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
= 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
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
is as
follows. Start with the collection of if-modules E ^ =
g W(B;K) as
before. There exists a collection of maps, the differentials, denoted c^ :

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
= fc
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