Spectral Sequences and Applications

Abstract

This chapter begins with the abstract properties of spectral sequences and their relation to the double complexes encountered earlier. Then in Section 15 comes the crucial transition to integer coefficients. Many, but not all, of the constructions for the de Rham theory carry over to the singular theory. We point out the similarities and the differences whenever appropriate. In particular, there is a very brief discussion of the Künneth formula and the universal coefficient theorems in this new setting. Thereafter we apply the spectral sequences to the path fibration of Serre and compute the cohomology of the loop space of a sphere. The short review of homotopy theory that follows includes a digression into Morse theory, where we sketch a proof that compact manifolds are CW complexes. In connection with the computation of π₃ (S ²), we also discuss the Hopf invariant and the linking number and explore the rather subtle aspects of Poincaré duality concerned with the boundary of a submanifold. Returning to the spectral sequences, we compute the cohomology of certain Eilenberg—MacLane spaces. The Eilenberg—MacLane spaces may be pieced together into a twisted product that approximates a given space. They are in this sense the basic building blocks of homotopy theory. As an application, we show that π₅ (S ³) =ℤ₂. We conclude with a very brief introduction to the rational homotopy theory of Dennis Sullivan. A more detailed overview of this chapter may be obtained by reading the introductions to the various sections. One word about the notation: for simplicity we often omit the coefficients from the cohomology groups. This should not cause any confusion, as H*(X)always denotes the de Rham cohomology except in Sections 15 through 18, where in the context of the singular theory it stands for the singular cohomology.