# Simplicial James-Hopf map and decompositions of the unstable Adams spectral sequence for suspensions

## TL;DR

This paper extends the classical James-Hopf invariant to a simplicial group context, enabling a functorial decomposition of the unstable Adams spectral sequence for suspensions using combinatorial group theory methods.

## Contribution

It introduces a simplicial group version of the James-Hopf invariant and provides a new decomposition of the spectral sequence related to the lower p-central series.

## Key findings

- Realized coalgebra idempotents at the simplicial set level
- Obtained a functorial decomposition of the spectral sequence
- Extended classical invariants to a simplicial setting

## Abstract

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