It is a result of Atiyah-Hirzebruch (1970) that the $\hat{A}$ genus of a spin manifold with a nontrivial $S^1$ action vanishes, and a result of Herrera and Herrera that the same result, if the manifold is not necessarily spin, but has finite $\pi_2,$ then the same result, is true. In the meantime, there is the result of Freedman and Meeks (Une obstruction élémentaire à l’existence d’une action continue de groupe dans une variété, C. R. Acad. Sci., Paris, Sér. A 286, 195-198 (1978)) that there are some cohomological/geometric obstructions (so the connected sum of a non-sphere and a torus admits no circle action) - this has been generalized in
Assadi, Amir; Burghelea, Dan, Examples of asymmetric differentiable manifolds, Math. Ann. 255, 423-430 (1981). ZBL0437.57021..
