22 LOWELL JONES

0 - P^+1 ® Z - kerO^ ® Z + Ht (U',b3(£')) , Z) ® Z - 0,

from which we conclude

Hi(tt,tb3(5,}),Z) ® Z ~ ^(( ^ ,b3(^),Z).

This last equality, 1.1, 1.5, and 1.16 imply that pof: (N-R,b3(£)) +

(C0b3(^ )) induces an isomorphism of integral homology groups as claimed.

This completes the proof of 1.18.

Lemma 1.19. The action ty: TL x£f • + £' of 1.5 can be chosen so that its

orbit space £ satisfies the following: Each 3b.C£Q) has a neighborhood

in biU0) homeomorphic to 3b-. UQ) * [0,1], with db± U0) ~ 3bt U0) x0;

each of the pairs

O b ^ ) x [0,1], ^b±a0) x {0,1})

and

0iC5o) " ^±aQ) x [0,1], ^b±a0) x 1)

is an oriented Poincare duality pair.

Proof of 1.19: Let £Q denote the union of all b.(£0)•in £Q such that

the homological dimension of the pair (b. (£0),3b. C£0)) is _k. The proof

is by induction over k. Suppose the conclusion of 1.19 is true for every

b^(.£0) in £ . For each pair (b.(£ ),3b.(£ )) having homological dimension

k+1 add 3bi(^Q) x [0,1] to ^ along the identification

8bi(^o) ~ 3bi(50) x 0.

Then add b.(£ ) to this extended £ by glueing along the identification

3bi(50) ~ 3V£0) x i.

k+1,

This gives £ for a new E , which satisfies the conclusion of 1.19

for all bi(C0) incj+1.

As an application of 1.19 we have the following result.

Lemma 1.20. Each subset b-(£ ) of £ is contained in a tubular neighbor-

o

hood D c £ homeomorphic to b.(E ) x [-1,1] , where I = m-dimension (b^CO),