The fundamental group of symplectic manifolds with Hamiltonian Lie group actions

Details The fundamental group of symplectic manifolds with Hamiltonian Lie group actions The fundamental group of symplectic manifolds with Hamiltonian Lie group actions

2006-05-04

arxiv.org/abs/math/0605133v3

Journal of Symplectic Geometry, Vol 4, No. 3, 345-372, 2007

Mathematics

Symplectic Geometry

23 pages

Scientific paper

Let $(M, \omega)$ be a connected, compact symplectic manifold equipped with a Hamiltonian $G$ action, where $G$ is a connected compact Lie group. Let $\phi$ be the moment map. In \cite{L}, we proved the following result for $G=S^1$ action: as fundamental groups of topological spaces, $\pi_1(M)=\pi_1(M_{red})$, where $M_{red}$ is the symplectic quotient at any value of the moment map $\phi$. In this paper, we generalize this result to other connected compact Lie group $G$ actions. We also prove that the above fundamental group is isomorphic to that of $M/G$. We briefly discuss the generalization of the first part of the results to non-compact manifolds with proper moment maps.

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