12 LOWELL JONES

Lemma 1.4. If cp: Znx£ - £ is a free PL action, leaving invariant each

block b . (£)oof££, thencc p extends toa

which has K a N for fixed point set.

bloc k b . (£ ) f , the n p extend s t o semi-free PL action cp: TL xN - N

Thus to prove Theorem 0.7 of the introduction it suffices to find a

free PL action cp:Z x-£- * £ which leaves the blocks of £ invariant. An

attempt to construct the desired free action on £ is made inthe

following two steps.

Outline of Step 1. In this step the block space £ will be replaced

by a finite CW complex V , containing subcomplexes b.(.£f) l£j£x which

satisfy the following.

1.5. (a) b-CO n b U) = u b U) if and only if b.(C') n b (£') =

! q j

e J

3 i q

u b.(5»)

j€J 3

(b) b^S) n bjCC) ^ 0 if and only if b±W) n b.(V) = 0.

(c) There is a homotopy equivalence f: £ -* • £' such that

fi, ^ is a homotopy equivalence b.(.£ ) -* • b.C£') for all j.

(d) There is action i| : 2 x£f - * £' which leaves

n

invariant eac h

subcomplexabfree,.cellular

•(£')

The following notation will be used in the rest of this paper.

1.6. (a) £Q is the orbit space of i| : %nxZl + £! •

(b) b.(^Q) is the orbit space of ip:Zn*b. C€'} " * b.(£').

(c) If 3b. (C) = U b.U), then 3b.(£'), 3b. (£ ) are defined

J

ieJ

x J J °

as U b.(£') and U b.(£ ) respectively.

i£J J i£J J °

Outline of Step 2. In this step an attempt is made to replace £ ,

and the b.(£ ),by a block space M having PL manifold blocks b.(M ),

such that the following conditions are satisfied.

1.7 (a) There is a homotopy equivalence h: M - * £0 so that hi, (u \

defines a homotopy equivalence b.(M ) • b.(£ ) for all l^jf_x.

(b) Let h: M + £f be the n-fold covering of h corresponding to

the covering £f - * £ . Let b.(M) be the n-fold covering of b.(MQ) . There

is PL homeomorphism r: £ •* • M satisfying r(.b.(£)) = b.(M) for all j.