The examples developed in Chapters 5 through 10 by no means exhaust the significant appearances of spectral sequences in mathematics. A recent search on the keyword spectral sequence in the database mathscinet delivered more than 2800 reviews in which the words are mentioned. In this chapter and the next, we present a kind of catalogue, by no means complete or self-contained, meant to offer the reader a glimpse of the scope of the applications of spectral sequences. (Similar catalogues are found in the books of [Griffiths-Harris78], [Benson91], [Weibel94], and the fundamental paper of [Boardman99].) I hope that the reader will find a useful example in this collection or at least the sense in which spectral sequences can be applied in his or her field of interest. The algebraic foundations supplied in Chapters 1, 2, and 3 are sufficient to understand the constructions found in the cited references.

In this chapter we concentrate on diverse applications of spectral sequences in algebraic and differential topology. The examples are organized loosely under the rubricks of spectral sequences associated to a mapping or space of mappings (§11.1), spectral sequences derived for the computation of generalized homology and cohomology theories (§11.2), other Adams spectral sequences (§11.3), spectral sequences that play a role in equivariant homotopy theory (§11.4), and finally, miscellaneous examples (§11.5).

Spectral sequences for mappings and spaces of mappings

The Leray-Serre spectral sequence is associated to a fibration, π : E → B.