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?

Answer 1

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.