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 wellknown statement.
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 wellknown statement. 


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. 

