By J. F. Davis
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Extra info for A Survey of Spherical Space Form Problem
This gives the desired complex on triangulating the orbit space of S(2 p '-I) under the action. Now, consider the case ps = pt. Then there is an eVIdent surjection S: A(m, pS+', 4J1T) ~ A(m, ps, 4J) (where 1T: Z/ ps+1 ~ Z/ ps is the surjection). Let Let be a free resolution for G'. 3 C=ZG@ZG'C', 252 OBSTRUCTIONS FOR P-GROUPS 253 then C is ZG-free but not acyclic. However for primes q other than p, Zq ® C is acyclic. Indeed the surjection ZqG' ~ ZqG splits, so ZqG is a projective (hence flat) ZqG' -module.
References for such material are given in [50, 4]. 30. A theorem of Wang [62, 4] implies that the reduced norm map nr:K,(QpG)~center(QpGris an isomorphism. PERIODIC RESOLUTIONS 251 Recall that a ring A is semilocal if A modulo its Jacobson radical is semisimple Artinian. ZpG is semilocal. 41  If A is semilocal, A·~ K](A) is a sur- jection. Often (ZpG)' can be studied by means of a filtration using the Jacobson radical. The structure of im K,(Z(l/IGi)G) is more difficult to describe. We shall do so however since this is the source of the more delicate number theory involved in the calculation of the Swan obstruction for the groups Q(2 n a, b, c).
6. This we see using the automorphisms of Q(2 n a, b, c) to convert a resolution g with X(g) = 0 to eg with X( eg) = 0 once more. See Davis  for an application of this idea. 16 260 J. F. DAVIS AND R. J. 18 Let 0=Q(8)xZ/3. Choose gEH 4 (6; Z) so that x(g) = [PI7]. 6, x(g),e O. However rQ(8)x(g) = 0 and rZ/3x(g) = O. This completes our discussion for p-hyperelementary groups. Now we turn to a brief discussion of the finiteness obstructions for all P-groups. 19 Let 0 be a P-group of period n. (a) (T2"(0) =0.