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arxiv: 2409.16962 · v4 · pith:TEQNQDXNnew · submitted 2024-09-25 · 🧮 math.AT · math.AG· math.KT

The geometric diagonal of the special linear algebraic cobordism

Egor Zolotarev This is my paper

Pith reviewed 2026-05-23 20:28 UTC · model grok-4.3

classification 🧮 math.AT math.AGmath.KT
keywords special linear algebraic cobordismmotivic Thom spectraP1-diagonalspecial unitary cobordismhermitian K-theoryPontryagin characteristic numbersmotivic Hopf elementAnderson-Brown-Peterson theorem
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The pith

The P1-diagonal of the homotopy groups of special linear algebraic cobordism equals the special unitary cobordism ring after inverting 2 and the exponential characteristic.

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

The paper constructs the motivic version of the c1-spherical cobordism spectrum and establishes its connection to other motivic Thom spectra. This connection is used to compute the P1-diagonal of the homotopy groups of the special linear algebraic cobordism spectrum MSL over a local Dedekind domain containing 1/2, after inverting the exponential characteristic of the residue field. The result gives a complete description in terms of the special unitary cobordism ring. The paper also determines the action of the motivic Hopf element eta, describes the localization away from 2, computes the 2-primary torsion subgroup, and constructs Pontryagin characteristic numbers valued in hermitian K-theory along with Chern numbers.

Core claim

Using the established connection between the motivic c1-spherical cobordism spectrum and other motivic Thom spectra, the P1-diagonal of the homotopy groups pi_{2*,*}(MSL) is computed over a local Dedekind domain k with 1/2 in k after inverting the exponential characteristic of the residue field of k, and the complete answer is expressed in terms of the special unitary cobordism ring. The action of the motivic Hopf element eta on this ring is determined, the localization away from 2 is obtained, and the 2-primary torsion subgroup is computed. Pontryagin characteristic numbers with values in hermitian K-theory are constructed, Chern numbers are built in this setting, and a motivic version of t

What carries the argument

The motivic c1-spherical cobordism spectrum, which supplies the connection to other motivic Thom spectra and permits the computation of the P1-diagonal in the homotopy groups of MSL.

If this is right

  • The action of the motivic Hopf element eta on the computed ring is fully determined.
  • The localization of the ring away from 2 is obtained explicitly.
  • The 2-primary torsion subgroup of the ring is computed.
  • Pontryagin characteristic numbers take values in hermitian K-theory.
  • A motivic version of the Anderson-Brown-Peterson theorem holds in this setting.

Where Pith is reading between the lines

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

  • The identification may transfer known computations from classical special unitary cobordism to the motivic special linear setting.
  • The construction of characteristic numbers in hermitian K-theory could produce new invariants for algebraic vector bundles.
  • The discussion of Calabi-Yau varieties in the SL-cobordism ring may link cobordism classes to geometric properties in algebraic geometry.
  • The result could support computations of similar diagonals for other algebraic cobordism spectra.

Load-bearing premise

The base ring is a local Dedekind domain containing 1/2, and inverting the exponential characteristic of the residue field preserves the essential structure of the homotopy groups.

What would settle it

An explicit calculation of one of the groups in pi_{2*,*}(MSL) over a specific local Dedekind domain that differs from the corresponding group in the special unitary cobordism ring after the same inversions.

read the original abstract

The motivic version of the $c_1$-spherical cobordism spectrum is constructed. A connection of this spectrum with other motivic Thom spectra is established. Using this connection, we compute the $\mathbb{P}^1$-diagonal of the homotopy groups of the special linear algebraic cobordism $\pi_{2*,*}(\mathrm{MSL})$ over a local Dedekind domain $k$ with $1/2\in k$ after inverting the exponential characteristic of the residue field of $k$. We discuss the action of the motivic Hopf element $\eta$ on this ring, obtain a description of the localization away from $2$ and compute the $2$-primary torsion subgroup. The complete answer is given in terms of the special unitary cobordism ring. An important component of the computation is the construction of Pontryagin characteristic numbers with values in the hermitian K-theory. We also construct Chern numbers in this setting, prove the motivic version of the Anderson-Brown-Peterson theorem and briefly discuss classes of Calabi-Yau varieties in the $\mathrm{SL}$-cobordism ring.

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

0 major / 2 minor

Summary. The paper constructs the motivic version of the c₁-spherical cobordism spectrum and establishes connections to other motivic Thom spectra. Using this connection, it computes the ℙ¹-diagonal of the homotopy groups π_{2*,*}(MSL) over a local Dedekind domain k with 1/2 ∈ k after inverting the exponential characteristic of the residue field, expressing the result in terms of the special unitary cobordism ring. Additional results include the action of the motivic Hopf element η, a description of the localization away from 2, the 2-primary torsion subgroup, constructions of Pontryagin numbers valued in hermitian K-theory and Chern numbers, a motivic Anderson-Brown-Peterson theorem, and a discussion of Calabi-Yau varieties in the SL-cobordism ring.

Significance. If the stated connections and computations hold, the work supplies an explicit, external reduction of the ℙ¹-diagonal of MSL to the special unitary cobordism ring after the indicated base change and inversion. This is a concrete computational advance in motivic homotopy theory. The constructions of Pontryagin numbers in hermitian K-theory and the motivic ABP theorem are reusable tools, and the absence of free parameters or self-referential definitions in the final description (as presented via external connections) strengthens the result.

minor comments (2)
  1. The introduction would benefit from an explicit statement of the main theorem (including the precise ring and the inversion of the exponential characteristic) before the discussion of auxiliary constructions.
  2. Notation for the motivic c₁-spherical cobordism spectrum is introduced without a dedicated symbol in the abstract; a consistent symbol throughout the text would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment. The referee's summary accurately captures the main results, including the computation of the ℙ¹-diagonal of π_{2*,*}(MSL), the connections to special unitary cobordism, the constructions of characteristic numbers, and the motivic Anderson-Brown-Peterson theorem. We appreciate the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs the motivic c₁-spherical cobordism spectrum, establishes a connection to other motivic Thom spectra, and uses this to compute the ℙ¹-diagonal of π_{2*,*}(MSL) over the specified base, expressed in terms of the special unitary cobordism ring. It also constructs Pontryagin numbers valued in hermitian K-theory, proves a motivic Anderson-Brown-Peterson theorem, and discusses related structures. These steps rely on external connections and constructions rather than reducing by the paper's own equations or self-citations to the target result. The derivation is self-contained against external benchmarks with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the standard framework of motivic homotopy theory and the existence of various Thom spectra; the paper introduces the motivic c1-spherical cobordism spectrum as a new object.

axioms (1)
  • standard math Standard axioms and constructions of motivic homotopy theory and algebraic cobordism spectra
    The entire computation is performed inside this established framework.
invented entities (1)
  • motivic version of the c1-spherical cobordism spectrum no independent evidence
    purpose: To establish connections with other motivic Thom spectra enabling the diagonal computation
    Explicitly constructed in the paper as the key new object.

pith-pipeline@v0.9.0 · 5726 in / 1446 out tokens · 32947 ms · 2026-05-23T20:28:05.618567+00:00 · methodology

discussion (0)

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