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