a question on oriented bundles and Euler class

Question

In Characteristic classes, J. Milnor, J. Stasheff, Prop. 9.7, it is proved that:

if the oriented vector bundle $\xi$ possesses a nowhere zero cross section, then the Euler class $e(\xi)=0$.

I want to ask

(1). whether the converse is true?

(2). In particular, if $\xi$ is a complex line bundle, whether the converse is true?

score 2 · Answer 1 · edited Apr 13 '17 at 12:19

2

See the answers to this question, which answer your question and tell you a good deal more about closely related stuff and intuition for questions like this one.

edited Apr 13 '17 at 12:19

Community

1

answered Feb 06 '15 at 09:58

John Hughes

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So the short answer is yes or no for 1) and 2)? – wonderich Jul 12 '18 at 20:43
Did you read the answer to the linked question by Akhil Mathew? It directly addresses this. – John Hughes Jul 12 '18 at 21:40
The 1) is no. The 2) is also no. True? – wonderich Jul 12 '18 at 21:51

a question on oriented bundles and Euler class

1 Answers1