Periodic Resolutions for Finite Groups

Abstract

In 113] I indicated a proof of the following theorem: THEOREM A. Let w be a finite group of order n. Let d be the greatest common divisor of n and p(n), p being Euler's p-function. Suppose 7r has periodic cohomology of period q. Then there exists a finite simplicial complex X of dimension dq -1 which has the homotopy type of a (dq -1)sphere and on which w acts freely' and simplicially. I do not know whether it is always possible to replace dq 1 by q 1 in this Theorem. This is, of course, the best possible value [3, Ch. XVI, ? 9]. This replacement is possible in special cases. In the appendix, I will consider the case wr 33, the symmetric group on three letters and will show that S3 can act freely and simplicially on a finite 3-dimensional homotopy 3-sphere X. The period of 33 is 4 so this is a best possible result. Note that a result of Milnor [71 shows that for some groups r, X cannot be a manifold. In particular, Milnor's result shows that 33 cannot act freely on any closed manifold with the 2 homology of a sphere. Consequently, the space X which I will construct in the appendix cannot be a manifold and so cannot be homeomorphic to S3. I will give here a complete proof of Theorem A as well as the proof of a mod C generalization. Before stating this generalization, it is necessary to give some definitions. Let 7C be any finite group. Let 9? be any set of primes. Then there is a 9f-period (possibly infinite) associated with the group r [3, Ch. XII, Ex. 11]. This is just the ordinary period if f? is the set of all primes. For any set of primes ?1?, let C?I2 be the Serre class [10] of finite abelian groups having no p-torsion for any prime p e 9?.