#### Volume 21, issue 4 (2017)

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

### Aravind Asok, Kirsten Wickelgren and Ben Williams

Geometry & Topology 21 (2017) 2093–2160
##### 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.

##### Keywords
$A^1$-homotopy, James construction
##### Mathematical Subject Classification 2010
Primary: 14F42, 19E15
Secondary: 55Q15, 55Q20, 55Q25