4 1 . THELERAY-SERRE SPECTRAL SEQUENCE
collapses at E
n
if all differentials vanish for r n, in which case E
n
= EQO. Note
also that if E^' = 0, then E ^ = 0 for all r n.
Each term E
r
can be visualized as the grid I? C R2. The point with coordinates
(i,j) represents E^'J, and the differentials are arrows between elements of the grid
pointing r steps to the right- and r— 1 steps downwards. As r increases, the arrows
become longer and their slope approaches 1.
EI
T?i-\-r,j-\-l—r
SLir
Typically, the E2-term contains lots of zeros. The first differentials are zero until
the arrows become long enough to reach from one non-zero entry to another one.
If the region in R2 containing the non-zero terms is bounded, the arrows become
too long after some time, and the spectral sequence collapses.
LEMMA
1.6. If R is a field, and i/dim(E
n
) oo7 then
dim(Eoo) dim(En).
Equality holds if and only if the spectral sequence collapses at n.
PROOF .
We have
dim(Er) = dim(im(Zr)) + dim(ker(dr)),
whence
dim(E
r+
i) = dim(Er) 2dim(im(dr)).

THEOREM
1.7 (The Leray-Serre spectral sequence).
Let
E
B
be a fibration over a path-connected space B. The fundamental group iri(B) acts
on the fibre F and hence on the cohomology
~R9(F;R).
If this action is trivial
(e.g. if B is 1-connected), then the fibration is called i^-simple. Suppose that this
is the case. Then there is an ^-^pGctral sequence which converges to the bigraded
module associated to some (bounded and convergent) filtration of W(E;R), with
EZ2'J ^Ul(B;W(F;R)).
PROOF. See Spanier [90] 9.4.9.
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