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

R«

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