Journal of Mathematical Sciences

, Volume 136, Issue 5, pp 4166–4200 | Cite as

The Thom isomorphism for nonorientable bundles

  • E. G. Sklyarenko


The classical theory of Thom isomorphisms is extended to nonorientable vector bundles. The properties of orientation sheaves of bundles and of the Thom and Euler classes τ and e with respect to projections, fiber maps, Cartesian products, and Whitney sums of bundles are studied. The validity of standard constructions used in the applications of the classes τ and e is confirmed. It is shown that the Thom isomorphisms, together with their form, are consequences of the Poincaré duality.


Manifold Exact Sequence Spectral Sequence Cohomology Class Normal Bundle 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Springer, New York (1982).Google Scholar
  2. 2.
    N. Bourbaki, Éléments de Mathématique. Fasc. II. Livre III: Topologie Géné rale. Chapitre 1: Structures Topologiques. Chapitre 2: Structures Uniformes, Hermann, Paris (1965).Google Scholar
  3. 3.
    G. E. Bredon, Sheaf Theory, 2nd edition, Springer (1997). (First edition: G. E. Bredon, Sheaf Theory, McGraw-Hill, New York (1967).)Google Scholar
  4. 4.
    H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton (1956).Google Scholar
  5. 5.
    H. Cartan, Cohomologie des Groups, Suite Spectral, Faisceaux, Seminaire, École Normal Sup., 1950–1951.Google Scholar
  6. 6.
    A. Dold, Lectures on Algebraic Topology, Springer, New York (1972).Google Scholar
  7. 7.
    R. Godement, Topologie Algébrique et Théorie des Faisceaux, Hermann, Paris (1958).Google Scholar
  8. 8.
    A. Grothendieck, “Sur quelques points d’algébre homologique,” Tohoku Math. J., 9, No. 2, 119–221 (1957).MATHMathSciNetGoogle Scholar
  9. 9.
    D. Husemoller, Fibre Bundles, McGraw-Hill, New York (1966).Google Scholar
  10. 10.
    C. U. Jensen, Les Foncteours Dérivés de lim et Leurs Applications en Théorie des Modules, Lect. Notes Math. Vol. 254, Springer, Berlin—New York (1972).Google Scholar
  11. 11.
    A. E. Kharlap, “Local homology and cohomology, homological dimension, and generalized manifolds,” Mat. Sb., 96, No. 3, 347–373 (1975).MathSciNetGoogle Scholar
  12. 12.
    V. I. Kuz’minov, “On derivative functors of the projective limit functor,” Sib. Mat. Zh., 8, No. 2, 333–345 (1967).MathSciNetGoogle Scholar
  13. 13.
    W. Massey, Homology and Cohomology Theory, Marcel Dekker, New York (1978).Google Scholar
  14. 14.
    J. Milnor and J. Stashe., Characteristic Classes, Princeton Univ. Press, Princeton (1974).Google Scholar
  15. 15.
    Yu. B. Rudyak, “On the Thom—Dold isomorphism for nonorientable bundles,” Dokl. Akad. Nauk SSSR, 255, No. 6, 1323–1325 (1980).MATHMathSciNetGoogle Scholar
  16. 16.
    E. G. Sklyarenko, “On the theory of generalized manifolds,” Izv. Akad. Nauk SSSR Ser. Mat. 35, No. 4, 831–843 (1971).MATHMathSciNetGoogle Scholar
  17. 17.
    E. G. Sklyarenko, “Homology and cohomology of general spaces,” in: Sovremennye Problemy Matematiki. Fundamental’nye Napravleniya [in Russian], Vol. 50, Moscow (1989), pp. 129–266.MATHMathSciNetGoogle Scholar
  18. 18.
    E. G. Sklyarenko, “Homology and cohomology relations between sets. Homology and cohomology of an entourage of a closed set,” Izv. Ross. Akad. Nauk Ser. Mat., 56, No. 5, 1040–1071 (1992).MATHGoogle Scholar
  19. 19.
    E. G. Sklyarenko, “On the nature of homological multiplications and duality,” Usp. Mat. Nauk, 49, No. 1, 141–198 (1994).MATHMathSciNetGoogle Scholar
  20. 20.
    E. G. Sklyarenko, “Hyper(co)homologies for left exact covariant functors and the theory of homologies of topological spaces,” Usp. Mat. Nauk, 50, No. 3, 109–146 (1995).MATHMathSciNetGoogle Scholar
  21. 21.
    E. G. Sklyarenko, “On cohomologies with supports,” Usp. Mat. Nauk, 51, No. 1, 167–168 (1996).MATHMathSciNetGoogle Scholar
  22. 22.
    E. G. Sklyarenko, “On homological multiplications,” Izv. Ross. Akad. Nauk Ser. Mat., 61, No. 1, 157–176 (1997).MATHMathSciNetGoogle Scholar
  23. 23.
    E. H. Spanier, Algebraic Topology, McGraw-Hill, New York (1966).Google Scholar
  24. 24.
    S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, N.J. (1964).Google Scholar
  25. 25.
    R. M. Switzer, Algebraic Topology—Homotopy and Homology, Springer, New York (1975).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • E. G. Sklyarenko
    • 1
  1. 1.Department of Higher Geometry and Topology, Faculty Mechanics and MathematicsMoscow State UniversityMoscowRussia

Personalised recommendations