COMBINATORIAL SYMMETRIES OF THE m-DIMENSIONAL BALL 29
(c) t: ^ -* » £Q is given by
t E (1 Ul U---Ul ) U (p)
^o . o S
where p: -£' -» - £ is the n-fold covering map; t.: b.(£') - • b.(£ ) is
given by t , ^ ^ .
Note, the Spivak fibration for each pair (b.(. £ ) 3^- (£ )) pulls back
along t. to the Spivak fibration for the pair (b.(£'),8b.(£')), and
each ti: (bi(^) , 3bi (^) ) + (bi(£Q), abi(£Q)) is a degree 1 map of
Poincare duality pairs. So t: £' - * F , together with this framing
information (in the Poincare duality category) for the map t, gives a
surgery problem (in the Poincare duality category) which will be denoted
by T. The problem x represents an element of L (£,Z ),because surgery
techniques can be extended to the Poincare duality category (see ).
Proposition 1.26. Let 3 e L (£,Z ) be the surgery obstruction represented
by the surgery problem $ of 1.21. The surgery problem T of 1.25
Proof of 1.26. Follows directly from the equalities a, b, c on pg.
Corollary 1.27. Surgery can be completed on $ in a way consistent with
the completion of surgery on $ in 1.22 if and only if surgery can be
completed on x.
Proof of 1.27. By 1.23, B lies in the image of
f: Lj((C,b3(C)),Zn) - l£te,Zn).
where f just forgets the completion of surgery over b-(£). Bv 0.1, 0.2,
and 3.5 in ,every element in LQ(U,b3(^)),Zn) has order prime to n.
So 3 = 0 *•* -nB = 0. Hence 1.27 follows from 1.26 and 1.24.