16 LOWELL JONES
By setting
X s b^R'),
X' = 3bi(R«),
Y E bi(K) x Sm"k_1,
rx,
E
*,
ry
E
$,
f E CXg,
1.8 may be applied to each i| : Z x3b.(R') + 3bi(R') to obtain an action
ij : Zn x b^R') - bi(R'). The hypotheses of 1.8 are satisfied for the
above choices of X, X', Y, r , , r , f, because of 1.10, 1.11, 1.12,
x y
1.13j (b),and lemma 1.14 (a) below. All of these extensions together
yield *: Z
n
x R]
+ 1
- Rj+1-
By the construction of ^: Z XR' -* R*, there is a homotopy equivalence
q: R' + R such that qi, ,' . is a homotopy equivalence b.(R') * b.(R)
for all blocks b^R') in R. We define V, and b^ 1 } as follows:
q *
(a) V is the mapping cylinder of the composite R - R = N-R.
(b) b.j(^!) E £', and b2(£f) is the mapping cylinder of the
a
composite
(RPI3N).'
- » RD3N c 3N-R.
(c) bjCC) Eb.(R') x 0 for 3j£x.
To complete the construction of ^: 2 x£' - * £' we must extend ty: 7L xR' - * Rf
to ]p: TL x£» - * £! so that i p leaves b2(£f) invariant (here we have identi-
fied R1 with R! x 0 in £'). First apply 1.9, with
X
E

Y
E
V
r
=
\b
x T
f E inclusion,
to extend \\: ZxR' + R' to |i:2 x^' + 5'. To do this we may have to replace
(mod R') the pair (£', b2(£')) by a homotopy equivalent finite CW complex
pair. That the hypothesis of 1.9 is satisfied for the above choice of
X,Y,r ,f is assured by the hypothesis of 0.6, lemma 1.14~(b)-below. Note
that \\): 7L x£f - £f may not leave b7(£') invariant, but leaves all other
bi(^f) invariant. To fix this we apply 1.8, with
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