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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
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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

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