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 x£ - * % 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