## Geometry & Topology

### 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 $A1$–homotopy theory. We use this model to establish an analog of G W Whitehead’s classical refinement of the Freudenthal suspension theorem in $A1$–homotopy theory: our result refines F Morel’s $A1$–simplicial suspension theorem. We then describe some $E1$–differentials in the EHP sequence in $A1$–homotopy theory. These results are analogous to classical results of G W Whitehead. Using these tools, we deduce some new results about unstable $A1$–homotopy sheaves of motivic spheres, including the counterpart of a classical rational nonvanishing result.

#### Article information

Source
Geom. Topol., Volume 21, Number 4 (2017), 2093-2160.

Dates
Revised: 7 July 2016
Accepted: 18 August 2016
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.gt/1508437638

Digital Object Identifier
doi:10.2140/gt.2017.21.2093

Mathematical Reviews number (MathSciNet)
MR3654105

Zentralblatt MATH identifier
1365.14027

Keywords
$A^1$-homotopy James construction

#### Citation

Asok, Aravind; Wickelgren, Kirsten; Williams, Ben. The simplicial suspension sequence in $\mathbb{A}^1\mskip-2mu$–homotopy. Geom. Topol. 21 (2017), no. 4, 2093--2160. doi:10.2140/gt.2017.21.2093. https://projecteuclid.org/euclid.gt/1508437638

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