The simplicial suspension sequence in $\mathbb{A}^1\mskip-2mu$–homotopy

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. 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 nonvanishing result.