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
Previous Page Next Page