COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL 27
Surgery can be completed on $ in a way consistent with the completion of
A Ah
surgery on $ in 1.22, if and only if 3 = 0. Elements of L ((£,b- (£)),Z )
are represented by elements of L (£,Z ) with specified completions of
surgery on their TL -covering surgery problems and on their b~( ) blocks
which are consistent with one another. For example, the surgery problem
$, together with the completions of surgery for b.,($) and $ given in 1.23,
A Ah
represents an element y e L C(£»t-(£)) ,Z). Surgery can be completed on
$ in a way that is consistent with the completions of surgery for b-($)
A b
and $ in 1.22 if and only if y = 0.
A
h
What we must prove is 3 = 0 = 3 = 0, where 3 L (£,Z ) is repre-
sented by $. Towards this end we first note that 3 has finite order
prime to n. To see this, note 0.1, 0.2 implies that BH*((£,b-(£)),Z)
is all torsion prime to n, where BH*((£,b-(£),Z) is the blocked homology
of the block space pair (£,b_(£;)) defined on pgs. 377-378 in [13]. It
follows, just as in the proof of Lemma 3.5 in [13],that the group
Ah A
L C(£b-(£)) ,Z ) is also all torsion prime to n. So y must have finite
A A Ah
order prime to n. Since y is mapped to 3 under the map L ((£,b-(£)),Z ) +
Ah
L (£,Z ) which forgets the completion of surgery over b^(^), it follows
A
that 3 has finite order prime to n.
h T
x h
Next consider the maps L..(£;,Z ), L- , (£, {!}), where x sends each
i
surgery problem to its TL -covering surgery problem, and i comes from the
inclusion {1} c TL^. Note that the composite
T
°i: L1(^,{1}) - L1(^,{1})
is multiplication by n. It follows that, modulo n-torsion, L.C^^l}) is
a retract of L^(5,Zn).
A
The result of the last paragraph will be used to show that 3 has
order dividing a power of n. This will complete the proof of 1.24. By
hypothesis of 1.24 there is a completion of surgery for $: let the map
h: M^ + C
o ^ o
represent this completion of surgery. Let g: V + £fx[0,l] be a blocked
surgery cobordism from the TL -covering of h, h = 3_g, to a map 3+g =
d+V-*-£'xl which represents the completion of surgery for $ given in 1.22.
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