The fundamental group of $G$-manifolds

TL;DR

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Contribution

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Abstract

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