COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL 21

b?(£JxB • Note the reduction y,•b3^0)pulls back along p°f: N-R -* • £

3

^ l °

to the stable Whitney inverse to Tang (N-R).K

rr

. . We wish to extend

\ 1

the PL reduction y. , . to a PL reduction y. , of yn so that (p°f)*(y1)

1|b3(50) _ X X

is the stable Whitney inverse to Tang (N-R). This could be accomplished

if it were known that

pof: (NTR,b3U)) + CCo.b3(50))

induces an isomorphism on the integral homology groups of these pairs.

We complete the proof of 1.18 by proving this. First note that the

cellular chaing complex for the pair (£',b-(£')), together with the action

\\): 7L *£' • •-* - V yields a finitely generated chain complex of free

Z(Z )-modules

0 * V-*

\-l ~

•• • * P2-iP l — Po + °

whose homology groups are the H.((£', b,(£')),Z). Because these homology

groups are of order relatively prime to n (by hypothesis of 0.6),we

deduce

Pi = ker(8i) + Pj for all i,

where ker(3-),P! are both projective Z(Z)-modules and 3..p, is a

monomorphism. We have the following exact sequence of Z(Z)-modules:

3 •

0 + P[+1 i+l ker^) + HjUe'.bjCe'^.Z) - 0,

which when tensored with Z over Z(Z ) yields the exact sequence

Tor1(Hi((C',b3(5')),Z),Z) *P£+1 »Z+ker(3i) »Z

+

Hi(ai1b3(C)),Z) 8Z

+

0

where Tor, and 8 are taken with respect the ring Z(Z ). Write

Then by 1.16,

H-((£f,b, (£f)),Z) = Z Z as a direct sum of cyclic abelian groups Z .

i ^

q

q q

Tor1(Hi((^',b3(^)),Z),Z) = Z Tor1(Zq,Z).

And by 8.1 on pg. 160 of [18],and [[8],1.0(a)] and the fact that q is

prime to n, we have Tor,(Z ,Z) = 0. So we have the exact sequence