COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL

7

where i is inclusions, I is the index, and j(1) equals the form with [1]

for matrix representative. Set crT(g.) equal the image of [X] under

W(Q) I WT(Q) + WT(Q)/rWT(Q). By replacing gi : M± -R/R by g±: 6M± + R/R in

the above constructions, we get the invariant oT[g. . ) € WT(Q).

1 1|6M. l

The collection of all invariants {aT(g.),alT(g., )| i=l,2,...,£}

1 1 1I5M-

represents a characteristic class

1

yT(K) € E H

k

((K,3K),W (Q)) ,

j J

which has been studied by the author (see [11], [10]), and by F. Latour

(see [17]). (In [17] the class YT(K) is denoted by X(K) and WT(Q)

is denoted by W(Q,Z)). In terms of this class 0.8 can be written

0.8' YT(K) = 0.

It can now be stated how condition 0.4 enteres into the proof of 0.7.

On the one hand condition 0.8' (or equivalently 0.8) will allow surgery

to be completed on all the (t-,6t-) (sections 3, 4), yielding the desired

action cp:2 Z xN -* • N. On the other hand each h^(K) can be computed in

terms of YT(K) (see [[10], appendix]), so 0.8! implies 0.4 holds. In

section 5 it is shown how to replace the stronger condition 0.8' by

the weaker condition 0.4, and still be able to complete surgery on each

(ti,6ti).

Here is how the condition Ory(6g.) = 0, aT(g.) = 0, is used to

arrange that the homological kernels of (^^t.) are simple. It is first

noted that the homological kernels of each block in t are simple (see

1.14-1.16 below). Then the condition o^dg^ i§M J = 0 and o^tg^) = 0 is

used, with surgery techniques developed in [9],to arrange that g. (K)

has two components X-.,X2, satisfying (see 3.4)

(a) g^C^i) i-s contained in the interior of a k-dimensional simplex

of K.

(b) X2-5X2 nas tne same Z -homology groups as a sphere of dim(X2)

minus r points, and 6X2 is a Z -homology sphere.