Differentiable group actions on homotopy spheres

Abstract

In several previous papers numerous results conserning smooth group actions on homotopy spheres not bounding parallelizable manifolds have been obtained [5, 13, 21, 23-26, 31, 32, 35, 39, 42-50]. This paper is a first step towards placing such results in a more unified setting; we shall concentrate our attention on semifiee actions to avoid book-keeping problems with various orbit types, postponing these considerations to later papers in the series. When combined with appropriate homotopy-theoretic machinery, this setting yields new proofs and substantial strengthenings of many known results. The basic underlying problem of this paper is the following: Given a homotopy (n +2k)-sphere 22, does it admit a semifree G-action with an n-dimensional fixed point set'? Our techniques show that the answer depends on the decomposition of the Pontrjagin-Thom invariants of 22 (which form a coset in the stable homotopy of spheres) in terms of lower-dimensional stable homotopy classes. In a subsequent paper we shall illustrate this by studying certain low-dimensional homotopy spheres in detail, obtained virtually complete information. On the other hand, the new non-existence theorems obtained in this paper also fall into this pattern indirectly; for if one knows that some specific composition operation (ordinary or higher order) on a stable homotopy class is nontriviat, such information often implies the class cannot be decomposed in the "right" way for an action to exist. Presumably some appropriate global information about the stable homotopy of spheres (perhaps resembling the Kahn-Priddy Theorem) is needed to fully answer questions such as the existence of smooth Zp actions on every homotopy sphere. The general viewpoint of this paper is derived from the work of Browder, Petrie, Rothenberg, and Sondow that classified certain semifree actions on homotopy spheres ([8, 10, 11, 39, 40]; also see [48]). A general motivating problem was the determination of the underlying differential structure on the ambient G-manifold from the classification information presented in the above papers; the insight needed to study this problem evolved from consideration of the examples in [45-47] where the differential structure was determined by entirely different methods. Our basic result (Theorem 3.4) establishes a strong relationship between the differentiable structure on 2" and an invariant that vaguely measures the knottedness of the fixed point set (the knot invariant of Section 2, which agrees with the internal normal invariant of [49]). This relationship exists because the