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? Secondly, it is really hard to imagine that some elements of cohomology groups are not homotopy invariant since homology groups are homotopy invariant.in the rational Pontrjagin classes. Novikov said that some combinatorial Pontrjagin classes are topological invariant.So how does this come?

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? \