l.A. ADDITIVE STRUCTUR E 5
Note that
E*'J
= 0 if i 0 or j 0. Such a first-quadrant spectral sequence is
always convergent because the arrows eventually stick out of the first quadrant.
Put FlW = FlW(E-1 R). There are short exact sequences
0 »
F£IT E2^0
- 0
0 FlW - F ^ J T - Ej^ 1 ' 1 - 0
0 FllW F2~2JT - E!T2'2 - 0
0 -
FXH?
- IT
E^z
- 0
We consider the 'edges' of the first quadrant, i.e. the X- and F-axis. Note that
°°
F
i
H
°
is a quotient of
F°H4
= W{E; R), and that
Eo,i
g
Eo.i
because all arrows which come from the left are zero. Since the base B is assumed
to be path connected, H°(B;H*(F]R)) = H*{F;R) and the following diagram
commutes, cp. Spanier [90] 9.5.
Ul(F;R) H*(E;B)

2 ^ D ^ o o
The projection E B can also be interpreted in terms of the Leray-Serre spectral
sequence. Note that there is a surjection
E'-°^EJ£
because all arrows starting on the X-axis are zero. Moreover,
E J ^
i
^ = F
i
I T C H \
If the fibre F is path connected, then H # ; H°(F; R)) 9* U9(B;R) and the dia-
gram
Ul(E; R) - JT(£; R)
u I
E*i° . E*'
commutes, cp. Spanier [90] 9.5.
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