Circle Actions on Homotopy Spheres Not Bounding Spin Manifolds

Smooth circle actions are constructed on odd-dimensional homotopy spheres that do not bound spin manifolds. Examples are given in every dimension for which exotic spheres of the described type exist. A result of H. B. Lawson and S.-T. Yau [15] implies that homotopy spheres not bounding spin manifolds do not admit effective smooth S3 or S03 actions. On the other hand, G. Bredon has shown that even-dimensional manifolds of this type can admit smooth circle actions [9] ; in fact, such examples exist in every dimension of the form 8k + 2, where k > 0. Since homotopy spheres that are not spin boundaries only occur in dimensions 8k + 1 and 8k + 2 ik> 0), it is natural to ask if smooth circle actions also exist on (8/V + l)-dimensional examples. We shall prove the answer is yes. Theorem. For every k > 1, there exists a homotopy i8k + l)-sphere not bounding a spin manifold that admits an effective smooth S1 action. We shall prove this result by explicitly constructing semifree circle actions (see [10]) using surgery-theoretic methods. The motivation for such an approach is that it yields an extremely simple proof of Bredon's result in the even-dimensional case (see Remark 3.11 below). Several technical facts make the odd-dimensional case more complicated; the most important are (i) the indecomposability of the (8& + l)-dimensional framed bordism classes of the homotopy spheres considered, and (ii) the need to calculate surgery obstructions for normal maps into certain simply connected 8/t-manifolds. Needless to say, the bulk of this paper is devoted to circumventing the first problem and studying the second. These are done using the machinery of Adams' JiX) papers (e.g., [2], [3]) and the validity of the Adams conjecture [19], Remarks. 1. It is natural to ask whether tori of rank > 2 can act smoothly on the manifolds considered; I do not know the answer to this question. 2. The exotic spheres considered here provide examples of odd-dimensional spin manifolds with smooth S1 actions but no Riemannian metric having positive scalar curvature (compare [15]). Received by the editors October 7, 1974. AMS (MOS) subject classifications (1970). Primary 57D65, 57E25; Secondary 57D20, 57D55, 55F50. (x)The author was partially supported by NSF Grant GP-19530A2. oo Copyright © 1975, American Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use