We study the structure of self-homeomorphism groups of fibered manifolds. A fibered topological space is a Birman–Hilden space whenever in each isotopic pair of its fiber-preserving (taking each fiber to a fiber) self-homeomorphisms the homeomorphisms are also fiber-isotopic (isotopic through fiber-preserving homeomorphisms). We prove in particular that the Birman–Hilden class contains all compact connected locally trivial surface bundles over the circle, including nonorientable ones and those with nonempty boundary, as well as all closed orientable Haken 3-manifold bundles over the circle, including nonorientable ones.

中文翻译：

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伯曼-希尔登捆绑包。二

我们研究纤维流形的自同胚群的结构。纤维拓扑空间是伯曼-希尔登空间，只要在其纤维保持（将每个纤维变为纤维）自同胚的每个同位素对中，同胚也是纤维同位素（通过纤维保持同态的同位素）。我们特别证明了 Birman-Hilden 类包含圆上所有紧连通局部平凡表面丛，包括不可定向的和非空边界的表面丛，以及圆上所有闭可定向哈肯 3 流形丛，包括不可定向的。