Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-13T09:18:19.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Vector bundles over (8k + 5)-dimensional manifolds

Published online by Cambridge University Press:  14 November 2011

Tze-Beng Ng
Tze-Beng Ng
Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore
Get access Rights & Permissions [Opens in a new window]

Extract

Suppose that M is a closed, connected and smooth manifold of dimension n = 8k + 5, with k ≧1. Let η be an n-plane bundle over M. Under suitable conditions on M, we derive necessary and sufficient conditions for the span of η to be ≧j, j = 5 or 6. We then apply the results to the tangent bundle of M. In particular, we prove a conjecture of E. Thomas, namely, if M is 3-connected mod 2, then span M ≧ 5 if, and only if, χ2(M) = 0. We prove that if also w8k(M) = 0, then span M≧6. We also derive some immersion theorems for M.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 111 , Issue 1-2 , 1989 , pp. 183 - 197
Copyright
Copyright © Royal Society of Edinburgh 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Adém, J. and Gitler, S.. Secondary characteristic classes and the immersion problem. Bol. Soc. Mat. Mexicana 8 (1963), 53–78.Google Scholar
2Atiyah, M. F.. Thorn complexes. Proc. London Math. Soc. (3) 11 (1961), 291–310.CrossRefGoogle Scholar
3Atiyah, M. F. and Dupont, J. L.. Vector fields with finite singularities. Acta Math. 128 (1972), 1–40.CrossRefGoogle Scholar
4Hughes, A. and Thomas, E.. A note on certain secondary cohomology operations. Bol. Soc. Mat. Mexicana 13 (1968), 1–17.Google Scholar
5Koschorke, U.. Vector Fields and other Vector Bundle Morphisms - A Singularity Approach. Lecture Notes in Mathematics 847 (Berlin: 1981).Google Scholar
6Mahowald, M. E.. The index of a tangent 2-field. Pacific J. Math. 58 (1975), 593–548.CrossRefGoogle Scholar
7Massey, W. S. and Peterson, F. P.. On the dual Stiefel-Whitney classes of a manifold. Bol. Soc. Mat. Mexicana (2) 8 (1963), 1–13.Google Scholar
8Maunder, C. R. F.. Cohomology operations of the Nth kind. Proc. London Math. Soc. (3) 13 (1963), 125–154.CrossRefGoogle Scholar
9Beng, Tze, Ng. The existence of 7-fields and 8-fields on manifolds. Quart. J. Math. Oxford Ser. (2) 30 (1979), 197–221.Google Scholar
10Beng, Tze, Ng. A note on the mod 2 cohomology of BŜOn〈16〉. Canad. J. Math. 37 (1985), 893–907.Google Scholar
11Beng, Tze, Ng. Vector bundles over (8k + 3)-dimensional manifolds. Pacific J. Math. 121 (1986), 427–443.Google Scholar
12Quillen, D. G.. The mod 2 cohomology rings of extra-special 2 groups and the spinor groups. Math. Ann. 194 (1971), 197–212.CrossRefGoogle Scholar
13Thomas, E.. Real and complex vector-fields on manifolds. J. Math. Mech. 16 (1967), 1183–1205.Google Scholar
14Thomas, E.. An exact sequence for principal fibration. Bol. Soc. Mat. Mexicana 12 (1967), 35–45.Google Scholar
15Thomas, E.. Postnikov invariants and higher order cohomology operations. Ann. of Math. (2) 85 (1967), 184–217.CrossRefGoogle Scholar
16Thomas, E.. The index of a tangent 2-fields. Comment. Math. Helv. 42 (1967), 86–110.CrossRefGoogle Scholar
17Thomas, E.. The span of a manifold. Quart. J. Math., Oxford Ser. (2) 19 (1968), 225–244.CrossRefGoogle Scholar
18Thomas, E.. Vector fields on manifolds. Bull. Amer. Math. Soc. 75 (1969), 643–683.CrossRefGoogle Scholar