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
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