Let X be a compact connected Riemann surface of genus g, with \(g\, \ge \, 2\), and let \({\text {G}}\) be a connected semisimple affine algebraic group defined over \({\mathbb {C}}\). Given any \(\delta \, \in \, \pi _1({\text {G}})\), we prove that the moduli space of semistable principal \({\text {G}}\)-bundles over X of topological type \(\delta \) is simply connected. More generally, if \({\text {G}}\) is a connected reductive complex affine algebraic group, then the fundamental group of the moduli space is isomorphic to \({\mathbb Z}^{2gd}\), where d is the complex dimension of the center of \({\text {G}}\). In contrast, the fundamental group of the moduli stack of principal \({\text {G}}\)-bundles over X of topological type \(\delta \) is shown to be isomorphic to \(H^1(X,\, \pi _1({\text {G}}))\), when \({\text {G}}\) is semisimple. We also compute the fundamental group of the moduli stack of principal \({\text {G}}\)-bundles when \({\text {G}}\) is reductive.

令X为g的紧连通黎曼曲面，具有\（g \，\ ge \，2 \），令\（{\ text {G}} \）为在\（上定义）的连通半简单仿射代数群。 {\ mathbb {C}} \）。给定任何\（\三角洲\，\在\，\ PI _1（{\文本{G}}）\） ，证明了的模空间半稳定主\（{\文本{G}} \） -bundles过拓扑类型\（\ delta \）的X被简单连接。更一般而言，如果\（{\ text {G}} \）是一个连通的还原复仿射代数群，则模空间的基群与\（{\ mathbb Z} ^ {2gd} \）同构，其中d是\（{\ text {G}} \）的中心的复数维。相反，在拓扑类型\（\ delta \）的X上，主\（{\ text {G}} \）- bundle的模堆栈的基组显示为与\（H ^ 1（X， \，\ pi _1（{\ text {G}}））\）时，\（{\ text {G}} \）是半简单的。当\（{\ text {G}} \}是可归约的时，我们还计算主体\（{\ text {G}} \）-束的模堆栈的基群。