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 •* 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
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