28 LOWELL JONES

Let a(g) € L.,(£,{1}) be represented by the problem of completing surgery

on each block of g, modulo gigy Let H: W - * £ x[0,l] be an arbitrary

blocked surgery cobordism of h: M -* • E , ending at another completion of

surgery for $, 9+H: 3+W - * £*1. The problem of completing surgery on H,

modulo H|aw, represents a surgery obstruction a(H) € L..(£;,Z ). Note, if

we could choose H so that under T: L^C^Z ) - * L,(£,{1}) a (H) gets mapped

to a(g), then 9+H: 9+W - £QX1 would represent a completion of surgery

for $ consistent with the completion of surgery for $ in 1.23 i.e.,

A h

3 would be zero. Any element of L,(£,Z ) can be realized in the form a(H).

A i n

So the obstruction 3 to completing surgery on $ consistent with the

completion of surgery on $ given in 1.22, corresponds to an element in

the quotient group

L£(£,{1}) / Image (x).

By the last paragraph, every element in this quotient group has order

A

dividing a power of n. This shows 3 has order dividing a power of n.

We have reduced the problem of proving 0.7 to completing surgery on

$ (cf. 1.22, 1.24). In the rest of this chapter it is shown that such a

completion of surgery exists if and only if there is a completion of

surgery for a related surgery problem t. This reduction is useful because

the blocks b• (t) of t all have bery simple homological kernels. In

Chapters 2,3,4,5 below it will be shown that surgery can be completed

on t. This will complete the proof of 0.7.

1.25. The surgery problem t.

Set

0O ^oE ^oU^o u'-'u V u (-5').

(n+l)-fold

where -£' equals £' but with the opposite orientation on each pair

(b^o^u*));

(b ) b±c^) = (bi(S0) u W u*"u biUo)} u t-bi^-)3

(n+l)-fold

where -b-(£') equals b.(£') but with the opposite orientation on the pair