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),
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