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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

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.

$A^1$-homotopy, James construction
Mathematical Subject Classification 2010
Primary: 14F42, 19E15
Secondary: 55Q15, 55Q20, 55Q25
Received: 6 August 2015
Revised: 7 July 2016
Accepted: 18 August 2016
Published: 19 May 2017
Proposed: Haynes Miller
Seconded: Peter Teichner, Mark Behrens
Aravind Asok
Department of Mathematics
University of Southern California
3620 S Vermont Ave
Los Angeles, CA 90089-2532
United States
Kirsten Wickelgren
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30308
United States
Ben Williams
Department of Mathematics
University of British Columbia
1984 Mathematics Road
Vancouver BC V6T 1Z2