4 1 . THELERAY-SERRE SPECTRAL SEQUENCE
collapses at E
if all differentials vanish for r n, in which case E
= EQO. Note
also that if E^' = 0, then E ^ = 0 for all r n.
Each term E
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.
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.
1.6. If R is a field, and i/dim(E
) oo7 then
Equality holds if and only if the spectral sequence collapses at n.
dim(Er) = dim(im(Zr)) + dim(ker(dr)),
i) = dim(Er) — 2dim(im(dr)).
1.7 (The Leray-Serre spectral sequence).
be a fibration over a path-connected space B. The fundamental group iri(B) acts
on the fibre F and hence on the cohomology
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
PROOF. See Spanier  9.4.9.