chern class of complex line bundle

Question

Let $\xi:=(\mathbb{C},E,p,B)$ be a complex line bundle, where $B$ is a manifold or CW-complex. How to determine whether the first Chern class $c_1(\xi)=0$ or non-vanishing?

Answer 1

If $B$ is a CW cplx there is an isomorphism of abelian groups: $$ (\{\text{iso classes of line bundles}\}, \otimes) \stackrel {c_1} \to (H^2(B),+)$$

Hence the second cohomology of your space classifies line bundles. The question translates to the question about triviality of your bundle. There are quite some methods to check that (especially for line bundles).

Answer 2

You can define the first Chern class as the Euler class of the underlying real bundle.(Every complex vector bundle has a natural orientation).

For any vector bundle, the Euler class is zero iff the bundle has a non-zero section. If the bundle is complex then rotation by i gives a second section so the bundle is actually trivial.