COMBINATORIAL SYMMETRIES OF THE nwDIMENSIONAL BALL

17

X E b2(g')

X' E 3b2(e')

Y = 5-

rx, s «

f = inclusion,

to extend ty: Z x3b2(^') - * 3b2(£f) to ijj : Z xFJ(^f) - * FJ(£'). Again, we may

have to replace b2(£') (mod 3b«(.£') ) by a homotopy equivalent finite CW

complex bT(^'). That the hypotheses of 1.8 are satisfied by the above

choice for X,X',Y,rx,,ry,f is assured by the hypothesis of 0.6, 1.9, and

lemma 1.14 (b) below. Let f: b2(£!) -* • b (£') be the map provided by 1.8,

satisfying

(a) 8b^U') = 3b2(Cf),

(b) £,3b (V) = identity,

(c) ^°f = foijj".

Modify the pair (£f,b2(£f)) by adding 5^"Uf)* [0,1] , along

b^Un xl = E^(S') I b2(C)

to £f to get a new £', and letting the new b2(£') be 3FT(£')x[0,1] u

bTf^^xO. The new b.(£'), 3jx, are the same as the old. Let the new

L J —

—

\p : Z x£f - £' be the union of the old if ; with i p x l - , .

Except for the proof of lemma 1.14, this completes the construction

of ty: TL x£' + £' as in 1.5, and hence step 1.

Lemma 1.14. Let \[: 7L xRl - * R! be as in 1.13j. Then \\) satisfies the

following:

(a) Suppose 3b.(R') is in R!. Then \p acts trivially on the integral

homology of 3bi(R').

(b) ty acts trivially on the integral homology of R and 3Rf (ifj=m).

The proof of 1.14 will be based on the following: