Basic Algebraic Topology

Chapter 13 Spectral Sequences

Spectral sequences were invented by Leray in the context of study of cohomology of sheaves. Soon it was taken up by other authors who realized its potential in application to a wide class of situations. At present, there are more than three dozen various spectral sequences. In this chapter, we introduce the basics of this wonderful tool with a single example, viz., the Leray–Serre spectral sequence of a fibration. We give both homology and cohomology versions, and include a variety of applications such as the generalised Wang sequence, Gysin sequence, generalised Hurewicz theorem and Whitehead theorem. As a culmination of all this and several other results developed in this book, we present Serre’s result on homotopy ...