The simplicial suspension sequence in A^1-homotopy

TL;DR

This paper develops an ${ m A}^1$-homotopy theoretical framework for the suspension sequence, refining classical theorems and analyzing differentials, leading to new insights into motivic spheres and their homotopy sheaves.

Contribution

It introduces an ${ m A}^1$-homotopy analog of the Whitehead refinement of the suspension theorem and analyzes EHP differentials, advancing the understanding of motivic homotopy sheaves.

Findings

Refined the ${ m A}^1$-homotopy suspension theorem.

Described $E_1$-differentials in the ${ m A}^1$-EHP sequence.

Established a rational non-vanishing result for motivic spheres.

Abstract

We study a version of the James model for the loop space of a suspension in unstable ${\mathbb A}^1$-homotopy theory. We use this model to establish an analog of G.W. Whitehead's classical refinement of the Freudenthal suspension theorem in ${\mathbb A}^1$-homotopy theory: our result refines F. Morel's ${\mathbb A}^1$-simplicial suspension theorem. We then describe some $E_1$-differentials in the EHP sequence in ${\mathbb A}^1$-homotopy theory. These results are analogous to classical results of G.W. Whitehead's. Using these tools, we deduce some new results about unstable ${\mathbb A}^1$-homotopy sheaves of motivic spheres, including the counterpart of a classical rational non-vanishing result.