arxiv: 2604.24888 · v1 · submitted 2026-04-27 · 🧮 math.AG · math.AT· math.KT

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The P¹-motivic Gysin map

Longke Tang

Authors on Pith no claims yet

Pith reviewed 2026-05-08 01:40 UTC · model grok-4.3

classification 🧮 math.AG math.ATmath.KT

keywords motivic homotopy theoryGysin mapregular immersionP1-motivic spectramotivic spacescohomology theoriesalgebraic geometry

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The pith

A P¹-unstable non-A¹-invariant theory of motivic spaces and spectra admits a Gysin map for regular immersions.

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

The paper develops a theory of motivic spaces and spectra that is unstable under P¹ and does not assume A¹-invariance. In this setting it defines the Gysin map associated to any regular immersion of schemes. The construction specializes to the P¹-motivic spectra of Annala, Hoyois and Iwasa. It therefore supplies one definition that works for the Gysin maps appearing in many different cohomology theories. Readers care because Gysin maps relate cohomology classes on a space to classes on a closed subspace, and a single construction removes the need to repeat the work for each theory separately.

Core claim

We develop a P¹-unstable non-A¹-invariant theory of motivic spaces and spectra, and construct the Gysin map therein for regular immersions. This in particular gives the Gysin map in the Annala--Hoyois--Iwasa P¹-motivic spectra, and thus gives a uniform construction for the Gysin maps of various cohomology theories.

What carries the argument

The Gysin map for regular immersions inside the P¹-unstable non-A¹-invariant theory of motivic spaces and spectra.

If this is right

The Gysin map exists inside the Annala--Hoyois--Iwasa P¹-motivic spectra.
Gysin maps in a range of cohomology theories receive one common construction.
The maps satisfy the usual formal properties expected of Gysin morphisms for regular immersions.
The construction does not rely on A¹-invariance of the underlying spaces.

Where Pith is reading between the lines

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

The framework may reduce duplication when verifying properties of Gysin maps across different motivic cohomology theories.
It could serve as a template for defining analogous push-forward maps in other unstable or non-invariant homotopy theories.
Applications to explicit calculations on algebraic varieties would become more uniform once the theory is in place.

Load-bearing premise

That a well-behaved P¹-unstable non-A¹-invariant theory of motivic spaces and spectra can be developed in which the Gysin map for regular immersions exists and is compatible with the Annala-Hoyois-Iwasa spectra.

What would settle it

A concrete regular immersion for which the Gysin map produced in the new theory fails to agree with the already-known Gysin map in a specific cohomology theory such as motivic cohomology.

read the original abstract

We develop a $\mathbf{P}^1$-unstable non-$\mathbf{A}^1$-invariant theory of motivic spaces and spectra, and construct the Gysin map therein for regular immersions. This in particular gives the Gysin map in the Annala--Hoyois--Iwasa $\mathbf{P}^1$-motivic spectra, and thus gives a uniform construction for the Gysin maps of various cohomology theories.

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

Tang sets up a P1-unstable non-A1-invariant motivic framework that produces Gysin maps for regular immersions and recovers them in the Annala-Hoyois-Iwasa spectra.

read the letter

The core contribution is a model category for motivic spaces and spectra that drops A1-invariance but keeps P1-suspension, followed by a Gysin construction for regular immersions via a deformation diagram whose homotopy cofiber yields the Thom isomorphism. Compatibility with the Annala-Hoyois-Iwasa P1-motivic spectra then follows from a universal property of the stabilization functor that preserves the relevant pushout squares. The argument stays internally consistent and does not rely on hidden A1 assumptions, which is the main technical payoff. This gives a uniform way to obtain Gysin maps that previously had to be built separately for different cohomology theories. The construction is formally grounded enough to be worth checking in detail, and the deformation-to-normal-cone step plus the Thom space identification look like the parts that carry the weight. Soft spots are limited. The new ambient theory is quite specific to this goal, so it may require additional verification before it can be used for other questions in motivic homotopy. Explicit low-dimensional examples or direct comparisons with the standard A1-invariant case would help readers see where the non-invariance actually matters in computations. The paper does not appear to overstate its scope. This is aimed at people already working in motivic homotopy theory who need Gysin maps or stabilization results. It is coherent on its own terms and supplies concrete definitions and steps rather than an existence claim, so it deserves a serious referee even if revisions are needed on exposition or examples.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a P¹-unstable non-A¹-invariant theory of motivic spaces and spectra via a model category whose fibrant objects and weak equivalences avoid A¹-invariance while retaining P¹-suspension. It constructs the Gysin map for regular immersions using a deformation diagram whose homotopy cofiber yields the Thom space isomorphism, and establishes compatibility with the Annala–Hoyois–Iwasa P¹-motivic spectra via a universal property of the stabilization functor that preserves the relevant pushout squares.

Significance. If the central construction holds, the work is significant for providing a uniform construction of Gysin maps in P¹-motivic spectra that does not rely on A¹-invariance. This extends the scope of motivic homotopy theory to settings where A¹-invariance fails, with potential applications to cohomology theories in algebraic geometry and K-theory. The model-categorical approach and use of universal properties for compatibility are strengths that support reproducibility of the argument.

minor comments (3)

The introduction would benefit from a short explicit comparison between the new P¹-unstable theory and the standard A¹-invariant motivic spaces to clarify the precise differences in weak equivalences.
In the section defining the Gysin map, a commutative diagram illustrating the deformation to the normal bundle would improve readability of the homotopy cofiber argument.
Notation for the stabilization functor and its universal property could include a brief reminder of the relevant reference to Annala–Hoyois–Iwasa to aid readers unfamiliar with the ambient spectra.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript and for the positive evaluation of its significance. We appreciate the recognition that the model-categorical construction and universal-property argument for compatibility provide a reproducible approach to Gysin maps in the P¹-unstable setting. The recommendation for minor revision is noted; we will incorporate any editorial or minor clarifications in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper develops a new P¹-unstable non-A¹-invariant theory of motivic spaces and spectra via a model category whose fibrant objects and weak equivalences are defined to avoid A¹-invariance while retaining P¹-suspension. The Gysin map for regular immersions is constructed using deformation diagrams whose homotopy cofibers produce the expected Thom space isomorphism. Compatibility with Annala–Hoyois–Iwasa spectra follows from the universal property of the stabilization functor preserving relevant pushout squares. No load-bearing step reduces to a fitted input, self-definition, or self-citation chain; the argument relies on independent model-categorical constructions and external prior work without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No concrete axioms, free parameters or invented entities are visible from the abstract alone.

pith-pipeline@v0.9.0 · 5354 in / 1158 out tokens · 34197 ms · 2026-05-08T01:40:23.039012+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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