20 LOWELL JONES

(bi(0) = dimension (3b.. (£)).- Then (3b.. (£Q) -bi UQ) , Sb^^)) is also an

oriented Poincare duality pair having the same homological dimension as

the pair (bi UQ) , db± (£Q)) .

Proof of 1.17: It will suffice to show that the normal (homotopy)

fibration for an embedding of the given pairs in a PL ball pair (B ,3B )

(L=large) is a spherical fibration (see [3] or [32]). If in the statement

of 1.17, £ were replaced by £ the normal fibrations would be spherical

because all the above pairs would be PL manifold pairs. This would also

be true for £' in place of £ , and hence for £f in place of £ , by 1.5(c).

Finally, note each of the (X,3X) E (b.(£»),3b,(£»)) or (3b.(£')-b±(£'),

3b-(£!)) is an n-fold covering of the pairs (X,3X ) = (b.(£ ),3b.(£ )) or

(3b.(£ )-b.(£ ), 3b-(£ )), and the normal fibration for an embedding

(X,3X) c (B ,3B ) is fiber homotopic equivalent to an n-fold covering of

the normal fibration for an embedding (X,3X) c (B ,3B ). So the normal

fibration for an embedding (X,3X) c (B ,3B ) must also be spherical.

This completes the proof of 1.17.

Lemma 1.18. (a) The Spivak fibration Y-I f°r the Poincare duality pair

f

P

(b-,(£ ),3b,(£ )) has a PL reduction, y-• Tne composition N-R - b1(£!)

b- i (£ ) pulls Y"I back to the stable Whitney inverse of Tang (N-R). Here

p is the n-fold covering map and f the map of 1.5(c).

(b) Moreover, it may be assumed that b-(F ) is a lens space with

fundamental group TL and YiIK rr °\ is equal the stable Whitney inverse

of Tang (b3(£o)).

Proof of 1.18: The action ty: 2 xb-(^) •+ b~(£f), whose orbit space is

b^(£, ), is constructed by using 1.8 to pull back the action j : Z xS

s

m-k-l

Qf 112 along the map g

. b^U) + s"1"*"1 of 1.10. Note, b3(£) =

S and recall g: b-(£) - S has degree prime to n. So by

[[22], Theorem V], b^(£ ) may be chosen to be some lens space. There

will be no loss of generality if we assume there is a neighborhood for

b,(£_) in £ homeomorphic to b,(£ )xB (see 1.19 for details). Thus

'3^0' ^o

,lumtumuitJUiV

^ "3^oJ

I has a PL reduction

YTTII

l

b

3^ IV*o

Y- . I ha s a P L reductio n Y coming from the PL structure of