The fundamental group of symplectic manifolds with Hamiltonian Lie group actions
The fundamental group of symplectic manifolds with Hamiltonian Lie group actions
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具有哈密顿李群作用的辛流形的基本群
DOI：
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发表时间：
2006
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通讯作者：
Hui Li
中科院分区：
文献类型：
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作者：
Hui Li
设$（M，\omega）$是一个连通的紧辛流形，其上有一个Hamilton $G$作用，其中$G$是一个连通的紧李群.让$\phi$是矩图。在L中，我们证明了G=S^1作用的下列结果：作为拓扑空间的基本群，$\pi_1（M）=\pi_1（M_{red}）$，其中$M_{red}$是矩映射$\phi$在任意值处的辛商.本文将这一结果推广到其它连通紧李群G作用上。我们还证明了上述基本群与M/G的基本群同构。我们简要地讨论了第一部分的结果推广到非紧流形与适当的时刻映射。
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.