Leibniz rule on higher pages of unstable spectral sequences

Abstract

A natural composition $\odot$ on all pages of the lower central series spectral sequence for spheres is defined. Moreover, it is defined for $p$-lower central series spectral sequence of a simplicial group. It is proved that $r$th differential satisfies a "Leibiz rule with suspension": $d^r(a\odot \sigma b)=\pm d^ra\odot b+a\odot d^r\sigma b,$ where $\sigma$ is the suspension homomorphism.