The fundamental group of G-manifolds (Q2853986)

Let \(M\) be a connected Hamiltonian manifold on which acts a connected compact Lie group \(G\) with a moment map \(\phi: M \rightarrow \mathfrak{g}^*\), where \(\mathfrak{g}^*\) denotes the dual of the Lie algebra of \(G\). In previous works [Proc. Am. Math. Soc. 131, No. 11, 3579--3582 (2003; Zbl 1066.53127); J. Symplectic Geom. 4, No. 3, 345--372 (2006; Zbl 1157.53361)], the author showed that if \(M\) is connected and compact, then \(\pi_1(M) \cong \pi_1(M/G) \cong \pi_1(\phi^{-1}(G.a)/G)\), for all \(a \in \phi(M)\).NEWLINENEWLINEIn the present paper, the author extends these results to the setting where \(M\) may not be compact: ``We prove that if there is a simply connected orbit \(G.x\), then \(\pi_1(M) \cong \pi_1(M/G)\); if additionally \(\phi\) is proper, then \(\pi_1(M) \cong \pi_1(\phi^{-1}(G.a))\), 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 \(\pi_1(M) \cong \pi_1(\phi^{-1}(G.a))\cong \pi_1(\phi^{-1}(a))\), where \(a=\phi(x)\). Furthermore, we prove that if \(\phi\) is proper, then \(\pi_1(M/\widehat{G}) \cong \pi_1(\phi^{-1}(G.a)/ \widehat{G})\) for all \(a \in \phi(M)\), where \(\widehat{G}\) is any connected subgroup of \(G\) which contains the identity component of each stabilizer group; in particular, \(\pi_1(M/G) \cong \pi_1(\phi^{-1}(G.a)/G)\) for all \(a \in \phi(M)\).''