ArticleOriginal scientific text

Title

On 4-fields and 4-distributions in 8-dimensional vector bundles over 8-complexes

Authors 1, 2

Affiliations

  1. Department of Algebra and Geometry, Masaryk University, Janáčkovo nám. 2a, 662 95 Brno, Czech Republic
  2. Institute of Mathematics, Academy of Sciences of the Czech Republic, Žižkova 22, 616 62 Brno, Czech Republic

Abstract

Let ξ be an oriented 8-dimensional spin vector bundle over an 8-complex. In this paper we give necessary and sufficient conditions for ξ to have 4 linearly independent sections or to be a sum of two 4-dimensional spin vector bundles, in terms of characteristic classes and higher order cohomology operations. On closed connected spin smooth 8-manifolds these operations can be computed.

Keywords

classifying spaces for groups, vector bundle, higher order cohomology operations, characteristic classes, Postnikov tower, distribution

Bibliography

  1. [AR] J. L. Arraut and D. Randall, Index of tangent fields on compact manifolds, in: Contemp. Math. 12, Amer. Math. Soc., 1982, 31-46.
  2. [AD] M. Atiyah and J. Dupont, Vector fields with finite singularities, Acta Math. 128 (1972), 1-40.
  3. [BH] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces I, Amer. J. Math. 80 (1958), 458-538.
  4. [BS] A. Borel and J. P. Serre, Groupes de Lie et puissances réduites de Steenrod, ibid. 75 (1953), 409-448.
  5. [CV1] M. Čadek and J. Vanžura, On the classification of oriented vector bundles over 9-complexes, Rend. Circ. Math. Palermo (2) Suppl. 37 (1994), 33-40.
  6. [CV2] M. Čadek and J. Vanžura, On the existence of 2-fields in 8-dimensional vector bundles over 8-com- plexes, Comment. Math. Univ. Carolin. 36 (1995), 377-394.
  7. [CV3] M. Čadek and J. Vanžura, Almost quaternionic structures on eight-manifolds, Osaka J. Math., to appear.
  8. [CS] M. C. Crabb and B. Steer, Vector-bundle monomorphisms with finite singularities, Proc. London Math. Soc. (3) 30 (1975), 1-39.
  9. [D] J. L. Dupont, K-theory obstructions to the existence of vector fields, Acta Math. 113 (1974), 67-80.
  10. [H] F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergeb. Math. Grenzgeb. 9, Springer, Berlin, 1959.
  11. [K1] U. Koschorke, Vector Fields and Other Vector Bundle Morphisms-a Singularity Approach, Lecture Notes in Math. 847, Springer, 1981.
  12. [K2] U. Koschorke, Nonstable and stable monomorhisms of vector bundles, preprint, 1995.
  13. [N1] T. B. Ng, 4-fields on (4k+2)-dimensional manifolds, Pacific J. Math. 129 (1987), 337-347.
  14. [N2] T. B. Ng, On the geometric dimension of vector bundles, span of a manifold and immersion of manifolds in manifolds, Exposition. Math. 8 (1990), 193-226.
  15. [Q] D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197-212.
  16. [R1] D. Randall, Tangent frame fields on spin manifolds, Pacific J. Math. 76 (1978), 157-167.
  17. [R2] D. Randall, On indices of tangent fields with finite singularities, in: Differential Topology (Siegen, 1987), Lecture Notes in Math. 1350, Springer, 1988, 213-240.
  18. [T1] E. Thomas, Seminar on Fiber Spaces, Lecture Notes in Math. 13, Springer, Berlin, 1966.
  19. [T2] E. Thomas, Postnikov invariants and higher order cohomology operations, Ann. of Math. 85 (1967), 184-217.
  20. [T3] E. Thomas, Fields of tangent k-planes on manifolds, Invent. Math. 3 (1967), 334-347.
  21. [T4] E. Thomas, Vector fields on manifolds, Bull. Amer. Math. Soc. 75 (1969), 643-683.
  22. [W] L. M. Woodward, The classification of orientable vector bundles over CW complexes of small dimension, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), 175-179.
Pages:
213-228
Main language of publication
English
Received
1996-07-01
Accepted
1997-08-01
Published
1998
Exact and natural sciences