COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL

15

Proof of 1.11: Suppose that c has been defined on all b.(R)with dimension

(b.(R)) £ 1. Consider extending c to some b.(R)with dimension (b.(R)) =

£+1. By induction c: 3b.(R) - 8b.(K) has already been defined. Because

b. (K) is a cone, we can extend C.M (inu ^u rm

t0 c:

b • (R) -* • b. (K).

Now here is how to use 1.8, 1.9, 1.10, 1.11 to complete the

construction of ty: ^nx£' -* - V •

Define an action

D: Znx(KxSm"k"1)

KxSm-k-1

by

1.12.

= l^xcj), where

LK

7L xS

n

m-k-1

S ' is induced by multiplication

2-rri

^m-k

by e n on complex (-^—)-dimensional space. Note that $ leaves the

subsets b-(K) x Sm invariant. Let K , R equal the union of all

b.(K), bi(R) where dimension (b-CR)) _ £. Apply 1.8, inducting over the

sequence

m-k

cxg

K ,

xSJm-k-

m- k

R

1

m-k+1

cxg

R

in-k+2

cxg

K -i . -i b CI K. -i , ^ X b

m-k+1 m-k+2

m-k- 1

• c R

Cxg

m-k- 1

KxS

(where g,c come from 1.10, 1.11), to pull the action f back to an action

\p: 2 Z xR1 - R1 which leaves the sets b.(R') invariant. In applying 1.8

it has been necessary to replace R and its blocks b.(_R), by a homotopy

equivalent finite CW complex R1 and subcomplexes b.(R')- In this

inductive construction the inductive hypothesis takes the form that

ip: 2Z xRt - R! has been constructed to satisfy the following.

1.13j. (a) Each bi(R') in R- is left invariant by

(b) \p acts trivially on the integral homology of each b-CR') in

Rt.

(C) (cxg)o\|j = $°(cxg).

To carry out the inductive step \\: 2 Z XR! - R! must be extended to

ty: z

n

xR' •* " Rl-+1 satisfying 1.13j+1. Let bi(R') denote any of the

specified subcomplexes in R.+i which are not contained in R..