Obstruction to a general S^1-action Question: Suppose you have a simply connected, closed, orientable, smooth manifold $M$. What are some restrictions to the existence of a smooth (non-trivial) $S^{1}$-action on $M$?
Note: 1.There are some well known restrictions in the case that we require this action to have isolated fixed points (or preserve some additional structures), but I am not aware of any obstruction in this generality. 


*I am aware of the classification of manifolds with $S^{1}$-actions in dimension $2$ and $3$. In higher dimensions (say $\geq 6$) it seems to become less clear.

*I am particularly interested in dimension 6. I would like to get a picture of which manifolds in this dimension (satisying the above restrictions), have a smooth $S^{1}$-action.
Edit: Out of curiosity, using Puppe's methods, is it possible to construct a simply connected, smooth, complex projective 3-fold such that the underlying smooth manifold has no circle actions?
Edit 2 It turns out that (the underlying smooth 6-manifold of) any smooth hypersurface in $\mathbb{C}\mathbb{P}^4$ of degree atleast 3 has no smooth (non-trivial) circle actions. It follows from the main result of https://arxiv.org/pdf/1108.5327.pdf.
 A: 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..
A: V. Puppe, in Simply connected manifolds without $S^1$-symmetry. Algebraic topology and transformation groups, Proc. Conf., Göttingen/FRG 1987, Lect. Notes Math. 1361, 261-268 (1988) proved the following:

There exist a simply connected, closed, oriented smooth 6-dimensional manifold $M$ such that no closed, orientable manifold with the same rational cohomology algebra as $M$ admits a non-trivial circle action.

