A Survey of Spherical Space Form Problem by J. F. Davis

By J. F. Davis

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19(c) can be extended to the case VI(b), P == 3 (mod 4). However here the period n is congruent to 4 modulo 8 and G contains sub-groups of type IlL. R. Lee [23] proved that such a group cannot act freely on a closed manifold M such that H*(M; Zj2) = H*(sn~l; Zj2). Thus, from the point of view of the space form problem we are interested in 0"2n( G), which is zero. SECTION 4 A Brief Review of Some Facts in Surgery Theory Let Eo be the classifying space for stable spherical fiberings [26, 30]. 2 where E PL classifies stable piecewise linearS n bundles.

These L~S( ) functors are contravariant with respect to subgroups He 7T, ()(). g. [15]). Using these maps Wall proved the basic result [58]. 12 Let M" be a closed compact oriented differentiable or PL manifold with finite fundamental group 7T, then if THEOREM N"~M" REVIEW OF SURGERY THEORY is a degree 1 normal map corresponding to a we have that E 265 [M n, G/( )], s(a) E im I*: L~(ZH) ~ L~(Z'TT) where H is the Sylow 2-subgroup of 'TT. Moreover, s( a) and only if rH(s(a)) ;f:. 0 in L~(ZH). ;f:.

Let X be a connected CW-complex. 25 X is finitely dominated if there is a finite complex Y and maps i: X --'> Y and r: Y --'> X such that r i is homotopic to the identity. X is homotopy finite if X is homotopic to a finite CW-complex. 26 X is finitely dominated if and only if the Z[1T 1X]-chain complex C*(X) is finitely dominated. X is homotopy finite if and only if C*(X) is homotopy finite. Define the Wall finiteness obstruction of a finitely dominated complex X to be £leX) = x( C*(X» E Ko(Z[ 1T 1X]).

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