# chern class of complex line bundle

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?

• Depends on what you know about the line bundle. Can you be more specific? – Qiaochu Yuan Feb 2 '15 at 6:58
• A complex line bundle is trivial iff its first Chern class vanishes, as far as I remember... – Najib Idrissi Feb 2 '15 at 16:51
• "A complex line bundle is trivial iff its first Chern class vanishes", Prof. Najib, could you give me some references? thanks. – Shiquan Feb 3 '15 at 2:46

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