Leibniz Rule on Higher Pages of Unstable Spectral Sequences

Abstract

A natural composition ⊙ on all pages of the lower central series spectral sequence for spheres is defined. Moreover, it is defined for the p-lower central series spectral sequence of a simplicial group. It is proved that the rth differential satisfies a ‘Leibniz rule with suspension’: dr(a ⊙ σ b) = ±dra ⊙ b + a ⊙ dr σ b, where σ is the suspension homomorphism.