Geometry & Topology

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

Aravind Asok, Kirsten Wickelgren, and Ben Williams

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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

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

Received: 6 August 2015
Revised: 7 July 2016
Accepted: 18 August 2016
First available in Project Euclid: 19 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15] 19E15: Algebraic cycles and motivic cohomology [See also 14C25, 14C35, 14F42]
Secondary: 55Q15: Whitehead products and generalizations 55Q20: Homotopy groups of wedges, joins, and simple spaces 55Q25: Hopf invariants

$A^1$-homotopy James construction


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.

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