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On P^1-stabilization in unstable motivic homotopy theory

Aravind Asok, Tom Bachmann, Michael J. Hopkins

TL;DR

This work develops a robust connection between unstable motivic homotopy theory and ${\mathbb P}^1$-stabilization by introducing weakly cellular, or $A$-cellular, structures in motivic contexts. Central to the approach is the refinement of cellularity via $S^{p,q}$-cellularity and the associated nullification framework, which yields precise control over the fiber of the stabilization map and yields a motivic Freudenthal-type suspension theorem under suitable cellularity hypotheses. The paper then leverages these tools to prove Murthy’s conjecture on splitting corank-1 vector bundles over smooth affine algebras in characteristic zero and to compute new unstable motivic homotopy groups of motivic spheres, alongside a suite of results on symmetric powers, the motivic Dold–Thom theorem, and the behavior of quotients and assembly maps with respect to cellularity. Collectively, these advances bridge unstable and stable motivic homotopy theories, enabling explicit obstructions and computations in geometric problems and advancing the computational reach of the motivic stable homotopy landscape.

Abstract

We analyze stabilization with respect to ${\mathbb P}^1$ in the Morel--Voevodsky unstable motivic homotopy theory. We introduce a refined notion of cellularity (a.k.a., biconnectivity) in various motivic homotopy categories taking into account both the simplicial and Tate circles. Under suitable cellularity hypotheses, we refine the Whitehead theorem by showing that a map of nilpotent motivic spaces can be seen to be an equivalence if it so after taking (Voevodsky) motives. We then establish a version of the Freudenthal suspension theorem for ${\mathbb P}^1$-suspension, again under suitable cellularity hypotheses. As applications, we resolve Murthy's conjecture on splitting of corank $1$ vector bundles on smooth affine algebras over algebraically closed fields having characteristic $0$ and compute new unstable motivic homotopy of motivic spheres.

On P^1-stabilization in unstable motivic homotopy theory

TL;DR

This work develops a robust connection between unstable motivic homotopy theory and -stabilization by introducing weakly cellular, or -cellular, structures in motivic contexts. Central to the approach is the refinement of cellularity via -cellularity and the associated nullification framework, which yields precise control over the fiber of the stabilization map and yields a motivic Freudenthal-type suspension theorem under suitable cellularity hypotheses. The paper then leverages these tools to prove Murthy’s conjecture on splitting corank-1 vector bundles over smooth affine algebras in characteristic zero and to compute new unstable motivic homotopy groups of motivic spheres, alongside a suite of results on symmetric powers, the motivic Dold–Thom theorem, and the behavior of quotients and assembly maps with respect to cellularity. Collectively, these advances bridge unstable and stable motivic homotopy theories, enabling explicit obstructions and computations in geometric problems and advancing the computational reach of the motivic stable homotopy landscape.

Abstract

We analyze stabilization with respect to in the Morel--Voevodsky unstable motivic homotopy theory. We introduce a refined notion of cellularity (a.k.a., biconnectivity) in various motivic homotopy categories taking into account both the simplicial and Tate circles. Under suitable cellularity hypotheses, we refine the Whitehead theorem by showing that a map of nilpotent motivic spaces can be seen to be an equivalence if it so after taking (Voevodsky) motives. We then establish a version of the Freudenthal suspension theorem for -suspension, again under suitable cellularity hypotheses. As applications, we resolve Murthy's conjecture on splitting of corank vector bundles on smooth affine algebras over algebraically closed fields having characteristic and compute new unstable motivic homotopy of motivic spheres.
Paper Structure (26 sections, 97 theorems, 193 equations)

This paper contains 26 sections, 97 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on unstable and stable motivic homotopy theory
  3. Motivic localization and the unstable connectivity theorem
  4. Stable motivic homotopy theory and effectivity
  5. Real-étale motivic homotopy theory
  6. Weak cellularity and nullity
  7. Weak cellularity and nullifications: basic properties
  8. Stable weak $S^{p,q}$-cellularity via $t$-structures
  9. Examples
  10. Weakly-cellular resolutions of spaces and consequences
  11. A weakly-cellular Whitehead tower
  12. Unstable cellularity of loop spaces and fibers
  13. Weak cellularity: comparing unstable and stable
  14. Further examples
  15. Equivariant geometry of symmetric powers
  16. ...and 11 more sections

Key Result

Theorem 1

Assume $k$ is a field that has characteristic $0$. If $\mathscr{X} \in \mathrm{Spc}(k)$ is a pointed motivic space and lies in $O(S^{p,q})$ with $p -q \ge 2, q \geq 2$, then the fiber of the stabilization map lies in $O(S^{a,2q})$ where $a = min(2p-1,p+2q-1)$.

Theorems & Definitions (237)

  • Theorem 1: See Theorem \ref{['thm:beyondthediagonal']}
  • Remark 2
  • Theorem 3: See Theorem \ref{['thm:murthy']}
  • Remark 4
  • Theorem 2.1.6: Morel
  • proof
  • Remark 2.1.8
  • Lemma 2.1.11
  • proof
  • Lemma 2.1.12
  • ...and 227 more