24

LOWELL JONES

b.(E ) of 1.19. So the PL reductions v. fit together, i.e. v.

i

v ^o

I

'l

V ^

y. + I, where +1 denotes Whitney sum with a trivial bundle, for any i,j

such that (b.(£ ),8b.(£ )) is a codimension zero Poincare duality pair

in Zb±Uo).

The above PL reduction information can be converted (using the usual

Thorn transversality arguments) into a collection of degree one normal maps

1.21 (a)

(bi(Mo),8bi(MQ))

*

h

i

h

i

-•CbiUo^V^))

from PL manifold pairs (bi(M ) , 8b. (M )) . Here T^ is the stable Whitney

inverse to Tang (b.(M )). Because the y. fit together, the h. can be

selected to fit together giving the PL blocked normal map

1.21 (b)

with b.(h) = h.. The blocked normal map 1.2. (b) is a blocked surgery

problem, denoted by $. Surgery is done on $ by doing surgery on the

h. (of 1.21(a)) one at a time. Surgery can be completed on $, if by

doing surgery on $ it can be arranged that all the h-: b.(M ) - * b.(£Q)

are homotopy equivalences. An element 3 € L (£,Z ) is represented by

$, and 3 = 0 if and only if surgery can be completed on $.

There are two other surgery problems, b,($), $ related to $.

The surgery problem b,($) is the block of $ corresponding to b-(£)

which was defined for 1.16 above.

The surgery problem $ is represented by the n-fold covering of 1.21(b),

which corresponds to the covering £' - £ . $ represents an element

3 € L£(S,{1).