On circle actions with exactly three fixed points

We study smooth, closed orientable $S^1$-manifolds $M$ with exactly $3$ fixed points. We show that the dimension of $M$ is of the form $4\cdot 2^a$ or $8\cdot (2^a+2^b)$ with $a,b\ge 0$ and $a\mathrm{\ne }b$. Moreover, under the extra assumption that $M$ is spin or unitary we determine exactly the dimension of $M$.