A simplicial James-Hopf map and decompositions of the unstable Adams spectral sequence for suspensions (Q2414160)

In this paper the authors construct the combinatorial James-Hopf maps $H_k$ in the simplicial group setting by using Milnor's simplicial group model $F[-]$ for the loop suspension $\Omega\Sigma$ and combinatorial group theory. By using them and the Cohen group they show that certain coalgebra idempotents can be realized at the based simplicial set $\text{sSet}_*$ level. As an application they obtain a functorial decomposition of the unstable Adams spectral sequence for suspensions.