# Where can I find a full proof of the Chern-Gauss-Bonnet theorem ?

Hello, I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the pullback of the Thom class by the zero section, but I would like a proof of the fact that this class gives the Euler characteristic when coupled to the fundamental class. Thanks in advance.

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It's in the last volume of Spivak's book, no? –  Mariano Suárez-Alvarez Mar 20 '12 at 23:25
What's wrong with Chern's paper? –  Igor Rivin Mar 20 '12 at 23:26
Quote from Chern: "It helps to be vague with bundles." –  Will Jagy Mar 21 '12 at 0:10
Check out Bryant's answer to this question: mathoverflow.net/questions/84521/… –  Ian Agol Mar 21 '12 at 13:03
@Agol: yes, that's probably the best reference... –  Igor Rivin Mar 21 '12 at 13:41

For a complete proof of the Gauss-Bonnet-Chern for arbitrary vector bundles (not just tangent bundles) see Section 8.3.2 of these notes. The proof is Chern's original proof, based on Chern-Weil theory, but the language is more modern.

For a purely topological proof, see Section 5.3 of these notes.

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I fixed the broken links above. –  Liviu Nicolaescu Mar 21 '12 at 14:23
Thanks! Looks perfect, although the proof in the first ref is basically the same as what I had previously seen, though my reference had a simpler approach, examining the case of 2-plane bundles and using the splitting principle to finish. –  Youloush Mar 22 '12 at 1:19