CHAPTER 1

The Leray-Serre spectral sequence

Our main tool from algebraic topology is the Leray-Serre spectral sequence.

It relates the cohomology rings of the fibre and the base of a fibration with the

cohomology of the total space. Although spectral sequences are standard devices

in topology, they tend to be somewhat intimidating to non-specialists. The aim of

this chapter is to give a basic introduction to the relevant notions and techniques.

Let us consider a specific example, the Leray-Serre spectral sequence with field

coefficients. Let K be a field, and let .F, B be topological spaces. The Kiinneth

Theorem asserts that the K-cohomology of the product E = F x B is given by

U.k(E;K)^

0

HX^;/0®HJ'(£;K).

2+7

= fc

It is convenient to visualize the /^-modules E M = VLl{F\K) 0 H3(B;K) as dis-

tributed on the lattice

Z2 CM2:

attached to the point (i,j) is the vector space E

u

.

Then

~H.k(E;

K) is obtained by adding up all vector spaces along the line i+j = k.

Now suppose that E is not a product, but the total space of a (jFf-simple) fibre

bundle F — E — B. Then the cohomology of E is 'smaller' than the cohomology

of the trivial bundle F x B. The recipe to obtain the K-modules

Hk(E;K)

is as

follows. Start with the collection of if-modules E ^ =

H%(F;K)

g W(B;K) as

before. There exists a collection of maps, the differentials, denoted c^ :

E1^3

—

E%2 ,3~ . These differentials should be visualized as arrows going from (i, j) to

(* + 2 , j - l ) .

E ^

They satisfy the relation ^ 2 ° ^ 2 = 0, so we can take their cohomology (kernel mod

image) at each point (i, j). Call the resulting K-module E^3. Note that this is a

quotient of a submodule of E^'*7, so its dimension is smaller. Again, these K-modules

should be viewed as distributed in the plane. There is another differential i3, this

time from (i,j) to (i + 3, j — 2). Now this process is iterated ad infinitum. The

arrows ^2,^3,^4,. • • become longer, and their slope approaches — 1. In the limit,

one obtains a collection of K-modules denoted E ^ . Similarly as in the Kiinneth

Theorem one has

2+7

= fc