Skip to main content
SpringerLink
Log in

Topological classification of complex vector bundles over 8-dimensional spin\(^{c}\) manifolds

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

In this paper, complex vector bundles of rank r over 8-dimensional spin\(^{c}\) manifolds are classified in terms of the Chern classes of the complex vector bundles, where \(r = 3\) or 4. As an application, we see that two rank 3 complex vector bundles over 4-dimensional complex projective space \(\mathbb {C}P^{4}\) are isomorphic if and only if they have the same Chern classes. Moreover, the Chern classes of rank 3 complex vector bundles over \(\mathbb {C}P^{4}\) are determined. Together with results of Thomas and Switzer, this completes the classification of complex vector bundles of any rank over \(\mathbb {C}P^4\).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Atiyah, M.F., Hirzebruch, F.: Riemann-Roch theorems for differentiable manifolds. Bull. Amer. Math. Soc. 65, 276–281 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atiyah, M.F., Rees, E.: Vector bundles on projective 3-space. Invent. Math. 35, 131–153 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bănică, C., Putinar, M.: On complex vector bundles on rational threefolds. Math. Proc. Cambridge Philos. Soc. 97(2), 279–288 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bănică, C., Putinar, M.: On complex vector bundles on projective threefolds. Invent. Math. 88(2), 427–438 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bănică, C., Putinar, M.: On the classification of complex vector bundles of stable rank. Proc. Indian Acad. Sci. Math. Sci. 116(3), 271–291 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Čadek, M., Vanžura, J.: On the classification of oriented vector bundles over \(9\)-complexes. Proc. Winter School Geometry and Physics, Zdíkov, January 1993, Suppl. Rend. Circ. Mat. Palermo, Ser. II 37, 33–40 (1994)

  7. Gilmore, M.E.: Complex Stiefel manifolds, some homotopy groups and vector fields. Bull. Amer. Math. Soc. 73, 630–633 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hatcher A. Spectral sequences, Chapter 5 of algebraic topology. 2004,http://pi.math.cornell.edu/~hatcher/AT/SSpage.html

  9. Hirzebruch, F.: Topological methods in algebraic geometry. Springer-Verlag, Berlin (1995)

    MATH  Google Scholar 

  10. James, I., Thomas, E.: An approach to the enumeration problem for non-stable vector bundles. J. Math. Mech. 14, 485–506 (1965)

    MathSciNet  MATH  Google Scholar 

  11. Li, B.H., Duan, H.B.: Spin Characteristic classes and reduced \(KSpin\) group of a low dimensional complex. Proc. Amer. Math. Soc. 113(2), 479–491 (1991)

    MathSciNet  Google Scholar 

  12. Milnor, J.W., Stasheff, J.D.: Characteristic classes. Princeton University Press, New Jersey (1974)

    Book  MATH  Google Scholar 

  13. Mimura, M., Toda, H.: Topology of Lie groups. I, II. In: Translations of Mathematical Monographs, American Mathematical Society, Providence (1991)

  14. Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Progress in Matheatics, 3. Birkhäuser, Boston, Mass., (1980

  15. Peterson, F.P.: Some remarks on Chern classes. Ann. of Math. 2(69), 414–420 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  16. Schneider, M.: Holomorphic vector bundles on \(P_n\). Séminaire Bourbaki (1978/79), Exp. No. 530, pp. 80–102, Lecture notes in Math., 770, Springer, Berlin (1980)

  17. Switzer, R.M.: Complex \(2\)-plane bundles over complex projective space. Math. Z. 168(3), 275–287 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Switzer, R.M.: Postnikov towers associated with complex \(2\)-plane and symplectic line bundles. Math. Z. 168(1), 87–103 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Thomas, E.: Seminar on fiber spaces. Springer-Verlag, Berlin-Heidelberg-New York (1966)

    Book  Google Scholar 

  20. Thomas, A.: Almost complex structures on complex projective spaces. Trans. Amer. Math. Soc. 193, 123–132 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  21. Whitehead, G.W.: Elements of homotopy theory. Springer-Verlag, New York-Berlin (1978)

    Book  MATH  Google Scholar 

  22. Woodward, L.M.: The classification of orientable vector bundles over \(CW\)-complexes of small dimension. Proc. Roy. Soc. Edinburgh 92A, 175–179 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wu, W.T.: Sur les classes caractéristiques des structures fibrées sphérique. Publ. Inst. Math. Univ. Strasbourg 11, pp. 5–89, Actualités Scientifiques et Industrielles, No. 1183, Hermann & Cie, Paris, (1952)

Download references

Acknowledgements

I would like to thank the referee for many detailed comments and helpful suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, Henan, China

    Huijun Yang

Authors
  1. Huijun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huijun Yang.

Ethics declarations

Conflict of interest

I states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, H. Topological classification of complex vector bundles over 8-dimensional spin\(^{c}\) manifolds. manuscripta math. 172, 57–74 (2023). https://doi.org/10.1007/s00229-022-01411-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-022-01411-0

Mathematics Subject Classification

Search

Navigation