  Let $(M, \omega)$ be a connected, compact symplectic manifold equipped with a
Hamiltonian $S^1$ action.
  We prove that, as fundamental groups of topological spaces,
$\pi_1(M)=\pi_1(\hbox{minimum})=\pi_1(\hbox{maximum})=\pi_1(M_{red})$, where
$M_{red}$ is the symplectic quotient at any value in the image of the moment
map $\phi$.
