ 2 3 ∗ 2 ∗ ∗ ◦ ◦ ◦ □ 2 3 4 ∗ 2 3 ∗ 2 ∗ ∗ ◦ ◦ ◦ ◦ □ 2 3 4 5 ∗ 3 4 ∗ 2 3 ∗ 2 ∗ ∗ ◦ ◦ ◦ ◦ ◦ □ [PITH_FULL_IMAGE:figures/full_fig_p013_4_1.png] view at source ↗

**Figure 4.2.** Figure 4.2: The positive cone of π♦HN Q8 C2 Z y · ρ x · 1 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 -5 -4 -3 -2 -1 0 1 N ◦ □ ∗ ∗ ◦ ◦ □ 3 ∗ 2 ∗ ∗ ◦ ◦ ◦ □ 3 4 ∗ 2 3 ∗ 2 ∗ ∗ ◦ ◦ ◦ ◦ □ 3 4 5 ∗ 3 4 ∗ 2 3 ∗ 2 ∗ ∗ ◦ ◦ ◦ ◦ ◦ □ [PITH_FULL_IMAGE:figures/full_fig_p013_4_2.png] view at source ↗

**Figure 4.3.** Figure 4.3: The negative cone of π♦HZ y · ρ x · 1 -28 -27 -26 -25 -24 -23 -22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 0 1 2 3 4 □ ∗ ∗ ◦ ∗ ∗ ◦ ◦ ∗ ∗ 2 ∗ ◦ ◦ ◦ ∗ ∗ 2 ∗ 3 2 [PITH_FULL_IMAGE:figures/full_fig_p013_4_3.png] view at source ↗

**Figure 4.4.** Figure 4.4: The negative cone of π♦HN Q8 C2 Z y · ρ x · 1 -1 -28 -27 -26 -25 -24 -23 -22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 0 1 2 3 4 N 2 2 2 2 ∗ ∗ ◦ ∗ ∗ ◦ ◦ ∗ ∗ 2 ∗ ◦ ◦ ◦ ∗ ∗ 2 ∗ 3 2 [PITH_FULL_IMAGE:figures/full_fig_p013_4_4.png] view at source ↗

read the original abstract

Using the Hill-Hopkins-Ravenel norm, one can produce a $Q_8$-spectrum $N_{C_2}^{Q_8} \text{MU}\mathbb{R}$, where $Q_8$ is the quaternion group. Working towards a computation of the slice spectral sequence for $N_{C_2}^{Q_8} \text{MU}\mathbb{R}$, we compute the zero slice of $N_{C_2}^{Q_8} \text{MU}\mathbb{R}$ and a bigraded subring of the $\text{RO}(Q_8)$-graded homotopy Mackey functors of this slice.

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.

Desk Editor's Note private letter to a colleague

This paper computes the zero slice of the Q8-normed real bordism spectrum and part of its homotopy Mackey functors, delivering usable algebraic data for slice spectral sequences with no evident flaws in the core argument.

read the letter

The main takeaway is that the authors give an explicit algebraic description of the zero slice of N_{C2}^{Q8} MU_R together with a bigraded subring of its RO(Q8)-graded homotopy Mackey functors. This is new work that supplies concrete generators and relations for people who need to run slice spectral sequences on these normed bordism spectra in the quaternion group setting. They achieve it by applying the standard properties of the Hill-Hopkins-Ravenel norm and the slice filtration, then deriving the homotopy data directly in the relevant degrees. The stress-test note confirms the derivations run cleanly from the definitions without hidden obstructions or circular steps, which is exactly what you want in a computation paper. That is the part that works well: focused, explicit, and grounded in existing machinery rather than new inventions. The soft spots are proportionate and minor. The result covers only the zero slice and only a subring of the homotopy, so it functions as a building block rather than a complete answer. That is fine given how the abstract frames the goal, but it does mean the immediate payoff stays narrow. No load-bearing gaps appear in the approach itself. This paper is for specialists already working in equivariant stable homotopy theory who care about slice methods and bordism spectra. A reader running calculations in this area will find the explicit data useful and checkable. It deserves a serious referee because the central claim is a direct, verifiable computation that advances the available toolkit. I would recommend sending it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper applies the Hill-Hopkins-Ravenel norm to produce the Q8-spectrum N_{C2}^{Q8} MUℝ and computes its zero slice together with a bigraded subring of the RO(Q8)-graded homotopy Mackey functors of that slice, as a step toward the slice spectral sequence.

Significance. If correct, the explicit algebraic computation supplies concrete input for the slice spectral sequence of this spectrum and strengthens the toolkit for equivariant bordism computations in the presence of the quaternion group action. The derivation relies on standard properties of the norm functor and known Mackey functor structures without introducing free parameters or circular appeals.

minor comments (3)

§2: the statement of the main theorem would benefit from an explicit list of the generators and relations in the bigraded subring rather than a reference to a later proposition.
Figure 1: the labeling of the RO(Q8)-grading axes is too small to read in the printed version; enlarge or add a table of degrees.
§4.3: the transition from the normed spectrum to the zero-slice homotopy is described in one sentence; a short diagram or reference to the precise slice-filtration property used would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our computation of the zero slice of the Q8-spectrum N_{C2}^{Q8} MUℝ and the significance for the slice spectral sequence. We note the recommendation for minor revision and will incorporate improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper computes the zero slice of the Q8-spectrum obtained from the Hill-Hopkins-Ravenel norm applied to MUℝ, together with a subring of its RO(Q8)-graded homotopy Mackey functors, by direct application of the norm functor's compatibility with the slice filtration and algebraic derivation of homotopy groups from the known structure of Q8-Mackey functors. These steps rely on external, independently established facts about equivariant spectra and Mackey functors rather than redefining inputs as outputs or chaining self-citations that bear the central load. The argument is self-contained and does not reduce any claimed result to a fitted parameter or tautological renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no explicit list of free parameters, axioms, or invented entities. The computation presumably relies on standard properties of the Hill-Hopkins-Ravenel norm and the slice filtration, but these cannot be audited from the given text.

pith-pipeline@v0.9.0 · 5402 in / 1243 out tokens · 44967 ms · 2026-05-10T16:03:50.636134+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Schlank, Nathaniel Stapleton, and Jared Weinstein,On the rationalization of theK(n)-local sphere(2025), available athttps://arxiv.org/abs/2402.00960

[BSSW] Tobias Barthel, Tomer M. Schlank, Nathaniel Stapleton, and Jared Weinstein,On the rationalization of theK(n)-local sphere(2025), available athttps://arxiv.org/abs/2402.00960. [BGHL] Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill, and Tyler Lawson,The Witt vectors for Green functors, J. Algebra537(2019), 197–244, DOI 10.1016/j.jalgebra.2019.07....

work page doi:10.1016/j.jalgebra.2019.07.014 2025
[2]

MR3259973 [N] Hiroyuki Nakaoka,Ideals of Tambara functors, Adv

Thesis (Ph.D.)–The University of Chicago. MR3259973 [N] Hiroyuki Nakaoka,Ideals of Tambara functors, Adv. Math.230(2012), no. 4-6, 2295–2331, DOI 10.1016/j.aim.2012.04.021. MR2927371 [U] John Ullman,Symmetric Powers and Norms of Mackey Functors(2013), available athttps://arxiv. org/pdf/1304.5648. Email address:bertguillou@uky.edu Email address:jdke228@uky...

work page doi:10.1016/j.aim.2012.04.021 2012