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

By J. F. Davis

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Extra resources for A Survey of Spherical Space Form Problem

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F. DAVIS AND R. J. MILGRAM 242 Let 1T be any group. o be a chain complex of projective Z1T-modules with H o( C) = Z. Let D = {D i , aL200 be a projective Z1T-resolution of Z. Let m = min{ i > 0 I H i ( C),e O}. Then there is a chain map This gives a cohomology class k(C)EH m +I (1T;H m (C» represented by Cl':Dm + , --'> Hm(C). Let X be a connected CW-complex and m = min{i > 111Ti(X),e O}. There metric interpretation of k(X). Let IS a geo- k( 1T 1 X, l)m --'> X be a map inducing the identity on 1T,.

16(a) we see that TQ (2 n ) is a quotient of Z/2 with possible generator [P3]. Further analysis ([16] or [57]) shows that in fact TQ (2 n ) = Z/2. Geometrically this means there are two homotopy classes of CWcomplexes X whose cover is homotopic to S3 and whose fundamental group is isomorphic to Q(2 n ). One homotopy J. F. DAVIS AND R. J. MILGRAM 246 type has a non-zero finiteness obstruction and hence is not the homotopy type of a manifold. D. The Swan Obstruction and Reidemeister Torsion A basic philosophy is to avoid doing computations in Ko at all costs-do them in Kt instead!

0 In [60,61] C. T. C. Wall defined his finiteness obstruction. We now review some of the major results of these papers. Let C = {Cj , a} be a chain complex of projective Z7Tmodules, with 7T any group. 23 C is finitely dominated if C is Z7T-chain homotopic to a finite projective complex {Pi. ah",j",o. e. ) C is homotopy finite if C is Z7T-chain homotopic to a finite free complex. 24 Suppose C is Z7T-chain homotopic to a finite projective complex {Pj, a}k",j",O' Then depends only on the chain homotopy type of C.

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