The simplicial suspension sequence in 𝔸1 –homotopy

Asok, Aravind¹ ; Wickelgren, Kirsten² ; Williams, Ben³

¹ Department of Mathematics, University of Southern California, 3620 S Vermont Ave, Los Angeles, CA 90089-2532, United States
² School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30308, United States
³ Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver BC V6T 1Z2, Canada

Geometry & topology, Tome 21 (2017) no. 4, pp. 2093-2160.

Résumé

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.

Détail

DOI : 10.2140/gt.2017.21.2093

Classification : 14F42, 19E15, 55Q15, 55Q20, 55Q25
Keywords: $A^1$-homotopy, James construction

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