8 LOWELL JONES

The surgery problem t pulls back along g. : X.. - » K, g. : X? -* K, to

Zr-surgery problems ti p ti

2

with ti = t^ ^ U t ^

2

(disjoint union).

The homology kernel of each block of t is simple, and (b) above assures

that this simple nature for the kernels pulls back along g7: X7 -* K,

so that the homological kernels for t .

7

and 6£.

7

are simple (see 3.2

1,

L I

, z

below). Since g.(X-.) c simplex of K, t. - . = X,xb_(£), where b^(t) is a

block of t having minimal dimension. Surgery can be completed on b-(.t)

because the obstruction lies in a surgery group L.(Z ) with i,n both odd

(see [1]). So it can be completed on t . , also.

1,1

This completes the outline of section 3.

Outline of Section 4. In this section it is shown that surgery can

be completed on (^,6^) if ^(Sii^M.) = ° crT(gi) = 0 (see 4.0). This

is done by a direct computation; based on the simple nature of the

homological kernels for £• and $#., established in section three under

the condition aTCgi) = 0, ^ ( g J ^ ) = 0.

The computations in this section can be viewed (philosophically,

at least) as the study of the relationship between surgery up to homotopy

equivalence and surgery up to some lesser equivalence relationship

l

W .

For example, letting "~" stand for rational homotopy equivalence can be

useful in the following way. In Step 1 of section 4 surgeries are

performed on (£.,6t.) changing the normal maps of (£-,$£•) into

— equivalences(see 4.1 below). On the other hand the inclusions

Lh(Z ) + L^(Q(2Z ))

of the homotopy equivalence surgery groups L. (7L ) into the /^-equivalence

surgery groups L.(Q(Z )) always have 2-primary groups for kernel and

cokernel (see [[30], 13A.4], [23], [6]). Thus the obstruction to

completing surgery (up to homotopy equivalence) on (£-,$£•) must have

order equal a power of 2. (This type of argument has been used in [13],

without the aid of the Characteristic Variety Theorem, to complete

surgery on t for n = even.) The author has not been able to determine

what "~" should be used in order to conclude that the obstruction to

completing surgery on (t-,6t.) vanishes. There is some negative evidence

to the existence of the right "~".