A simplicial James–Hopf map and decompositions of the unstable Adams spectral sequence for suspensions

Abstract

We use combinatorial group theory methods to extend the definition of the classical James–Hopf invariant to a simplicial group setting. This allows us to realize certain coalgebra idempotents at an ${sSet}_{\ast }$ level and obtain a functorial decomposition of the spectral sequence, associated with the lower $p$–central series filtration on a free simplicial group.