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)).

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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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