Birman-Hilden bundles. II. (English. Russian original) Zbl 1535.57044
Sib. Math. J. 65, No. 2, 351-362 (2024); translation from Sib. Mat. Zh. 65, No. 2, 358-373 (2024).
Summary: 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.
For Part I, see [the author, Sib. Math. J. 65, No. 1, 106–117 (2024; Zbl 1533.57064); translation from Sib. Mat. Zh. 65, No. 1, 125–139 (2024)].
For Part I, see [the author, Sib. Math. J. 65, No. 1, 106–117 (2024; Zbl 1533.57064); translation from Sib. Mat. Zh. 65, No. 1, 125–139 (2024)].
MSC:
| 57R22 | Topology of vector bundles and fiber bundles |
| 55R99 | Fiber spaces and bundles in algebraic topology |
Keywords:
fiber bundle; fibering; fiber-preserving; fiberwise; locally trivial bundle; fiber-preserving self-homeomorphism; mapping class group; isotopy; homotopy; homotopy equivalence; manifoldCitations:
Zbl 1533.57064References:
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