The simplicial EHP sequence in $\mathbb{A}^{1}$–algebraic topology

Abstract

We give a tool for understanding simplicial desuspension in ${\mathbb{A}}^1$–algebraic topology: we show that $X\to \mathrm{\Omega }\left(S^1\wedge X\right)\to \mathrm{\Omega }\left(S^1\wedge X\wedge X\right)$ is a fiber sequence up to homotopy in $2$–localized ${\mathbb{A}}^1$ algebraic topology for $X={\left(S^1\right)}^m\wedge {\mathbb{G}}_m^{\wedge q}$ with $m>1$. It follows that there is an EHP spectral sequence $${\mathbb{Z}}_{\left(2\right)}\otimes {\pi }_{n+1+i}^{{\mathbb{A}}^1}\left(S^{2n+2m+1}\wedge {\left({\mathbb{G}}_m\right)}^{\wedge 2q}\right)\Rightarrow {\mathbb{Z}}_{\left(2\right)}\otimes {\pi }_i^{{\mathbb{A}}^1,s}\left(S^m\wedge {\left({\mathbb{G}}_m\right)}^{\wedge q}\right).$$