The surgery problem t pulls back along g. : X.. - » K, g. : X? -* K, to
Zr-surgery problems ti p ti
with ti = t^ ^ U t ^
(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 .
and 6£.
are simple (see 3.2
, 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.
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 $#., established in section three under
the condition aTCgi) = 0, ^ ( g J ^ ) = 0.
The computations in this section can be viewed (philosophically,
at least) as the study of the relationship between surgery up to homotopy
equivalence and surgery up to some lesser equivalence relationship
W .
For example, letting "~" stand for rational homotopy equivalence can be
useful in the following way. In Step 1 of section 4 surgeries are
performed on (£.,6t.) changing the normal maps of (£-,$£•) 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 "~".
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