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On the Whitehead theorem for nilpotent motivic spaces

Aravind Asok, Tom Bachmann, Michael J. Hopkins

TL;DR

The paper extends Whitehead-type results to nilpotent motivic spaces by developing a robust motivic group-theoretic framework (strongly and very strongly A^1-invariant sheaves, functorial central series, abelianization, and A^1-lower central series) and by establishing refined Moore–Postnikov factorizations for nilpotent morphisms. It proves a Whitehead theorem in the motivic setting, with connectivity results that transfer between unstable and S^1-stable contexts, and provides a detailed relative Hurewicz theory within motivic homotopy. These tools enable principal refinements of Postnikov towers and yield powerful applications, including an unstable motivic periodicity phenomenon realized via affine Grassmannians and James constructions. The results collectively deepen the understanding of nilpotence and connectivity in motivic homotopy theory and open pathways to new unstable phenomena grounded in stabilization and geometric constructions.

Abstract

We improve some foundational connectivity results and the relative Hurewicz theorem in motivic homotopy theory, study functorial central series in motivic local group theory, establish the existence of functorial Moore--Postnikov factorizations for nilpotent morphisms of motivic spaces under a mild technical hypothesis and establish an analog of the Whitehead theorem for nilpotent motivic spaces. As an application, we deduce a surprising unstable motivic periodicity result.

On the Whitehead theorem for nilpotent motivic spaces

TL;DR

The paper extends Whitehead-type results to nilpotent motivic spaces by developing a robust motivic group-theoretic framework (strongly and very strongly A^1-invariant sheaves, functorial central series, abelianization, and A^1-lower central series) and by establishing refined Moore–Postnikov factorizations for nilpotent morphisms. It proves a Whitehead theorem in the motivic setting, with connectivity results that transfer between unstable and S^1-stable contexts, and provides a detailed relative Hurewicz theory within motivic homotopy. These tools enable principal refinements of Postnikov towers and yield powerful applications, including an unstable motivic periodicity phenomenon realized via affine Grassmannians and James constructions. The results collectively deepen the understanding of nilpotence and connectivity in motivic homotopy theory and open pathways to new unstable phenomena grounded in stabilization and geometric constructions.

Abstract

We improve some foundational connectivity results and the relative Hurewicz theorem in motivic homotopy theory, study functorial central series in motivic local group theory, establish the existence of functorial Moore--Postnikov factorizations for nilpotent morphisms of motivic spaces under a mild technical hypothesis and establish an analog of the Whitehead theorem for nilpotent motivic spaces. As an application, we deduce a surprising unstable motivic periodicity result.
Paper Structure (7 sections, 33 theorems, 49 equations)

This paper contains 7 sections, 33 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Motivic localization and group theory
  3. Motivic localization and group theory
  4. Connectivity of fibers and cofibers
  5. Abelianization and ${\mathbb A}^1$-lower central series
  6. Principal refinements of Moore--Postnikov factorizations
  7. Applications

Key Result

Theorem 1

If $k$ is a perfect field and $f: \mathscr{X} \to \mathscr{Y}$ is a morphism of nilpotent motivic $k$-spaces such that $\Sigma^{\infty}_{S^1}f$ is an equivalence, then $f$ is an equivalence.

Theorems & Definitions (87)

  • Theorem 1: See Theorem \ref{['thm:whitehead']}
  • Theorem 2: See Theorem \ref{['thm:exoticequivalence']}
  • Remark 3
  • Theorem 2.1: Morel
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5
  • ...and 77 more