COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL

23

satisfying the following:

(a) b.(£ ) corresponds to the subset b-(^ )xixOxOx...xOof b.(£ )

]£if i-2 or if b±U0) *• V^o 3 '

(b) bi(^0) corresponds to the subset b.(£ ) xlxlxOxOxOx. . . xO of

)x[-i,l]A

iff

i^2 but

b±(0)

i

(c) Each of the triples

b

i

( ^

0

) x [ - l , l ] ^ i i^ 2 bu t b U c b

2

U

Q

)

Cb1C50)"D, 3D-3b

1

(5

0

), 3b

1

(^

0

)-D)

and

(D,aDfl3b

1

(^

o

), 3D-ab

1

(^

0

))

is a Poincare duality triple. Here 3D = (3bi(SQ)x[-1,1]£) u 0*^ ) x

3[-l,l]V

The Surgery Problem $. Recall that a PL surgery problem is specified

by a finite Poincare duality space X, a PL reduction for the Spivak

fibration for the space X, and the fundamental group data given by a

map X - * K(TT,1). The corresponding surgery group is then denoted by

L~J(TT) or L (IT)depending on whether X is a simple Poincare duality space

and surgery is to be completed up to simple homotopy equivalence, or X is

just a Poincare duality space and surgery is to be completed up to homotopy

equivalence. We shall only be interested in the L (IT)setting in this

q

paper. Here q equals the homological dimension of X. More complicated

surgery groups L (£,TT) have been defined in [[13],3.3], and similar

surgery groups have been studied by W. Browder and F. Quinn [4]. Each

element & G L (£,TT) is represented by conglomerate of surgery problems

$, one over each block b^(£) of £ denoted by b.($), with dimension

(ti($)) = dimfb. CO)+cL« If 3 = 0 this means that surgery can be completed

on all of the b.($) simultaneously. By way of further reviewing these

concepts lemmas 1.17-1.20 will be used to construct a surgery problem $

representing an element 3 € L (£,Z ).

Construction of $. Let y- denote the Spivak fibration of

(bi(£0),3bi(C0)). By 1.18 y1 has a PL reduction y^. By 1.20 the PL

reduction Y", induces a PL reduction y~ . for all 2ix. The 7-Ii2 are

' 1 i — — ' —

in effect induced by y^ and the collarings 3b.(£ ) x [0,1] for 3b.(£ ) in