Non-linear circle actions on the 4-sphere and twisting spun knots

CIRCLE actions on spheres form one of the most important problems in transformation groups. The aim of this paper is to study this problem in dimension 4. We answer a question of Montgomery and Yang7], and show that there are infinitely many non-linear circle actions on S⁴. Moreover, if the 3-dimensional Poincaré conjecture is true, these actions plus the linear ones are the only possible circle actions on S⁴. The proof of this assertion involves identifying some homotopy 4-spheres. It is closely related to the work, twisting spun knots, of Zeeman14]. We give a different treatment of this subject. This new setting yields new proofs and substantial strengthenings of some known results. In particular, we answer two questions of Zeeman14, pp. 493–494, Questions 3 and 4].