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Birman–Hilden Bundles. II

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Abstract

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.

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Notes

  1. In the case of a manifold \( N \) locally trivially fibered over the circle with compact fiber \( M \) the subgroup \( Fib_{1}(N) \) is closed regardless of the bundle lying in the Birman–Hilden class. We can justify this by using the local contractibility (Chernavskii’s Theorem [27]) of the underlying space of the group of self-homeomorphisms of \( M \): since \( Homeo(M) \) is locally simply-connected, we conclude that \( Fib_{1}(N) \) is locally path-connected.

  2. A fiber-preserving self-homeomorphism of a fibered space is called fiberwise whenever it takes each fiber to the same fiber.

  3. For the case of a closed orientable surface of genus above \( 1 \) this fact is already justified in [34, 49].

  4. Thus, in the case of a closed surface of negative Euler characteristic both alternative hypotheses of Theorem 4 hold: \( Homeo_{1}(X)\subset Map_{1}(X,X) \) induces an isomorphism of fundamental groups and \( Homeo_{1}(X) \) is simply-connected.

  5. Henceforth in similar cases the mentioned statements, as a rule, deal only with isomorphisms of groups, but it is clear from the constructions that these isomorphisms are induced by the embeddings of spaces in question.

  6. The statements of Theorems 6.3 and 6.4 in [35] refer to homotopies and isotopies without restrictions on the boundary, but in their proofs presented in [35] the situation reduces precisely to the pointwise boundary fixing cases, and the required property is established.

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Acknowledgments

The author is grateful to Yu.S. Belousov, I.A. Dynnikov, S.S. Podkorytov, and E.A. Fominykh for useful discussions and to the referee for useful remarks.

Funding

The study was supported by the Russian Science Foundation grant no. 22–11–00299, https://rscf.ru/en/project/22-11-00299/.

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  1. Steklov Mathematical Institute, Moscow, Russia

    A. V. Malyutin

  2. Steklov Mathematical Institute, St. Petersburg Branch, St. Petersburg, Russia

    A. V. Malyutin

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Translated from Sibirskii Matematicheskii Zhurnal, 2024, Vol. 65, No. 2, pp. 358–373. https://doi.org/10.33048/smzh.2024.65.210

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Malyutin, A.V. Birman–Hilden Bundles. II. Sib Math J 65, 351–362 (2024). https://doi.org/10.1134/S0037446624020101

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