Chapter 5. Spectral Sequences

This material was originally supposed to be a chapter of my Algebraic Topology book, but to get that book published sooner I split the spectral sequence chapter off with the idea of expanding it to a separate book later. This has not happened (for the usual reasons -- lack of time and energy) so I have reverted to the idea of having it as an extra chapter of the Algebraic Topology book. The current version of this chapter, about 110 pages dating mostly from around 2004, can be downloaded here.

The intention is for this to be an introduction to spectral sequences focusing on the most important ones in algebraic topology. Foremost is the Serre Spectral Sequence, for which there are both homology and cohomology versions. This together with some of its applications takes up more than half of the chapter in its current form. After this the Adams Spectral Sequence is introduced (just the additive structure so far), with some background material on spectra and with an application to computing a few stable homotopy groups of spheres. After this there are a few other miscellaneous topics listed below.

Table of Contents

5.1. The Serre Spectral Sequence

Exact couples. The Serre Spectral Sequence for Homology. Serre Classes. Generalizations and Further Properties. The Serre Spectral Sequence for Cohomology. Rational Homotopy Groups. Localization of Spaces. Cohomology of Eilenberg-MacLane Spaces. Computing Homotopy Groups of Spheres.

5.2. The Adams Spectral Sequence

Spectra. Constructing the Adams Spectral Sequence. Computing a Few Stable Homotopy Groups of Spheres.

Additional Topics

5.A. Whitehead's Exact Sequence [a nice application of exact couples]

5.B. The Bockstein Spectral Sequence [not yet included]

5.C. The Mayer-Vietoris Spectral Sequence [not yet included]

5.D. The EHP Spectral Sequence

5.E. Eilenberg-Moore Spectral Sequences [following the geometric constructions due to Larry Smith and Luke Hodgkin, without any applications yet]