$\pi_1$ of Hamiltonian $S^1$ manifolds

Abstract

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 $\phi$.