The fundamental group of $G$-manifolds

Abstract

Let $G$ be a connected compact Lie group, and let $M$ be a connected Hamiltonian $G$-manifold with equivariant moment map $ϕ$. We prove that if there is a simply connected orbit $G\cdot x$, then $π_1(M)\congπ_1(M/G)$; if additionally $ϕ$ is proper, then $π_1(M)\congπ_1(ϕ^{-1}(G\cdot a))$, where $a=ϕ(x)$. We also prove that if a maximal torus of $G$ has a fixed point $x$, then $π_1(M)\congπ_1(M/K)$, where $K$ is any connected subgroup of $G$; if additionally $ϕ$ is proper, then $π_1(M)\congπ_1(ϕ^{-1}(G\cdot a))\congπ_1(ϕ^{-1}(a))$, where $a=ϕ(x)$. Furthermore, we prove that if $ϕ$ is proper, then $π_1\big(M/\Gh\big)\congπ_1\big(ϕ^{-1}(G\cdot a)/\Gh\big)$ for all $a\inϕ(M)$, where $\Gh$ is any connected subgroup of $G$ which contains the identity component of each stabilizer group. In particular, $π_1(M/G)\congπ_1(ϕ^{-1}(G\cdot a)/G)$ for all $a\inϕ(M)$.