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Algebraic vector bundles and $p$-local A^1-homotopy theory

Aravind Asok, Jean Fasel, Michael J. Hopkins

TL;DR

The paper addresses the question of when topological vector bundles on smooth varieties admit motivic (algebraic) lifts in the ${\mathbb A}^1$-homotopy framework, focusing on cellular varieties and Rees bundles. It introduces a $p$-local A$^1$-homotopy toolkit, including Milnor–Witt $K$-theory actions and a weight-shifting operator $\\tau$, to realize motivic lifts of topological generators $\\alpha_1$ and $\\alpha_1^2$; it then constructs explicit maps between quadrics and, via clutching, nontrivial rank-$2$ algebraic vector bundles on Jouanolou devices $X_{2p-1}$ that lift topological Rees bundles. The results yield nontrivial algebraic vector bundles on high-dimensional affine varieties and provide polynomial maps between quadrics whose realizations correspond to known topological classes, revealing a rich interaction between motivic homotopy theory and algebraic vector bundle theory. The work also hints at broader conjectures about surjectivity of $[X, \mathrm{Gr}_n]_{\mathbb A^1}$ onto topological bundles for cellular $X$ and connects to the Wilson space viewpoint on algebraic cobordism.

Abstract

Using techniques of A^1-homotopy theory, we produce motivic lifts of elements in classical homotopy groups of spheres; these lifts provide polynomial maps of spheres and allow us to construct ``low rank'' algebraic vector bundles on ``simple'' smooth affine varieties of high dimension.

Algebraic vector bundles and $p$-local A^1-homotopy theory

TL;DR

The paper addresses the question of when topological vector bundles on smooth varieties admit motivic (algebraic) lifts in the -homotopy framework, focusing on cellular varieties and Rees bundles. It introduces a -local A-homotopy toolkit, including Milnor–Witt -theory actions and a weight-shifting operator , to realize motivic lifts of topological generators and ; it then constructs explicit maps between quadrics and, via clutching, nontrivial rank- algebraic vector bundles on Jouanolou devices that lift topological Rees bundles. The results yield nontrivial algebraic vector bundles on high-dimensional affine varieties and provide polynomial maps between quadrics whose realizations correspond to known topological classes, revealing a rich interaction between motivic homotopy theory and algebraic vector bundle theory. The work also hints at broader conjectures about surjectivity of onto topological bundles for cellular and connects to the Wilson space viewpoint on algebraic cobordism.

Abstract

Using techniques of A^1-homotopy theory, we produce motivic lifts of elements in classical homotopy groups of spheres; these lifts provide polynomial maps of spheres and allow us to construct ``low rank'' algebraic vector bundles on ``simple'' smooth affine varieties of high dimension.

Paper Structure

This paper contains 4 sections, 17 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Proofs of the main results
  3. Mod $n_{\epsilon}$ Moore spaces and weight shifting
  4. Building vector bundles and maps

Key Result

Theorem 4

For every prime number $p$, the bundle $\xi_p$ lifts to a rank $2$ algebraic vector bundle on $X_{2p-1}$.

Theorems & Definitions (50)

  • Remark 2
  • Remark 3
  • Theorem 4: See Theorem \ref{['thm:reesbundles']}
  • Conjecture 5
  • Remark 6
  • Theorem 7: See Proposition \ref{['prop:tautatesuspensionofeta']} and Theorem \ref{['thm:nonconstantmaps']}
  • Remark 8
  • Theorem 2.1.1: MField
  • Definition 2.1.2
  • Example 2.1.3
  • ...and 40 more