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..
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