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 $\varphi$. We prove that if there is a simply connected orbit $G\cdot x$, then ${\pi }_1(M)\cong {\pi }_1(M/G)$; 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 ${\pi }_1(M)\cong {\pi }_1(M/K)$, where $K$ is any connected subgroup of $G$; if additionally $\varphi$ is proper, then ${\pi }_1(M)\cong {\pi }_1({\varphi }^{-1}(G\cdot a))\cong {\pi }_1({\varphi }^{-1}(a))$, where $a=\varphi (x)$. Furthermore, we prove that if $\varphi$ is proper, then $π_1\big(M/\Gh\big)\congπ_1\big(ϕ^{-1}(G\cdot a)/\Gh\big)$ for all $a\in \varphi (M)$, where $\Gh$ is any connected subgroup of $G$ which contains the identity component of each stabilizer group. In particular, ${\pi }_1(M/G)\cong {\pi }_1({\varphi }^{-1}(G\cdot a)/G)$ for all $a\in \varphi (M)$.

References in corpus (3)

• Gradient flow of the norm squared of a moment map
• The fundamental group of symplectic manifolds with Hamiltonian Lie group actions
• The fundamental group of $S^1$-manifolds

Cited by in corpus (3)

• $S^1$-invariant symplectic hypersurfaces in dimension $6$ and the Fano condition
• The fundamental groups of contact toric manifolds
• The fundamental groups of presymplectic Hamiltonian G-manifolds