26 LOWELL JONES

g: V - b3U') x [0,1]

be a surgery cobordism beginning at d_g = h, and ending at a homotopy

equivalence 3+g: 3+V - b-(£!)xl which represents a completion of surgery

for b~($) consistent with the completion of surgery for $ given in 1.22.

Let cr(g) e L denote the simply connected surgery obstruction to m~,((l))

K

completing surgery on g modulo gl

v

For the moment let H,: W -•b-(£ ) x

[0,1] denote any surgery cobordism of h- ending at a homotopy equivalence

3+H3: a+W - b3(S0)xl. Let

a

(H3) € L^_k(Zn) denote the Z^surgery

obstruction to completing surgery on H- modulo H_i . Let tr: L , (Z ) -*»

j^w m~K n

L , ((I)) denote the transfer homorphism (pg. 168, [30]). Note that

the surgery cobordism, H-, will have the desired property if tr(a(H-)) =

a(g)- Note also that H, can be constructed so that cr(H-) is any desired

element of L , (2n), and recall that the transfer marp tr: L , -*-

m-K v y '

m-kv(2n)

L , ({l}) is an epimorphism when n = odd (see [[30], 13A.4]). Thus there

is a completion of surgery for b-($) consistent with the completion of

surgery for §.

Proposition 1.24. If there is a completion of surgery for $, then there

is a completion o£ surgery for $ consistent with the completion of

surgery for $ in 1.22.

Proof of 1.24. A completion of surgery for $ is represented by a map

h: U • * E

o o

such that each h.: b.(M ) -* • b.(£ ) is a homotopy equivalence.

The surgery groups L ((€,b-(£)),Z ) have been defined on pgs. 377-

379 of [13]. For example the surgery problem $ and the completion of

surgery on b3(J) given in 1.23, represent an element Y € L ((£,b3(£)) ,Z ).

The completion of surgery on b3($) given in 1.22 can be extended to a

completion of surgery on all of $ if and only if Y = 0.

A A

There are also the surgery groups L (' € ,Z ) and L ((£,b-(£)) ,Z ).

A CI 11 O 11

n

V»

Elements of L (€,Z ) are represented by elements of L C£Z ) having

specified completions surgery ontogetheZ their -coveringcompletion surgejry problems.fo

For example the

surgerof

y proble m j , r with the

* A A,

surgery for $ specified in 1.22, represents an element 3 of L (£,Z ),