An orientable two plane bundle is trivial over a surface with boundary

Question

Why an orientable two plane bundle is trivial over a surface with boundary?

When, I read a lecture notes "Legendrian and transversal knots by Jhon B Etnyre", I seen this result(he just state it). I don't know how the problem will go.

Can any one fix it???

Thanks advance.

Answer 1

Complex line bundles are (topologically) determined by their first Chern class. $H^2(\Sigma,\Bbb Z)=0$ for a surface with boundary.