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arxiv: 2606.05020 · v1 · pith:ZHSPPWOOnew · submitted 2026-06-03 · 🧮 math.AT · math.AG· math.KT

Slices of the special linear algebraic cobordism spectrum

Ahina Nandy , Oliver R\"ondigs , Egor Zolotarev This is my paper

Pith reviewed 2026-06-28 03:22 UTC · model grok-4.3

classification 🧮 math.AT math.AGmath.KT
keywords special linear algebraic cobordismslice spectral sequenceAdams-Novikov spectral sequenceMilnor-Witt stemshermitian K-theoryspecial unitary cobordismrational decomposition
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The pith

The slices of MSL after inverting the exponential characteristic are expressed using the second page of the Adams-Novikov spectral sequence for MSU.

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

The paper computes the slices of the special linear algebraic cobordism spectrum MSL after inverting the exponential characteristic e of the base field. These slices are identified with the second page of the Adams-Novikov spectral sequence for the special unitary cobordism spectrum MSU as previously determined by Novikov. The identification is applied to compute the first three Milnor-Witt stems of the homotopy groups of MSL[e^{-1}] in terms of very effective hermitian K-theory. A decomposition of the rational special linear algebraic cobordism spectrum is also established over any quasi-compact quasi-separated scheme.

Core claim

The slices of MSL[e^{-1}] are expressed in terms of the second page of the Adams-Novikov spectral sequence for the special unitary cobordism spectrum MSU, which was explicitly determined by Novikov.

What carries the argument

The slice spectral sequence for MSL combined with the relation to the Adams-Novikov spectral sequence for MSU after inverting e.

If this is right

  • The first three Milnor-Witt stems of the homotopy groups of MSL[e^{-1}] are determined in terms of very effective hermitian K-theory.
  • The rational special linear algebraic cobordism spectrum decomposes over any qcqs scheme.
  • The identification is applicable in the range where the low-degree computations are performed.

Where Pith is reading between the lines

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

  • The method supplies a route to determine additional Milnor-Witt stems of MSL[e^{-1}] in higher degrees.
  • The slice identification may connect computations in algebraic cobordism to classical topological data in other settings.

Load-bearing premise

The slice spectral sequence for MSL converges in the relevant range and the identification with the ANSS data for MSU applies after inverting e.

What would settle it

An independent computation of one of the first three Milnor-Witt stems of the homotopy groups of MSL[e^{-1}] that disagrees with the value predicted from the second page of the ANSS for MSU would falsify the claim.

read the original abstract

Let $F$ be a field of exponential characteristic $e$. We compute the slices of $\mathbf{MSL}[e^{-1}]$, where $\mathbf{MSL}$ is the special linear algebraic cobordism spectrum defined by Panin and Walter. The answer is expressed in terms of the second page of the Adams-Novikov spectral sequence for the special unitary cobordism spectrum, which was explicitly determined by Novikov. Its applicability is demonstrated by computations with the slice spectral sequence for $\mathbf{MSL}$, which determine the first few Milnor-Witt stems of its homotopy groups (up to the third) in terms of very effective hermitian $K$-theory. We also establish a decomposition of the rational special linear algebraic cobordism spectrum over an arbitrary qcqs scheme.

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.

Referee Report

2 major / 2 minor

Summary. The paper computes the slices of the special linear algebraic cobordism spectrum MSL inverted at the exponential characteristic e over a field F, expressing the result in terms of the E2-page of the classical Adams-Novikov spectral sequence for the topological special unitary cobordism spectrum MSU as computed by Novikov. It applies the slice spectral sequence to determine the first three Milnor-Witt stems of the homotopy groups in terms of very effective hermitian K-theory and establishes a decomposition of the rational MSL over an arbitrary qcqs scheme.

Significance. If the central identification holds and the relevant convergence assumptions are verified, the result provides a direct link between Panin-Walter's algebraic cobordism and classical topological computations, enabling explicit motivic calculations that reduce to established data. The explicit description of low-degree Milnor-Witt stems and the rational decomposition are concrete outputs that would be useful for further work in hermitian K-theory and motivic homotopy.

major comments (2)
  1. [Main result on slices (abstract and the section stating the identification)] The central claim that the slices of MSL[e^{-1}] are given exactly by the ANSS E2-page for MSU (as in the abstract) requires that e-inversion commutes with the slice filtration and introduces no motivic corrections or changes to the coefficient rings. The manuscript provides no explicit comparison or proof of this isomorphism as graded objects, which is load-bearing for reducing the computation to Novikov's data.
  2. [Application to Milnor-Witt stems via the slice spectral sequence] The demonstration that the slice spectral sequence computes the first three Milnor-Witt stems assumes convergence in the relevant range after e-inversion. No explicit bounds, vanishing results, or verification of the convergence assumptions are supplied, which directly affects the reliability of the explicit description in terms of very effective hermitian K-theory.
minor comments (2)
  1. The notation distinguishing the algebraic MSL from its topological counterpart and the precise meaning of 'very effective hermitian K-theory' could be clarified on first use for readers outside the immediate subfield.
  2. The rational decomposition result is stated without a section reference or outline of the argument; adding a brief pointer to its location would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the two major comments point by point below, agreeing that additional explicit verifications are warranted for clarity and rigor.

read point-by-point responses

  1. Referee: [Main result on slices (abstract and the section stating the identification)] The central claim that the slices of MSL[e^{-1}] are given exactly by the ANSS E2-page for MSU (as in the abstract) requires that e-inversion commutes with the slice filtration and introduces no motivic corrections or changes to the coefficient rings. The manuscript provides no explicit comparison or proof of this isomorphism as graded objects, which is load-bearing for reducing the computation to Novikov's data.

    Authors: We agree that an explicit comparison establishing that e-inversion commutes with the slice filtration (with no additional motivic corrections) is necessary to make the identification fully rigorous as graded objects. The manuscript derives the slices from the definition of MSL and known properties of the slice functor, but does not include a dedicated verification step. In the revised version we will add a short subsection providing this graded comparison, confirming the coefficient rings match Novikov's E2-page without alteration. revision: yes

  2. Referee: [Application to Milnor-Witt stems via the slice spectral sequence] The demonstration that the slice spectral sequence computes the first three Milnor-Witt stems assumes convergence in the relevant range after e-inversion. No explicit bounds, vanishing results, or verification of the convergence assumptions are supplied, which directly affects the reliability of the explicit description in terms of very effective hermitian K-theory.

    Authors: We acknowledge that the current text relies on general convergence properties of the slice spectral sequence after e-inversion without spelling out explicit bounds or vanishing lines in the low-degree range. While the slice descriptions themselves imply the necessary vanishings for stems up to 3, this is not stated explicitly. The revised manuscript will include a paragraph verifying the convergence assumptions (via the known connectivity of the slices) and stating the precise range in which the spectral sequence computes the Milnor-Witt stems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; identification uses independent external computation

full rationale

The paper's central claim expresses the slices of MSL[e^{-1}] in terms of the E2-page of the classical Adams-Novikov spectral sequence for MSU, as explicitly computed by Novikov (an external, pre-existing result independent of the present authors). This identification is applied after e-inversion using the Panin-Walter definition of MSL and known properties of the slice spectral sequence; no step reduces a derived quantity to a fitted parameter, self-citation chain, or definitional renaming within the paper itself. The subsequent computations of Milnor-Witt stems are applications of this external input rather than circular re-derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The computation rests on the standard properties of the slice filtration in motivic homotopy theory and the known ANSS computation for MSU; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The slice spectral sequence for MSL converges to the homotopy groups in the relevant range after inverting e.
    Invoked to derive the Milnor-Witt stems from the slices.
  • domain assumption The identification of slices with the second page of the ANSS for MSU holds after base change or localization.
    Central to expressing the answer in terms of Novikov's computation.

pith-pipeline@v0.9.1-grok · 5669 in / 1430 out tokens · 28596 ms · 2026-06-28T03:22:26.154745+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On the metalinear algebraic cobordism spectrum

    math.AT 2026-06 unverdicted novelty 7.0

    MML ≅ MSL ⊕ Σ^{2,1} MGL after fixing a retraction, enabling computations of low Milnor-Witt stems, geometric diagonal, slices, and 2-inverted modules over MML.

Reference graph

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