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