Skip to main content
SpringerLink
Log in

On continuous maps between Grassmann manifolds

  • Published:
Proceedings of the Indian Academy of Sciences - Mathematical Sciences Aims and scope Submit manuscript

Abstract

LetG n,k denote the Grassmann manifold ofk-planes in ℝn. We show that for any continuous mapf: G n,k→Gn,l the induced map inZ/2-cohomology is either zero in positive dimensions or has image in the subring generated by w1(γn, k), provided 1≤l<k≤[n/2] andn≥k+2l-1. Our main application is to obtain negative results on the existence of equivariant maps between oriented Grassmann manifolds. We also obtain positive results in many cases on the existence of equivariant maps between oriented Grassmann manifolds.

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

References

  1. Adams J F, Vector fields on spheres,Ann. Math. (2)75 (1962). pp. 603–632

    Article  Google Scholar 

  2. Borel A, La cohomologie mod 2 de certains espaces homogènes,Comm. Math. Helv. 27 (1953) pp. 165–197

    Article  MATH  MathSciNet  Google Scholar 

  3. Greub W, Halperin S and Vanstone R,Connections, curvature, and cohomology vol. II, (NY: Academic Press) 1973

    MATH  Google Scholar 

  4. Husemoller D, Fibre bundles, (NY: McGraw-Hill) 1966

    MATH  Google Scholar 

  5. Korbas J.,Vector fields on (O(n 1+…+nk)/(O(n1)×·×O(nk)), (in Slovak Ph.D. Thesis, Praha, Czechoslovakia, 1985

  6. Koschorke U, Vector fields and other vector bundle morphisms—a singularity approach,Lec. Notes Math 847 (Springer-Verlag) 1981

  7. Lam K Y, Sectioning vector bundles over real projective spaces,Quart. J. Math. (2)23 (1972) pp. 97–106

    Article  MATH  Google Scholar 

  8. Lam K Y, A formula for the tangent bundle of flag manifolds and related manifolds,Trans. Am. Math. Soc. 213 (1975) pp. 305–314

    Article  MATH  Google Scholar 

  9. Milnor J W and Stasheff J D, Characteristic classes,Ann. Math. Stud. 76 (1974) (N J: Princeton)

  10. Osborn H, Vector bundles, vol. 1, (1982) (N Y: Academic Press)

    MATH  Google Scholar 

  11. Patterson R R, The square-preserving algebra endomorphisms ofH *(BO;Z/2),Quart. J. Math (2)29 (1978), pp. 225–240

    Article  MATH  MathSciNet  Google Scholar 

  12. Steenrod N and Epstein D B A, Cohomology operations,Ann. Math. Stud. 50 (1962) (N Y: Princeton)

  13. Stong R E, Cup products in Grassmannians,Top. Appl. 13 (1982) pp. 103–113

    Article  MATH  MathSciNet  Google Scholar 

  14. Zvengrowski P, A 3-fold vector product in ℝn,Comm. Math. Helv. 40 (1966) pp. 149–152

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814-73, Bratislava, CSFR

    J Korbaš

  2. School of Mathematics, SPIC Science Foundation, 92 GN Chetty Road, T. Nagar, 600 017, Madras, India

    P Sankaran

Authors
  1. J Korbaš
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P Sankaran
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Korbaš, J., Sankaran, P. On continuous maps between Grassmann manifolds. Proc. Indian Acad. Sci. (Math. Sci.) 101, 111–120 (1991). https://doi.org/10.1007/BF02868020

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02868020

Keywords

Search

Navigation