Let $M$ be a compact manifold with an $\mathbb S^1$-action that fixes a point on $M$.
Is it correct that $\pi\_1(M/S^1)=\pi\_1(M)$?


I believe this is correct and is a corollary of some well-known statement. 



The fundamental group of an $S^1$-quotient
Suppose $S^1$ acts on $S^2$ by rotation around one axis. This action commutes with the antipodal map and hence gives an action on $\mathbb{R}P^2$. But $\mathbb{R}P^2/S^1\cong [0,1]$ and hence this cannot be true.


