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