Comultiplication in the Serre Spectral Sequence

TL;DR

This paper demonstrates that the homological Serre spectral sequence with field coefficients possesses a coalgebra structure, with comultiplication on the $E^2$ page derived from standard homology comultiplication, supported by example computations.

Contribution

It establishes the coalgebra structure of the Serre spectral sequence and identifies the comultiplication on the $E^2$ page as induced by homology comultiplication.

Findings

Spectral sequence with field coefficients is a coalgebra.

Comultiplication on $E^2$ is induced by homology.

Examples illustrate the co-Leibniz rule.

Abstract

We show the homological Serre spectral sequence with coefficients in a field is a spectral sequence of coalgebras. We also identify the comultiplication on the $E^2$ page of the spectral sequence as being induced by the usual comultiplication in homology. At the end, we provide some example computations highlighting the use the co-Leibniz rule.