On 2-distributions in 8-dimensional vector bundles over 8-complexes

Martin Čadek; Jiří Vanžura

Colloquium Mathematicae (1996)

  • Volume: 70, Issue: 1, page 25-40
  • ISSN: 0010-1354

Abstract

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It is shown that the 2 -index of a 2-distribution in an 8-dimensional spin vector bundle over an 8-complex is independent of the 2-distribution. Necessary and sufficient conditions for the existence of 2-distributions in such vector bundles are given in terms of characteristic classes and a certain secondary cohomology operation. In some cases this operation is computed.

How to cite

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Čadek, Martin, and Vanžura, Jiří. "On 2-distributions in 8-dimensional vector bundles over 8-complexes." Colloquium Mathematicae 70.1 (1996): 25-40. <http://eudml.org/doc/210394>.

@article{Čadek1996,
abstract = {It is shown that the $ℤ_2$-index of a 2-distribution in an 8-dimensional spin vector bundle over an 8-complex is independent of the 2-distribution. Necessary and sufficient conditions for the existence of 2-distributions in such vector bundles are given in terms of characteristic classes and a certain secondary cohomology operation. In some cases this operation is computed.},
author = {Čadek, Martin, Vanžura, Jiří},
journal = {Colloquium Mathematicae},
keywords = {secondary cohomology operation; classifying spaces for groups; vector bundle; characteristic classes; Postnikov tower; distribution},
language = {eng},
number = {1},
pages = {25-40},
title = {On 2-distributions in 8-dimensional vector bundles over 8-complexes},
url = {http://eudml.org/doc/210394},
volume = {70},
year = {1996},
}

TY - JOUR
AU - Čadek, Martin
AU - Vanžura, Jiří
TI - On 2-distributions in 8-dimensional vector bundles over 8-complexes
JO - Colloquium Mathematicae
PY - 1996
VL - 70
IS - 1
SP - 25
EP - 40
AB - It is shown that the $ℤ_2$-index of a 2-distribution in an 8-dimensional spin vector bundle over an 8-complex is independent of the 2-distribution. Necessary and sufficient conditions for the existence of 2-distributions in such vector bundles are given in terms of characteristic classes and a certain secondary cohomology operation. In some cases this operation is computed.
LA - eng
KW - secondary cohomology operation; classifying spaces for groups; vector bundle; characteristic classes; Postnikov tower; distribution
UR - http://eudml.org/doc/210394
ER -

References

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  2. [CS] M. C. Crabb and B. Steer, Vector bundle monomorphisms with finite singularities, Proc. London Math. Soc. (3) 30 (1975), 1-39. Zbl0294.57015
  3. [CV1] M. Čadek and J. Vanžura, On the existence of 2-fields in 8-dimensional vector bundles over 8-complexes, Comment. Math. Univ. Carolin. (1995), to appear. Zbl0921.57016
  4. [CV2] M. Čadek and J. Vanžura, On the classification of oriented vector bundles over 9-complexes, Proceedings of the Winter School Geometry and Physics 1993, Suppl. Rend. Circ. Math. Palermo (2) 37 (1994), 33-40. Zbl0853.57026
  5. [H] F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergeb. Math. Grenzgeb. 9, Springer, Berlin, 1959. Zbl0101.38301
  6. [K] U. Koschorke, Vector Fields and Other Vector Bundle Morphisms - a Singularity Approach, Lecture Notes in Math. 847, Springer, 1981. Zbl0459.57016
  7. [M] M. H. de Paula Leite Mello, Two plane sub-bundles of nonorientable real vector bundles, Manuscripta Math. 57 (1987), 263-280. Zbl0602.55009
  8. [Q] D. Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197-212. Zbl0225.55015
  9. [D] D. Randall, CAT 2-fields on nonorientable CAT manifolds, Quart. J. Math. Oxford (2) 38 (1987), 355-366. Zbl0628.57015
  10. [T1] E. Thomas, Fields of tangent 2-planes on even dimensional manifolds, Ann. of Math. 86 (1967), 349-361. Zbl0168.21401
  11. [T2] E. Thomas, Complex structures on real vector bundles, Amer. J. Math. 89 (1966), 887-908. Zbl0174.54802
  12. [T3] E. Thomas, Postnikov invariants and higher order cohomology operations, Ann. of Math. 85 (1967), 184-217. Zbl0152.22002
  13. [T4] E. Thomas, Fields of tangent k-planes on manifolds, Invent. Math. 3 (1967), 334-347. Zbl0162.55402

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