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 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

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 ),
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