It is a result of [Atiyah-Hirzebruch (1970)][1]  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)][2]) 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

<cite authors="Assadi, Amir; Burghelea, Dan">_Assadi, Amir; Burghelea, Dan_, [**Examples of asymmetric differentiable manifolds**](http://dx.doi.org/10.1007/BF01450714), Math. Ann. 255, 423-430 (1981). [ZBL0437.57021](https://zbmath.org/?q=an:0437.57021).</cite>.


  [1]: https://zbmath.org/?q=an:0193.52401
  [2]: https://zbmath.org/?q=an:0373.57021