Free 𝑆¹ actions and the group of diffeomorphisms

Abstract

Let S 1 {S^1} act linearly on S 2 p − 1 × D 2 q {S^{2p - 1}} \times {D^{2q}} and D 2 p × S 2 q − 1 {D^{2p}} \times {S^{2q - 1}} and let f : S 2 p − 1 × S 2 q − 1 → S 2 p − 1 × S 2 q − 1 f:{S^{2p - 1}} \times {S^{2q - 1}} \to {S^{2p - 1}} \times {S^{2q - 1}} be an equivariant diffeomorphism. Then there is a well-defined S 1 {S^1} action on S 2 p − 1 × D 2 q ∪ f D 2 p × S 2 q − 1 {S^{2p - 1}} \times {D^{2q}}{ \cup _f}{D^{2p}} \times {S^{2q - 1}} . An S 1 {S^1} action on a homotopy sphere is decomposable if it can be obtained in this way. In this paper, we will apply surgery theory to study in detail the set of decomposable actions on homotopy spheres.