${\pi }_1$ 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, ${\pi }_1(M)={\pi }_1(minimum)={\pi }_1(maximum)={\pi }_1(M_{red})$, where $M_{red}$ is the symplectic quotient at any value in the image of the moment map $\varphi$.

4 pages