Unique Path Lifting from Homotopy Point of View

Abstract

The paper introduces some notions extending the unique path lifting property from a homotopy viewpoint and studies their roles in the category of fibrations. First, we define some homotopical kinds of the unique path lifting property and find all possible relationships between them. Moreover, we supplement the full relationships of these new notions in the presence of fibrations. Second, we deduce some results in the category of fibrations with these notions instead of unique path lifting such as the existence of products and coproducts. Also, we give a brief comparison of these new categories to some categories of the other generalizations of covering maps. Finally, we present two subgroups of the fundamental group related to the fibrations with these notions and compare them to the subgroups of the fundamental group related to covering and generalized covering maps.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Brazas, J.: A characterization of the unique path lifting property for the whisker topology. Talk at the Joint Meeting of the German Mathematical Society and the Polish Mathematical Society, Poznan, Poland (2014). www2.gsu.edu/jbrazas

    Google Scholar 

  2. 2.

    Brazas, J.: Generalized covering space theories. Theory Appl. Categ. 30, 1132–1162 (2015)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Brazas, J.: Semicoverings: a generalization of covering space theory. Homology Homotopy Appl. 14, 33–63 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Brazas, J.: Semicoverings, coverings, overlays, and open subgroups of the quasitopological fundamental group. Topology Proc. 44, 285–513 (2014)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Brodskiy, N., Dydak, J., Labuz, B., Mitra, A.: Covering maps for locally path-connected spaces. Fund. Math. 218, 13–46 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Fischer, H., Zastrow, A.: A core-free semicovering of the Hawaiian Earring. Topology Appl. 160, 1957–1967 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Fischer, H., Zastrow, A.: Generalized universal coverings and the shape group. Fund. Math. 197, 167–196 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Fischer, H., Repovs, D., Virk, Z., Zastrow, A.: On semilocally simply connected spaces. Topology Appl. 158, 397–408 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Klevdal, C.: A galois correspondence with generalized covering spaces. Undergratuate honor theses university of colorado (2015)

  10. 10.

    Kowkabi, M., Mashayekhy, B., Torabi, H.: When is a local homeomorphism a semicovering map? https://doi.org/10.1007/s40306-017-0205-4

  11. 11.

    Mashayekhy, B., Pakdaman, A., Torabi, H.: Spanier spaces and covering theory of non-homotopically path Hausdorff spaces. Georgian Math. J. 20, 303–317 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Pakdaman, A., Torabi, H., Mashayekhy, B.: On the existence of categorical universal coverings. Italian J. Pure Appl. Math. 37, 289–300 (2017)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Spanier, E.H.: Algebraic Topology. McGraw-Hill, New York (1966)

    Google Scholar 

  14. 14.

    Virk, Z., Zastrow, A.: A homotopically Hausdorff space which does not admit a generalized universal covering space. Topology Appl. 160, 656–666 (2013)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the referee for his/her careful reading and useful suggestions. This research was supported by a grant from Ferdowsi University of Mashhad-Graduate Studies (No. 31685).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Behrooz Mashayekhy.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tajik, M., Mashayekhy, B. & Pakdaman, A. Unique Path Lifting from Homotopy Point of View. Acta Math Vietnam 43, 257–273 (2018). https://doi.org/10.1007/s40306-017-0219-y

Download citation

Keywords

  • Homotopically lifting
  • Unique path lifting
  • Fibration
  • Fundamental group
  • Covering map

Mathematics Subject Classification (2010)

  • 55P05
  • 57M10
  • 57M05