COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL
5
surgery problem back along the composition
(where f is the classifying map for t noted in the outline of Step 1),
and amalgamate this pull-back first over 6M- (the codimension 1 singular
set of M.) and then over M. to get a Z -surgery problem (£.,$£.). It
follows directly from the Characteristic Variety Theorem (see [[14] , 1.4])
that surgery can be completed on t (block by block) if surgery can be
completed on each (t. ,5t-).
In the general case, when K may not be a PL manifold, there is a more
indirect procedure for constructing (£.,$£.) which will give (up to
normal Z -cobordism equivalence) the same (t.,6£.) as constructed in
the last paragraph for K a PL manifold. Begin by choosing a character-
istic variety for the quotient space R/R, denoted {g,:M.+R/R|i=l,2,...,£},
consisting of mappings from oriented smooth manifolds or smooth Z -
manifolds. Let K* - ' denote the first barycentric subdivision of the
triangulation of K by T (recall T is a triangulation of N which also
triangulates K c N). First putting g.: 6M. - R/R into transverse position
to every simplex of K^ , and then extending this to a transversality of
g-: M- •+ R/R to every simplex of K* * \ we obtain "correspondences"
6ci: 6n± + K^ , ci: n
±
- * K ^ as described in [[9],1.2]. Here 6 n±»n±
are the block space structures for (g., ) (K) , g. (K) having for
blocks (g.I.w ) (A),g- (A) where A is any simplex of K^ ' ; and
6ciCCgil6M ) " 1 ^ A ^ = A ci(gT1(A)) = A. Note that K(1) is the "base
1
i
space", for the blocked space structure £ (see [[9],pg. 490]). Since
the blocks of £ and t are in a one-one correspondence in a way that is
consistent with boundary operators, it follows that £ ; and t have the
same base space, K^ ^. Thus t can be pulled back along 5c- and c. (see
[[9], pg. 491]) to get blocked surgery problems 5ci#(t), ci#(t), having
ordinary surgery problems and Z -surgery problems as blocks respectively.
The Z -surgery problem (t.,St.) is obtained by amalgamating the blocks
of 6c.#(£) to get 6t. and by amalgamating the blocksof c^#(t) to get t^.
Surgery can be completed on t if it can on all the (t.,61^) (see below).
This fact is not an immediate consequence of the Characteristic Variety
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