**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.

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