The simplicial suspension sequence in 𝔸1 –homotopy

Abstract

We study a version of the James model for the loop space of a suspension in unstable 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 A 1 -homotopy theory: our result refines F. Morel's A 1 -simplicial suspension theorem. We then describe some E 1 -differentials in the EHP sequence in A 1 -homotopy theory. These results are analogous to classical results of G.W. Whitehead's. Using these tools, we deduce some new results about unstable A 1 -homotopy sheaves of motivic spheres, including the counterpart of a classical rational non-vanishing result.

Motivic spheres and the image of the Suslin–Hurewicz map

“…Granted the results mentioned in the preceding paragraph, Suslin's conjecture for general n may be interpreted as a statement about the structure of A 1 -homotopy sheaves of BGL n . Once reformulated in terms of A 1 -homotopy sheaves, we establish Suslin's conjecture in degree 5 by showing that the relevant A 1 -homotopy sheaf computation may be related to a computation of an unstable A 1 -homotopy sheaf of a motivic sphere, refining a key computation of [AWW17]. In particular, we establish the following result, which is one of the main results of Section 3.…”

“…For example, we will write R Zar for the Zariski fibrant replacement functor with respect to the injective Zariski local model structure on Spc k , and R Nis for the corresponding construction in the Nisnevich local model structure. Our notation for A 1 -homotopy sheaves follows that of [AWW17] on which this paper builds.…”

“…We note here that in [RSØ19] the Hermitian K-theory spectrum is denoted KQ; we have chosen to follow Schlichting's notation since it was consistent with [AWW17]. Furthermore, the computation in [RSØ19] does not look exactly like this, but their statement implies this one because the map out of π A 1 4 (Ω ∞ P 1 Σ ∞ P 1 S 3+3α ) in their description is the unit map from the sphere spectrum to the spectrum KQ.…”

Unstable motivic homotopy theory

“…The conjecture is proved for n = 4, again by analysis of a group in the 1-stem: π A 1 3 (A 3 − {0}). In spite of considerable attention, [AF13], [AF17], [AWW17], the groups π A 1 n (A n − {0}) have not been calculated for n ≥ 4, and the problem remains open.…”

Desuspensions of S1 ∧ (ℙℚ1 −{0,1,∞})

“…There are topological obstructions to desuspension coming from James-Hopf maps, and they generalize to the setting of A 1 -homotopy theory [24,1] as was known to Morel. They do not a priori have a relationship with the Galois action on πé t 1 (P 1 Q −{0, 1, ∞}).…”