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.