COMBINATORIAL SYMMETRIES OF THE m-DIMENAIONSAL BALL
13
Note that if this second step could be carried out then the deck trans-
formation for the covering M - * M would pull back along r: £ - + M to give
a PL free action tp: TL - * % leaving the blocks of £ invariant. Soby
1.4 there would be P: TL XN - N, a semi-free PL action with K = N for
fixed point set.
The second step is begun by noting that each subcomplex pair
(b- ),^b. )) is a Poincare duality space pair and that the Spivak
fibrations for all these spaces have PL reductions which fit together in
a coherent way. Moreover these PL reductions pull back along p°f:£ •*• £
to the stable Whitney inverse of Tang(£). Here f: £ -* S! comes from 1.5
and p: £' -*• £ is the n-fold covering map. This information specifies
a surgery problem $ which represents an element 3 lying in abstract
surgery group. If 3 = 0, then surgery could be completed to getthe
desired block space M of 1.7 (see 1.23 and 1.24 below). In showing that
3=0 under the hypothesis of 0.7,it is necessary to discuss another
geometric representation for 3 other than the spaces £ , b.(£ ) lfj£x
together with the PL reductions of their Spivak fibrations. We shall
denote this new surgery problem by t and study it in chapters 2,3,4,5
below.
Completion of Step 1. We return now to Step 1 to carry outthe
details. The construction of \\): ^ x£! -**£', as in 1.5,will utilize the
following two propositions which are proven on pgs. 368-375 in [13].
Proposition 1.8. Let X,Y be connected, finite CW complexes satisfying
TT,(X) = TT.(Y) = 0 for i = l,2, and let f: X * Y be a map which induces
an isomorphism on Zn~homology groups H*(X',Zn) = H*(Y,Z ). Let X! bea
subcomplex of X; and let r : 7L^Y Y, rx,:Z^X1 + X' be free cellular
actions which commute with f. ,, such that r , r , act trivially onthe
integral homology groups of Y and X'. Then, after replacing X (modX1)
by a homotopy equivalent finite CW complex (ifneed be), r . extends toa
free cellular action r : TL xX -* X which acts trivially on the integral
x n J
homology of X, and satisfies for = r of.
x y
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