Semifree Actions of Finite Groups on Homotopy Spheres

We show that for any finite group the group of semifree actions on homotopy spheres of some fixed even dimension is finite, provided that the dimension of the fixed point set is greater than 2. The argument shows that for such an action the normal bundle to the fixed point set is equivariantly, stably trivial. 0. Introduction. A group G is said to act semifreely on a space X if every point is either fixed by every element of G or fixed only by the identity. The classification of smooth semifree actions of finite groups on homotopy spheres has been discussed by Browder and Pétrie [3] and Rothenberg [6]. We briefly summarize the basic scheme. Given a finite group G we fix a representation p: G -* 0(d) such that p restricted to the unit sphere Sd~ ' is fixed point free. A (G, p)-manifold M is a smooth manifold together with a smooth, semifree action of G on M such that the fixed point set F is nonempty and locally the representation of G on the normal bundle of F in M is equivalent to p. In a natural way this defines a reduction of the structure group of the normal bundle to Z(p), the centralizer of p(G) in 0(d). A (G, p)-orientation is a specific reduction of the structure group of the normal bundle to Z(p). We then define CN(p) to be the set of (G, p)-oriented A-cobordism classes of (G, p)-oriented manifolds which are homotopy N-spheres with fixed point set a homotopy (N — d)-sphere. The set CN(p) has the structure of an abelian group under the connected sum operation. The object of the present paper is to prove the following qualitative result about the groups CN (p). Theorem A. Let p: G -» O (2d) be a fixed point free representation of a finite group G and suppose that 2N — 2d > 2. Then C2N(p) is a finite group. We point out that the condition that p has even dimension is a restriction only when | G | = 2. This case is already well understood [3], [6]. The essential ingredient for proving this theorem is well known. According Received by the editors September 16, 1977. AMS (MOS) subject classifications (1970). Primary 57E25; Secondary 57D85. 'Partially supported by NSF Grant #MCS76-05973. © American Mathematical Society 1979