The Simplicial EHP Sequence in A1-Algebraic Topology

We give a tool for understanding simplicial desuspension in A1-algebraic topology: we show that $X \to \Omega (S^1 \wedge X)\to \Omega (S^1 \wedge X \wedge X)$ is a fiber sequence up to homotopy in 2-localized A1-algebraic topology for $X=(S^1{)}^m\wedge G_m^{\wedge q}$ with $m>1$. It follows that there is an EHP sequence spectral sequence $$Z_{(2)}\otimes {\pi }_{n+1+i}^{A1}(S^{2n+2m+1}\wedge (G_m{)}^{\wedge 2q})\Rightarrow Z_{(2)}\otimes {\pi }_i^{A1,s}(S^m\wedge (G_m{)}^{\wedge q})\mathrm{.}$$