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Birman–Hilden Bundles. I

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Abstract

A topological fibered 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 present a series of sufficient conditions for a fiber bundle over the circle to be a Birman–Hilden space.

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Notes

  1. As usual, we sometimes call the total space of a bundle just a bundle for brevity.

  2. Indeed, the definition of the direct product topology implies that a locally trivial bundle is an open mapping. This implies that the self-bijection on the base induced by a fiber-preserving self-homeomorphism of the total space of a locally trivial bundle carries open sets into open sets, i.e., it is a self-homeomorphism.

  3. For the definition of a special isotopy, see Section 2.

  4. Since we assume that \( G(X) \) is a normal subgroup of \( Homeo(X) \), the subgroup \( G(X^{\prime}) \) and accordingly the containment of \( g\in Homeo(X^{\prime}) \) in \( G(X^{\prime}) \) are well-defined regardless of the choice of a homeomorphism between \( X^{\prime} \) and \( X \).

  5. A space is Lindelöf or finally-compact whenever each open cover has an at most countable subcover.

References

  1. Birman J.S. and Hilden H.M., “On the mapping class groups of closed surfaces as covering spaces,” in: Advances in the Theory of Riemann Surfaces, Princeton University Press, Princeton (1971), 81–115 (Ann. Math. Stud.; vol. 66).

  2. Birman J.S. and Hilden H.M., “Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin’s braid group,” Bull. Amer. Math. Soc., vol. 78, no. 6, 1002–1004 (1972).

    Article  MathSciNet  Google Scholar 

  3. Birman J.S. and Hilden H.M., “Lifting and projecting homeomorphisms,” Arch. Math. (Basel), vol. 23, 428–434 (1972).

    Article  MathSciNet  Google Scholar 

  4. Birman J.S. and Hilden H.M., “On isotopies of homeomorphisms of Riemann surfaces,” Ann. Math. (2), vol. 97, 424–439 (1973).

    Article  MathSciNet  Google Scholar 

  5. Birman J.S. and Hilden H.M., “Erratum to ‘Isotopies of homeomorphisms of Riemann surfaces’,” Ann. Math. (2), vol. 185, 345 (2017).

    Article  MathSciNet  Google Scholar 

  6. Zieschang H., “On the homeotopy group of surfaces,” Math. Ann., vol. 206, 1–21 (1973).

    Article  MathSciNet  Google Scholar 

  7. Maclachlan C. and Harvey W.J., “On mapping-class groups and Teichmüller spaces,” Proc. Lond. Math. Soc., vol. 30, 496–512 (1975).

    Article  Google Scholar 

  8. Berstein I. and Edmonds A.L., “On the construction of branched coverings of low-dimensional manifolds,” Trans. Amer. Math. Soc., vol. 247, 87–124 (1979).

    Article  MathSciNet  Google Scholar 

  9. Fuller T., “On fiber-preserving isotopies of surface homeomorphisms,” Proc. Amer. Math. Soc., vol. 129, no. 4, 1247–1254 (2001).

    Article  MathSciNet  Google Scholar 

  10. Aramayona J., Leininger C.J., and Souto J., “Injections of mapping class groups,” Geom. Topol., vol. 13, no. 5, 2523–2541 (2009).

    Article  MathSciNet  Google Scholar 

  11. Winarski R.R., “Symmetry, isotopy, and irregular covers,” Geom. Dedicata, vol. 177, 213–227 (2015).

    Article  MathSciNet  Google Scholar 

  12. Ghaswala T. and Winarski R.R., “Lifting homeomorphisms and cyclic branched covers of spheres,” Michigan Math. J., vol. 66, no. 4, 885–890 (2017).

    MathSciNet  Google Scholar 

  13. Atalan F. and Medetogullari E., “The Birman–Hilden property of covering spaces of nonorientable surfaces,” Ukrainian Math. J., vol. 72, no. 3, 348–357 (2020).

    Article  MathSciNet  Google Scholar 

  14. Margalit D. and Winarski R.R., “Braids groups and mapping class groups: The Birman–Hilden theory,” Bull. Lond. Math. Soc., vol. 53, no. 3, 643–659 (2021).

    Article  MathSciNet  Google Scholar 

  15. Kolbe B., Evans M.E., “Isotopic tiling theory for hyperbolic surfaces,” Geom. Dedicata, vol. 212, 177–204 (2021).

    Article  MathSciNet  Google Scholar 

  16. Dey S., Dhanwani N.K., Patil H., and Rajeevsarathy K., Generating the Liftable Mapping Class Groups of Regular Cyclic Covers. arXiv:2111.01626v1 (2021).

    Google Scholar 

  17. Vogt E., “Projecting isotopies of sufficiently large \( P^{2} \)-irreducible \( 3 \)-manifolds,” Arch. Math. (Basel), vol. 29, no. 6, 635–642 (1977).

    Article  MathSciNet  Google Scholar 

  18. Ohshika K., “Finite subgroups of mapping class groups of geometric \( 3 \)-manifolds,” J. Math. Soc. Japan, vol. 39, no. 3, 447–454 (1987).

    MathSciNet  ADS  Google Scholar 

  19. Steenrod N.E., The Topology of Fibre Bundles, Princeton University, Princeton (1951) (Princeton Math. Ser.; vol. 14).

    Book  Google Scholar 

  20. Artin E., “Theorie der Zöpfe,” Abh. Math. Sem. Univ. Hamburg, vol. 4, 47–72 (1925).

    Article  MathSciNet  Google Scholar 

  21. Morton H.R., “Infinitely many fibred knots having the same Alexander polynomial,” Topology, vol. 17, no. 1, 101–104 (1978).

    Article  MathSciNet  Google Scholar 

  22. Burde G. and Zieschang H., Knots, De Gruyter, Berlin (1985) (De Gruyter Stud. Math.; vol. 5).

    Google Scholar 

  23. Kassel C. and Turaev V., Braid Groups. 3rd ed., Springer, New York (2008) (Grad. Texts Math.; vol. 247).

    Book  Google Scholar 

  24. Husemoller D., Fibre Bundles. 3rd ed., Springer, New York (1994) (Grad. Texts Math.; vol. 20).

    Book  Google Scholar 

  25. Edwards R.D. and Kirby R., “Deformations of spaces of imbeddings,” Ann. Math. (2), vol. 93, 63–88 (1971).

    Article  MathSciNet  Google Scholar 

  26. Epstein D.B.A., “Curves on 2-manifolds and isotopies,” Acta Math., vol. 115, 83–107 (1966).

    Article  MathSciNet  Google Scholar 

  27. Chernavskii A.V., “Local contractibility of the group of homeomorphisms of a manifold,” Math. USSR Sb., vol. 8, no. 3, 287–333 (1969).

    Article  Google Scholar 

  28. Brown M., “Locally flat imbeddings of topological manifolds,” Ann. Math. (2), vol. 75, 331–341 (1962).

    Article  MathSciNet  Google Scholar 

  29. Schechter E., Handbook of Analysis and Its Foundations, Academic, San Diego (1997).

    Google Scholar 

  30. Hatcher A., Algebraic Topology, Cambridge University, Cambridge (2002).

    Google Scholar 

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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. The author is also grateful to the referee for valuable comments.

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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Authors and Affiliations

  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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  1. A. V. Malyutin
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Correspondence to A. V. Malyutin.

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Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2024, Vol. 65, No. 1, pp. 125–139. https://doi.org/10.33048/smzh.2024.65.111

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Malyutin, A.V. Birman–Hilden Bundles. I. Sib Math J 65, 106–117 (2024). https://doi.org/10.1134/S0037446624010117

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