Advertisement

Journal of Mathematical Sciences

, Volume 197, Issue 6, pp 741–752 | Cite as

Strong Homology Groups of Continuous Maps

  • A. Beridze
Article
  • 32 Downloads

Abstract

In this paper, we define coherent morphisms of chain maps and homology groups of morphisms of this type. We construct strong homology groups of continuous maps of compact metric spaces and prove that for each continuous map f : X → Y , there exists a long exact homological sequence. Moreover, we show that for each inclusion i : A → X of compact metric spaces, there exists an isomorphism \( {{\bar{H}}_n}(i)\approx {{\bar{H}}_n}\left( {X,A} \right) \).

Keywords

Exact Sequence Topological Space Chain Complex Homology Group Inverse System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Baladze, “On the shape of a mapping,” in: Proc. Int. Topological Conf. Baku, 1987, Baku (1989), pp. 35–43.Google Scholar
  2. 2.
    V. Baladze, “On coshapes of topological spaces and continuous maps,” Georgian Math. J., 16, No. 2, 229–242 (2009).MATHMathSciNetGoogle Scholar
  3. 3.
    V. Baladze, “On coshape invariant extensions of functors,” Proc. A. Razmadze Math. Inst., 150, 1–50 (2009).MATHMathSciNetGoogle Scholar
  4. 4.
    V. Baladze, “The coshape invariant and continuous extensions of functors,” Topology Appl., 158, No. 12, 1396–1404 (2011).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    V. Baladze, On the spectral (co)homology exact sequences of maps, Preprint, Batumi (2011).Google Scholar
  6. 6.
    A. Beridze, Strong homology groups of continuous maps, Preprint, Batumi (2011).Google Scholar
  7. 7.
    D. A. Edwards and P. Tulley McAuley, “The shape of a map,” Fund. Math., 96, No. 3, 195–210 (1977).MATHMathSciNetGoogle Scholar
  8. 8.
    S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton Univ. Press, Princeton, New Jersey (1952).MATHGoogle Scholar
  9. 9.
    S. Mardešič and J. Segal, Shape Theory. The Inverse System Approach, North-Holland Math. Library, 26, North-Holland, Amsterdam–New York (1982).Google Scholar
  10. 10.
    S. Mardešič, Strong Shape and Homology, SpringerMonogr. Math., Springer-Verlag, Berlin (2000).MATHGoogle Scholar
  11. 11.
    L. D. Mdzinarishvili, “On total homology,” in: Geometric and Algebraic Topology, Banach Center Publ., 18, PWN, Warsaw (1986), pp. 345–361.Google Scholar
  12. 12.
    N. E. Steenrod, “Regular cycles of compact metric spaces,” Ann. Math. (2), 41, 833–851 (1940).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsShota Rustaveli State UniversityBatumiGeorgia

Personalised recommendations