The fundamental group of $G$-manifolds

Let $G$ be a connected compact Lie group, and let $M$ be a connected Hamiltonian $G$-manifold with equivariant moment map $\varphi$. We prove that if there is a simply connected orbit $G\cdot x$, then ${\pi }_1(M)\cong {\pi }_1(M\mathrm{/}G)$; if additionally $\varphi$ is proper, then ${\pi }_1(M)\cong {\pi }_1({\varphi }^{-1}(G\cdot a))$, where $a=\varphi (x)$. We also prove that if a maximal torus of $G$ has a fixed point $x$, then ${\pi }_1(M)\cong {\pi }_1(M\mathrm{/}K)$, where $K$ is any connected subgroup of $G$; if additionally $\varphi$ is proper, then $\pi_1(M)\cong\pi_1(\phi^{-1}(G\cdot a))\cong\pi_1(\phi^{-1}(a))$,where$a=\phi(x)$. Furthermore, we prove that if $\varphi$ is proper, then ${\pi }_1\left(M\mathrm{/}\text{Gh}\right)\cong {\pi }_1({\varphi }^{-1}(G\cdot a)\mathrm{/}\text{Gh})$ for all $a\in \varphi (M)$, where $\text{Gh}$ is any connected subgroup of $G$ which contains the identity component of each stabilizer group. In particular, ${\pi }_1(M\mathrm{/}G)\cong {\pi }_1({\varphi }^{-1}(G\cdot a)\mathrm{/}G)$ for all $a\in \varphi (M)$.