[Math] 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).