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
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