14 LOWELL JONES

Proposition 1.9. X,Y,f: X + Y are as in 1.8. Let r : Z xX - + X be a

1 x n

free cellular action which acts trivially on the integral homology of X.

Then, after replacing Y by a homotopy equivalent finite CW complex (if

need be), there will be a free cellular action r : Z xY - • Y which acts

J y n

trivially on the integral homology of Y and satisfies r of = for .

y x

The following lemma is also needed.

Lemma 1.10. There is a map g: £ - S (k=dimension(K),m=dimension(N))

s a mod n such that for each block b. (£) in £ the restriction g., - i

homotopy equivalence.

Proof of 1.10: By the hypothesis of 0.6 N-R is a simply connected space

having the same Z -homology as the m-k-1 dimensional sphere S . Use

the Spanier-Whitehead dual of the Serre-Hurewicz theorem to choose a

mod-n homotopy equivalence g: N-R -* • S . Note, also by the hypothesis

of 0.6, that each inclusion b.(£) c £ is a mod-n homotopy equivalence.

So g: £ - Sm~ " satisfies the conclusions of 1.10.

To be consistent with the previous notation, we will set

b^R) = e

bi(R) = e n R

bt(K) = e n K

where e is the dual cell in C with b.(£) = e n R and 3lx. Set

9b, (R) = U b . (R)

1

j£JJ

3b, (R)

E

U b (R)

jej J

3b. (K) E U b (K)

jGJ J

where 3b. (C) = U b.(.£) and 3£ix.

1 i€J 3

Lemma 1.11. There is a retraction map c: R -* • K, satisfying

cCb^R)) = bi(K) 3ix.