COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL

25

Proposition 1.22. There is a completion of surgery for $ realizing the

map f: £ - * V of 1.5(c). If in addition there is a consistent completion

of surgery for $ then there is a semi-free PL action cp: TL xN -•N having

K c N for fixed point set.

Proof of 1.22: By 1.18(a) there is a completion of surgery for $

realizing the map f: £ •* - V of 1.5(c). A consistent completion of surgery

for $ realizes a free PL group action cp: TL x£ •* • C as in'1.4. So by 1.4

the proof of 1.22 is completed.

We have the following corollary to 1.22.

Corollary 1.22(a). Let K c N be as in 0.7. Suppose there is a PL

semi-free action cp:-Z *R - R on a regular neighborhood R of K in N,

having K c R for fixed point set. Then c p extends to a PL semi-free action

(p: TL xN - • N having K c N for fixed point set.

Lemma 1.23. There are completions of surgery for b,($) and for $ which

are consistent with one another. The completion of surgery for $ realizes

the map f: K + V of 1.5(c).

Proof of 1.23: By 1.18(a) there is a completion of surgery for $

realizing the map f: £ -* • C! of 1.5(c). By 1.18(b) there is a completion

of surgery for b-($).

It remains to find a completion of surgery for b-($) consistent with

that for $. Let

v w - w

be a homotopy equivalence representing a completion of surgery for b-(£).

The TL -covering of h3,

h3: b3(M) + b3(£'),

represents a completion of surgery for b-($) which may or may not agree

with the completion of surgery for $. It will now be argued that there

is a surgery cobordism H-: W -* • b-(£ ) x [0,1], beginning at 3_H3 = h^,

and ending at a homotopy equivalence 3+H3: B+W - b3(£Q)xl,"which repre-

sents a completion of surgery for b3(£) consistent with the completion

of surgery for $ given in 1.22. Let