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 $$Z_{ (2)} \otimes \pi_{ n+1+i}^{ A1} (S^{ 2n+2m+1} \wedge (G_m)^{ \wedge 2q}) \Rightarrow Z_{ (2)} \otimes \pi_{ i}^{ A1, s} (S^{ m} \wedge (G_m)^{ \wedge q}).$$
Categories: math.AT math.AG