Leibniz rule on higher pages of unstable spectral sequences

TL;DR

This paper introduces a natural composition operation on spectral sequence pages for spheres and simplicial groups, proving a Leibniz rule with suspension for differentials, advancing understanding of spectral sequence algebraic structures.

Contribution

It defines a composition on spectral sequence pages and proves a Leibniz rule with suspension for differentials, extending algebraic tools for spectral sequences in topology.

Findings

Defined a composition on spectral sequence pages for spheres and simplicial groups

Proved the Leibniz rule with suspension for differentials

Enhanced algebraic understanding of spectral sequence structures

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.