It is well known [l, Chapter XVI, §9, Application 4] that a finite group which acts freely 2 on a finite dimensional homology ^-sphere must have periodic cohomology with period k + 1. I will outline here a proof of a converse result : A ny finite group with periodic cohomology can act freely on a finite simplicial homotopy sphere. More precisely, THEOREM 1. Let w be a finite group having periodic cohomology of period q. Let n be the order of 7r. Let d be the greatest common divisor of n and

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... c/> being Ruler's $-function. Then IT acts freely and simplicially on a finite simplicial homotopy (dq-1)-sphere X of dimension dq-1. Furthermore y if X is not required to be a finite complex, we can replace dq -1 by q-1. Note that a result of Milnor [3] shows that for some groups 7r, X cannot be a manifold. The proof of this theorem can be reduced to pure algebra by using a remark of Milnor (unpublished) to the effect that any free resolution of Z over TT can be realized geometrically provided this is so in low dimensions. Thus our main problem is to prove the following result. THEOREM 2. Let w, q, d be as in Theorem 1. Then w has a periodic free resolution of period dq. Also, T has a projective resolution of period q. By a periodic free resolution of period k, I mean a Tate complex (or complete resolution [l, Chapter XII, §2]) for TT having an automorphism of degree k. The existence of such a resolution is easily seen to be equivalent to the existence of an exact sequence with all Wi being free over the group ring Zir. It is this sequence (1) to which we apply Milnor's construction. As a first step in constructing such a sequence, we choose a free resolution 1 Sponsored by the Office of Ordnance Research, U. S. Army under contract DA-11-022-ORD-2911. 2 A group is said to act freely on a space if no element of the group other than 1 fixes any point of the space. 368