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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?

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

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  • $\begingroup$ Dear Prof. Dan, could you give some references? $\endgroup$ – Shiquan Feb 3 '15 at 2:47
  • $\begingroup$ Perfect reference for characteristic classes is always Milnor - Stasheff $\endgroup$ – Daniel Valenzuela Feb 3 '15 at 3:13
  • $\begingroup$ Hatcher also has a proof of this in his book on vector bundles and K-theory. $\endgroup$ – user98602 Feb 3 '15 at 18:05
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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.

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