The simplicial EHP sequence in 𝔸1–algebraic topology

Abstract

We give a tool for understanding simplicial desuspension in A 1 -algebraic topology: we show that X → Ω(S 1 ∧ X) → Ω(S 1 ∧ X ∧ X) is a fiber sequence up to homotopy in 2-localized A 1 algebraic topology for X = (S 1 ) m ∧ G ∧q m with m > 1. It follows that there is an EHP sequence spectral sequence

Localization and nilpotent spaces in A^1-homotopy theory

“…Versions of some facts here about R-localization were worked out in [WW19,§3], though they mainly analyze localization functors for the analogs of simply connected or, more generally, simple spaces. Moreover, it was observed there that various inequivalent versions of the theory exist, just as in the classical situation.…”

“…After reviewing some aspects of R-locality for sheaves of groups, we then introduce a notion of R-local weak equivalence. We choose definitions that are well-adapted to stalkwise analysis, following [WW19].…”

“…The natural analog of "homology" in the context of A 1 -homotopy theory is the A 1 -homology theory studied by F. Morel in [Mor12, §6.2]. For spaces that are simply connected, there is a tight connection between A 1 -homotopy and A 1 -homology and so previous approaches to localization, e.g., [WW19,§3] use equivalent alternative approaches circumventing discussion of A 1 -homology.…”

The simplicial suspension sequence in 𝔸1 –homotopy

“…However, since the A 1 -EHP sequences of Theorem 3.2.1 are truncated, some algebraic manipulation is required to form an exact couple (for example extending the sequences to the left by a kernel and then zeros), and the resulting spectral sequence will differ from the A 1 -EHP spectral sequence. Nevertheless, it is shown in [WW14] that after localizing at 2, the exact sequences of Theorem 3.2.1 are "low-degree portions" of suitable long exact sequences, and these long exact sequences yield the 2-primary A 1 -EHP sequence (with the expected convergence properties).…”

“…We analyze the low-degree portion of this sequence in Theorem 3.3.13 and give a more explicit description of the sequence in the first degree where the suspension map fails to be an isomorphism. When X is a motivic sphere, it is shown in [WW14] that the exact sequences of Theorem 1 can be extended to all degrees after localizing at 2. By suitably varying the input sphere, these sequences can be strung together to obtain the EHP spectral sequence converging to the 2-local S 1 -stable A 1 -homotopy sheaves of spheres.…”