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

By J. F. Davis

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27. (See [13]). The type 11 2-hyperelementary groups act freely on S8k+7. 7 [27] A P-group G acts freely on some sphere if and only if G contains no dihedral subgroups. 8 Suppose G is a group whose Sylow 2-subgroup is cyclic or generalized quaternionic. Then the following are equivalent: (a) Every element of order 2 is central, (b) G has a unique element of order 2, (c) G contains no dihedral subgroups. LEMMA Proof (a) implies, (b) since G does not contain Z/2(f)Z/2; that (b) implies (c) is clear.

Some Comments on Kt The previous section was motivated by the philosophy that calculations in K 1 are easier than calculations in Ko. In this section we discuss K,. Some familiarity with the groups K,(A) for A a semilocal ring, a semisimple algebra, or maximal order in a semisimple algebra are needed. 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.

MILGRAM forms with quadratic refinements, modulo hyperbolic forms (not much changed from Wall's definition). e. K j is Z7T-free, the quadratic form vanishes on K j and Kt = K j (significantly different from Wall's definition). The relations needed are fairly complex though. See [41, 7] for a discussion. Using these definitions a concerted attack was mounted on the structure of the L~(Z7T) groups for 7T finite by Bak, Pardon, and Carlsson-Milgram and others [3, 35, 36, 7, 8]. of Hambleton and Milgram [18] which allow the effective calculation of L~(Z7T) for 7T a finite 2-group.

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