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.