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

The Leray–Serre and the Eilenberg–Moore spectral sequence 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. When the preludes degenerate early enough, they appear to echo Deligne’s décalage, but in general, this is an illusion. We discuss several principal fibrations to illustrate the possible cases and give applications, in particular, to Lie groups, torus bundles, and generalizations.

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