COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL 11

are unions of some of the blocks in £.

Because c p is a PL action, T can always be chosen so that c p is a

simplicial action, which shall be assumed henceforth. The reader can

easily verify the following:

Lemma 1.2. The action cp: TL xN - N leaves invariant each block b.(£)of £.

Lemma 1.3. The semi-free action cp: TL xN - * N can be reconstructed from

the free action tp: TL x£ - £.

n

Proof of 1.3: Let

Xj ^ (TN^RT) U (Ue),

where the union Ue is taken over all cells e of C which are of dimension

_ j. The proof proceeds by induction, over the nested sequence

N^R = X c Xm - c • -c N, m-,

k

m-k+-

1 *

where k = dimension (K). Let e be any cell of C dual to a k-dimensional

simplex of K. To extend cp: TL x (eflR) - » eflR to cp: TL xe - * e cone the first

t- ^ n y n

action, noting that cone (eflR) = e, and that c p leaves eflR invariant

because eflR = b.(.£) for some 3jx. Doing this for all e with dimension

fe) = m-k,togives , the extension of cp: TL x£ -+ r to cp: TL xX , - + X , . To

v n ^ ^ n m- k m- k

extend this last action to cp: Z xX , . -* X , _, , note that for any cell

n m-k+1 m-k+1 J

e in C dual to a simplex in K of dimension k - 1 , e DX , = de. So cone

r m-k

(enxm~,) = e, allowing cp:Znx (eflXm ,) - » eC\X , to be extended to

K m- K m- K

cp: TL xe - • e by coning. All these extensions give cp: TL xX , ,

n

- • X , ,-.

n ' ° ° n m-k+1 m-k+1

In the same way this last action extends to cp: ZxX , ^~ - » X , ^~ by

' m-k+z m-k+z J

coning, and then to cp: TL*X * X , ,_-, etc.,and finally to cp: TL xN-»N.

&

' n

m-,+_,_--

k 3 m-k+3 ' J n

Now let us return to the problem stated in the introduction. Let

K c N be a compact, PL, subset of N as in 0.7, T, R, R, £, b.(£)is

described above. The same argument used to prove 1.3 above yields the

following: