6 LOWELL JONES
Theorem, but can be deduced from it by a careful comparison of the"#"-
pullback construction (of [[9],pg. 491]) and the ordinary "*" block
bundle pull back construction for block bundles (see 2.5 below).
This completes the outline for section 2 below.
Outline of Section 3. In this section it is proven that if
g.: M. R/R satisfies a condition (to be described in a minute) that it
can be arranged that the homological kernels of the (£.,6t.) have a very
simple form (see 3.2 below). Such an arrangement is clearly of advantage
when trying to complete surgery on the (t.,6t.)-
The condition which g.: M. - * R/R must satisfy is that certain
invariants
WT(Q)
0.8. aT(g.) , aT(g., ) W (Q),
=
T X
r-WT(Q)
T
"1|6Mi
T
of the Z -bordism class represented by g- must vanish, where WT(Q) denotes
the torsion subgroup of the Witt group of non-singular symmetric forms
defined on finite dimensional vector spaces over Q (see [7]).
To define aT(gi) we first note (by 0.1, 0.2, n=odd) that (K,3K) is
a rational homology manifold pair which is integrally orientable. It
follows that by pulling gT (K) apart at (g.. )" (K),a rational homology
K _ ,
manifold L is obtained such that 3L equals r copies of Cg.. ) (K).
By choosing integral orientations [K] e Hk((K, 3K) , Z) , [R] Hm(.(R, 3R) , Z) ,
a unique orientation [L] H ((L,3L),Z) is obtained consistent with [K],
[R], and the orientation [Mi] Hm ((Mi,6Mi),Z) for Mi, where mi = dimC^)
i
and I = dim(L). If dim(L) f 0 mod 4 set aT(gi) = 0. If dim(L) = 4A,
set V equal the cokernel of H2j,(3L,Q) c H2 (L,Q), and let
X: H2£(L,Q)xH (L,Q) - Q denote the intersection pairing with respect
to [L]. By Lefschetz duality, X: V*V - Q is well defined and non-singular
pairing, so it represents [X] W(Q) . Let n: W(Q) -*YT(Q) denote the
projection determined by the split exact sequence
0 + WT(Q) i W(Q)^J^Z
+
0
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