Preludes to the Eilenberg-Moore and the Leray-Serre spectral sequences

TL;DR

This paper explores the relationship between the Leray-Serre and Eilenberg-Moore spectral sequences, introducing a joint refinement involving preludes that clarify their degeneration properties and connections.

Contribution

It introduces a joint tri-graded refinement of both spectral sequences using preludes, revealing their degeneration behavior and a local-to-global property.

Findings

One prelude always degenerates from its second page.

The other prelude degenerates for all base spaces iff it does so for contractible bases.

The refinement clarifies the relationship between the spectral sequences.

Abstract

The Leray-Serre and the Eilenberg-Moore spectral sequences are fundamental tools for computing the cohomology of a group or, more generally, of a space. We describe the relationship between these two spectral sequences when both of them share the same abutment. There exists a joint tri-graded refinement of the Leray--Serre and the Eilenberg--Moore spectral sequence. This refinement involves two more spectral sequences, the preludes from the title, which abut to the initial terms of the Leray--Serre and the Eilenberg--Moore spectral sequence, respectively. We show that one of these always degenerates from its second page on and that the other one satisfies a local-to-global property: It degenerates for all possible base spaces if and only if it does so when the base space is contractible.