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Existence Theorem for Coverings of Serre Bundles

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Abstract

In this paper, we study coverings of the Serre bundle category. A covering mapping of one such bundle onto another is understood as a morphism of the indicated category that consists of covering mappings of total spaces and bases. Earlier, the author associated each such covering with a subsequence of the homotopy sequence of the base bundle. The conjugacy class of this subsequence was also shown to be an invariant of the corresponding covering. The main result of this study is the existence theorem for a covering with a specified invariant. The local triviality of the base bundle is proved here to imply a similar property for the covering bundle.

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REFERENCES

  1. T. A. Gonchar and E. I. Yakovlev, “Coverings in the category of principal bundles,” Russ. Math. 65, 22–43 (2021). https://doi.org/10.3103/S1066369X21040034

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Gromol, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Großen, Lecture Notes in Mathematics, Vol. 55 (Springer, Berlin, 1968). https://doi.org/10.1007/978-3-540-35901-2

  3. E. I. Yakovlev and T. A. Gonchar, “Geometry and topology of some fibered Riemannian manifolds,” Russ. Math. 62, 69–85 (2018). https://doi.org/10.3103/S1066369X18020081

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Scott, “The geometries of 3-manifolds,” Bull. London Math. Soc. 15, 401–487 (1983). https://doi.org/10.1112/blms/15.5.401

    Article  MathSciNet  MATH  Google Scholar 

  5. A. V. Lapteva and E. I. Yakovlev, “Index vector-function and minimal cycles,” Lobachevskii J. Math 22, 35–46 (2006).

    MathSciNet  MATH  Google Scholar 

  6. J. Erickson and A. Nayyeri, “Minimum cuts and shortest non-separating cycles via homology covers,” in Proc. Twenty-Second Annu. ACM-SIAM Symp. on Discrete Algorithms (Society for Industrial and Applied Mathematics, 2011), pp. 1166–1176. https://doi.org/10.1137/1.9781611973082.88

  7. N. I. Zhukova, “Attractors and an analog of the Lichnérowicz conjecture for conformal foliations,” Sib. Math. J. 52, 436–450 (2011). https://doi.org/10.1134/S0037446611030062

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Kh. Aranson and V. Z. Grines, “On the representation of minimal sets of currents on two-dimensional manifolds by geodesics,” Math. USSR Izv. 12, 103–124 (1978). https://doi.org/10.1070/IM1978v012n01ABEH001842

    Article  MATH  Google Scholar 

  9. L. Lerman and E. Yakovlev, “On interrelations between divergence-free and Hamiltonian dynamics,” J. Geom. Phys 135, 70–79 (2019). https://doi.org/10.1016/j.geomphys.2018.09.002

    Article  MathSciNet  MATH  Google Scholar 

  10. E. I. Yakovlev, “Invariants of coverings of Serre fibrations,” Russ. Math. 66, 59–71 (2022). https://doi.org/10.3103/S1066369X22030100

    Article  MATH  Google Scholar 

  11. W. S. Massi, Algebraic Topology: An Introduction (Harcourt, Brace & World, New York, 1967).

    Google Scholar 

  12. V. A. Rokhlin and D. B. Fuks, Introductory Course of Topology (Nauka, Moscow, 1977).

    MATH  Google Scholar 

  13. V. Grines, Yu. Levchenko, V. Medvedev, and O. Pochinka, “The topological classification of structural stable 3-diffeomorphisms with two-dimensional basic sets,” Nonlinearity 28, 4081–4102 (2015). https://doi.org/10.1088/0951-7715/28/11/4081

    Article  MathSciNet  MATH  Google Scholar 

  14. V. Z. Grines, E. Ya. Gurevich, and E. D. Kurenkov, “Topological classification of gradient-like flows with surface dynamics on 3-manifolds,” Math. Notes 107, 173–176 (2020). https://doi.org/10.1134/S0001434620010162

    Article  MathSciNet  MATH  Google Scholar 

  15. V. Z. Grines, E. Ya. Gurevich, and O. V. Pochinka, “On the number of heteroclinic curves of diffeomorphisms with surface dynamics,” Regular Chaotic Dyn. 22, 122–135 (2017). https://doi.org/10.1134/S1560354717020022

    Article  MathSciNet  MATH  Google Scholar 

  16. D. D. Shubin, “Topology of ambient manifolds of non-singular Morse–Smale flows with three periodic orbits,” Izv. Vyssh. Uchebn. Zaved., Nelineinaya Dinamika 29, 863–868 (2021). https://doi.org/10.18500/0869-6632-2021-29-6-863-868

    Article  Google Scholar 

  17. D. B. A. Epstein, “Periodic flows on three-manifolds,” Ann. Math. 95, 66–82 (1972). https://doi.org/10.2307/1970854

    Article  MathSciNet  MATH  Google Scholar 

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Funding

The work was supported by the Russian Science Foundation, grant no. 21-11-00010, except for Sections 3 and 4, the research in which was supported by the Laboratory of Dynamic Systems and Applications of the National Research University Higher School of Economics, and by the Ministry of Science and Higher Education of Russian Federation, agreement no. 075-15-2022-1101.

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Authors and Affiliations

  1. National Research University, Higher School of Economics, 603155, Nizhny Novgorod, Russia

    E. I. Yakovlev

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  1. E. I. Yakovlev
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Correspondence to E. I. Yakovlev.

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Translated by A. Ivanov

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Yakovlev, E.I. Existence Theorem for Coverings of Serre Bundles. Russ Math. 67, 76–84 (2023). https://doi.org/10.3103/S1066369X23030088

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