The fundamental group of symplectic manifolds with Hamiltonian Lie group actions

Let $(M, \omega)$ be a connected, compact symplectic manifold equipped with a Hamiltonian $G$ action, where $G$ is a connected compact Lie group. Let $\phi$ be the moment map. In \cite{L}, we proved the following result for $G=S^1$ action: as fundamental groups of topological spaces, $\pi_1(M)=\pi_1(M_{red})$, where $M_{red}$ is the symplectic quotient at any value of the moment map $\phi$. In this paper, we generalize this result to other connected compact Lie group $G$ actions. We also prove that the above fundamental group is isomorphic to that of $M/G$. We briefly discuss the generalization of the first part of the results to non-compact manifolds with proper moment maps.