Question

I am studying characteristic classes recently and find some quesions about Pontrjagin classes. Firstly the definition of Pontrjagin classes is not that natural. When we talk about Pontrjagin classes we mean the characteristic classes of complexification of real bundles. But as everyone knows, the Pontrjagin classes can be determined uniquely by the Chern classes. Then why should we use them instead of Chern classes since Pontrjagin classes will lose infromation in the dimension lower than 4? \

Answer 1

The odd Chern classes of the complexified bundle are of order 2 and are determined by the Stiefel-Whitney classes of the original real bundle $\xi$ by the formula $$ c_{2k+1}(\xi\otimes \Bbb C) = \beta(w_{2k}(\xi)w_{2k+1}(\xi)) , $$ where $\beta$ is the Bockstein. So there isn't any new information in the odd Chern classes of the complexification.